WILLIAMS COLLEGE LIBRARIES
COPYRIGHT ASSIGNMENT AND INSTRUCTIONS FOR A STUDENT THESIS
Your unpublished thesis, submitted for a degree at Williams College and administered by the Williams College Libraries, will be made available for research use. You may, through this form, provide instructions regarding copyright, access, dissemination and reproduction of your thesis. The College has the right in all cases to maintain and preserve theses both in hardcopy and electronic format, and to make such copies as the Libraries require for their research and archival functions.
_The faculty advisor/s to the student writing the thesis claims joint authorship in this work.
_ 1/we have included in this thesis copyrighted material for which llwe have not received permission from the copyright holder/s. If you do not secure copyright permissions by the time your thesis is submitted, you will still be allowed to submit. However, if the necessary copyright pennissions are not received, e-posting of your thesis may be affected. Copyrighted material may include images (tables, drawings, photographs, figures, maps, graphs, etc.), sound files, video material, data sets, and large portions of text.
l. COPYRIGHT An author by law owns the copyright to his/her work, whether or not a copyright symbol and date arc placed on the piece. Please choose one of the options below with respect to the copyright in your thesis.
__ Ilwe choose not to retain the copyright to the thesis, and hereby assign the copyright to Williams College. Selecting this option will assign copyright to the College. If the author/swishes later to publish the work, he/she/they will need to obtain permission to do so from the Libraries, which will be granted except in unusual circumstances. The Libraries will be free in this case to also grant permission to another researcher to publish some or all of the thesis. If you have chosen this option, you do not need to complete the next section and can proceed to the signature line.
~I/we choose to retain the copyright to the thesis tBHt-period·o-r===::.. years, or until my/our death/s, whichever is the earlier, at which time the copyright shall be assigned to Williams College without need of further action by me/us or by my/our heirs, successors, or representatives of my/our estate/s.
Selecting this option allows the author/s the flexibility of retaining his/her/their copyright for a period of years or for life.
Signatures Redacted
Entanglement and Bosonic Character of Pairs
of Distinguishable Fermions
by
Christina P. Knapp
Professor William K. Wootters, Advisor
A thesis submitted in partial ful.llment
of the requirements for the
Degree of Bachelor of Arts with Honors
in Physics
WILLIAMS COLLEGE
Williamstown, Massachusetts
May 17, 2013
Acknowledgements
My deepest gratitude goes to my advisor, Professor William Wootters, for his guidance and support over the last year. His enthusiasm and encouragement has made this thesis an incredibly positive experience. I could not imagine a topic better suited to my interests or an advisor better able to inspire and motivate my research. I am also extremely grateful to Professor David Tucker-Smith for his careful reading of this entire thesis and his helpful suggestions. I would like to thank Julian Hess for his patient help with LaTex questions, saving me many hours of frustration and troubleshooting. Finally, thank you to my family, friends, and fellow majors who have endured my thesis obsession and supported me throughout this process.
Abstract
Almost since the birth of quantum mechanics physicists have realized that composite particles made of even numbers of fermions exhibit uniquely bosonic properties. The best examples are atoms such as hydrogen, sodium, and rubidium that are able to condense into Bose Einstein condensates. In 2005, C.K. Law proposed that the ability of such systems to ignore the internal structure of the composite particle and avoid the Pauli exclusion principle is due to the entanglement of the constituent particles. We investigate Law’s hypothesis for systems of distinguishable fermion pairs. We de.ne two measures of bosonic character, both related to the condensation ability (maximum occupation number) of a single fermion pair within the system. We discuss three inequalities relating entanglement of the fermions within the pair to the bosonic character of the pair. We present evidence in support of the claim “a pair of fermions exhibits bosonic character if and only if the fermions within the pair are highly entangled.”
Contents
ExecutiveSummary ......................... v
ImportantDe.nitions ........................ ix
1 Introduction ............................ 1
2 Background ............................ 5
2.1 History............................... 5
2.2 MathematicalTools........................ 7
2.2.1 Creation and Annihilation Operators . . . . . . . . . . 7
2.2.2 DensityMatrices ..................... 11
2.2.3 SchmidtDecomposition ................. 13
2.2.4 Entanglement of Identical Particles . . . . . . . . . . . 16
2.3 Setup................................ 17
2.3.1 OurSystem ........................ 17
2.3.2 MeasuresofBosonicCharacter . . . . . . . . . . . . . 17
2.3.3 MeasuresofEntanglement ................ 19
2.3.4 Single-PairStateMatrix ................. 19
3 Entanglement Implies Bosonic Character . . . . . . . . . . . 23
= M(F)-1
3.1 1 - NPcN-1 ......................... 24
3.2 1 - NPd = M.(d)-1 ........................ 28
N-1
3.3 Remarks.............................. 29
4 Bosonic Character Implies Entanglement . . . . . . . . . . . 31
4.1 SystemofTwoPairs ....................... 32
4.2 SpecialInitialSingle-PairStates ................. 35
5 NumericalWork.......................... 37
M(F)-1
5.1 = 1 - NP ......................... 37
N-1
5.1.1 Broken-LineBound .................... 37
5.1.2 Dimension of Initial Single-Pair State . . . . . . . . . . 39
5.1.3 BoundaryStates ..................... 40
iii
5.2 P = Pc ............................... 42 6 Conclusions............................. 45 A Details of the Single-Pair State Matrix . . . . . . . . . . . . 49
A.1 Simpli.cationoftheMatrix ................... 49
A.2 DerivationoftheMatrix ..................... 51
B Single-Pair State Matrix for Special |a) ............ 55
B.1 |a) =(a, a, ..., a, 0)........................ 55
B.2 |a) =(e, e, ..., a),e . 0...................... 58
References ............................... 61
Executive Summary
There are many physical systems in which composite particles made of even numbers of fermions act like and are classi.ed as ideal bosons. One example is an atomic Bose Einstein condensate: atoms composed of fermions (electrons, protons, and neutrons) are cooled into a single quantum state, thereby making the atoms exhibit distinctly bosonic behavior. What attribute of such a system allows us to ignore the internal composition of the composite particle and accurately approximate it as an ideal boson?
In 2005 C.K. Law proposed that the bosonic behavior of a pair of fermions is due to the entanglement of the constituent particles [5]. To understand the motivation behinds Law’s hypothesis, consider the example of a hydrogen BEC. Recall that hydrogen is composed of a proton and an electron. Were we to have a system of N electrons, we would certainly not be able to condense the particles into a single state. The Pauli exclusion principle tells us that identical fermions cannot occupy the same state. Similarly a system of N protons could not condense. What makes a system of N hydrogen atoms di.erent, when the system still contains N electrons and N protons?
The key is that the proton and electron within a hydrogen atom are in an entangled state. While we might be able to write down a de.nite wavefunction for the atom, neither the electron nor the proton has a de.nite state of its own. As a concrete example, consider the following entangled wavefunction for particles A and B (not corresponding to any speci.c physical system):
1
(1) (1) (2) (2)
.AB = v (.f+ .f).
AB AB
2We are able to say that the A, B pair occupies the pair state described by .AB, but we cannot say anything de.nite about the state of particle A or of particle
B. The individual particles are in a mixture of two states. Suppose A and B are distinguishable fermions. Forcing another A, B pair into the same state would not necessarily violate the Pauli exclusion principle. A measurement on
(1)
the A’s would tell us that one A fermion has wavefunction .and the other
A
(2) (1) (2)
has wavefunction ., assuming .and .are orthogonal.
A AA
In the same way, condensing hydrogen atoms into the same state does not force the multiple electrons into the same state. The electrons do not
v
occupy de.nite states in the .rst place. Law’s idea was that entanglement allows for more identical fermions within a pair state. In the above example wavefunction, we could not have three A, B pairs with wavefunction .AB.A measurement on the the A’s would force two identical fermions into the same state. However, if we had a two-particle wavefunction
1
(1) (1) (2) (2) (3) (3)
= v (.f+ .f+ .f)
FAB ABABAB
3
we could conceivably .t a third A, B pair into the state. The more entangled the pair, the more available single-particle states there are within the pair state. In this sense, there is more “room” to add another composite particle as it does not force two identical fermions into the same single-particle state. In the case of a hydrogen BEC, each atom’s wavefunction extends over a region much larger than an atom. This means that the sum analogous to the one shown above would have many more terms. Fermion pairs that are able to condense into the same state are bosonic in the sense that the pair is not obeying the Pauli exclusion principle.
Papers by Chudzicki et al [1] and Tichy et al [11] have investigated Law’s hypothesis, providing evidence of a connection between bosonic behavior and entanglement. A 2011 paper by Ramanathan et al addressed limitations of such a connection [8]. In this thesis we further examine Law’s hypothesis for a system of pairs of distinguishable fermions.
The system in question consists of N pairs, each pair made of one A-type and one B-type fermion. We de.ne two measures of bosonic character, both of which look at the ability of the fermion pairs to condense into a single-pair state. Our measures are:
M(F) - 1 M'(d) - 1
and ,
N - 1 N - 1
where the quantities M(F) and M'(d) are analogous to occupation number for a fermion pair state. The di.erence in the two measures involves the particular pair state we consider. M(F) is analogous to the occupation number of the maximally occupied single-pair state (.x N-pair state |F) and .nd the single-pair state maximally occupied by |F)). M'(d) corresponds to the maximum possible occupation number for a .xed single-pair state (associated with pair operator d) over all possible N-pair states.
We measure the entanglement using the purity of a single fermion pair. There are several single-pair states whose purity we could reasonably use as an indication of the entanglement of the system. The three states we consider are: the initial single-pair state used to generate the N-pair state |F), the maximally occupied single-pair state of |F), and the .xed single-pair state used when calculating M'(d).
Our work naturally separates into two sections. The .rst section presents evidence that entanglement of the fermion pairs sets a lower bound on the bosonic character of the system. Our results in this section are in the form of
N - 1
the two inequalities:
1 - NPd = M'(d) - 1 N - 1 (1)
1 - NPc = M(F) - 1 (2)
where Pd in inequality 1 is the purity of the .xed single-pair state associated with operator d and Pc in inequality 2 is the purity of the initial single-pair state used to generate the N-pair state |F). We analytically prove inequalities 1 and 2 in Chapter 3.
According to inequality 1, the purity of some .xed single-pair state together with the number of pairs in the system sets a lower bound on the ability of the system to condense into that particular state. That is, highly entangled states can be highly occupied. By using M'(d) we consider the optimal condensation ability of the single-pair state: we choose our system to be in the N-pair state that maximally occupies the .xed single-pair state.
Inequality 2 relates the purity of the initial single-pair state to the bosonic character of the maximally occupied state. We see that the more strongly entangled the single-pair state we use to create our N-pair state, the more highly occupied the maximally occupied single-pair state must be.
The second half of the thesis supports the claim that high bosonic character of the fermion pairs ensures the pairs are strongly entangled. Again, our results are in the form of an inequality:
M(F) - 1 N
= 1 - NP for M(F) = , (3)
N - 12
where P is the purity of the single-pair state maximally occupied by the N-pair state |F). In Chapter 4 we prove inequality 3 for the case of a system of two fermion pairs (N = 2) as well as when we consider special forms of the initial single-pair state used to generate |F). We also give extensive numerical evidence of inequality 3 for N =2, 3, 4, and 5 in Chapter 5.
According to inequality 3, for a given N-pair state |F), the bosonic character and entanglement of the maximally occupied single-pair state have the opposite relation as that of the .xed single-pair state in inequality 1. For the maximally-occupied single-pair state bosonic character sets a lower bound on the entanglement. That is, the pair state that is most highly occupied must be highly entangled.
The .rst half of this thesis presents evidence supporting Law’s hypothesis. The second half extends Law’s hypothesis to “a system of fermion pairs will act bosonic if and only if the fermions within each pair are highly entangled.” That is, the implication goes in both directions. In our concluding chapter we further discuss the implications of each of our inequalities and describe several of the remaining open problems.
Important De.nitions
General De.nitions
N = the number of fermion pairs in the system (4)
aj = annihilation operator for fermion A in the jth state (5)
c =akakbk, annihilation operator for initial fermion pair (6) k
|F)= c †N |0), unnormalized state of N fermion pairs of special form (7)
|O)= unnormalized state of N fermion pairs of arbitrary form (8)
Bosonic Character
(F|d†d|F)
M(F) = max , where d corresponds to the maximally (9)
d (F|F)occupied single-pair state of |F)(O|d†d|O)
M'(d) = max , where |O) is the N-pair state maximally (10)
|O\ (O|O)
occupying the single-pair state associated with d
†
(F|cc|F)
Mc = (11)
(F|F)
Entanglement
P =.k2, where the .’s are the reduced density matrix eigenvalues (12)
k
P =ßk4, purity of the maximally occupied pair state in |F) (13)
k
Pc =aj 4, purity of the initial pair state associated with operator c (14) j
Pd purity of the pair state associated with operator d (15)
ix
Chapter 1
Introduction
Elementary particles fall into two categories: bosons and fermions. In particle physics a boson is usually a force carrier while a fermion is a mass constituent (particle upon which the force acts). More fundamentally the di.erence has to do with which statistics the particle obeys: bosons satisfy Bose-Einstein statistics while fermions obey Fermi-Dirac statistics. It turns out that the statistical behavior of an elementary particle is deeply connected to the particle’s spin by the spin-statistics theorem in quantum .eld theory [10]. This results yields yet another distinction: bosons have integer spin while fermions have half-integer spin.
In ordinary quantum mechanics we distinguish between bosons and fermions by the behavior of their wavefunctions under the exchange of identical particles. Bosons have symmetric wavefunctions: exchanging identical bosons does not change the wavefunction. Let |.) describe a state of two identical bosons:
|.) = cjk|j)|k),
j,k
where the sum is over the modes available to each boson. The symmetry of the wavefunction requires
cjk = ckj.
Fermions on the other hand have antisymmetric wavefunctions: exchange of identical fermions results in an overall minus sign for the wavefunction. If |.) is now a two-fermion state:
|.) = ajk|j)|k)
j,k
(again the sum is over the modes available to each fermion) then antisymmetrization requires
ajk = -akj.
It immediately follows that ajj =0.
1
This important result is the Pauli exclusion principle: two identical fermions cannot inhabit the same quantum state. In this thesis we are particularly focused on the Pauli exclusion principle as a way of distinguishing between bosons and fermions.
Not only do bosons not obey the Pauli exclusion principle, they violate it in the most egregious way. A system of many bosons is able to condense so that every boson is in the same quantum state. There is no theoretical limit to the number of bosons occupying a particular state. Such a system is called a Bose-Einstein condensate (BEC) and is especially interesting because it exhibits (relatively) large-scale quantum mechanical e.ects.
Consider the following paradox.
Experiments have made a BEC out of hydrogen [4]. In doing so, the experiments manipulate a system of many hydrogen atoms so that every hydrogen atom is in the same state. Recall that a hydrogen atom consists of a proton and an electron, both of which are fermions. If every hydrogen atom is in the same state, all the electrons are in the same state. But doesn’t that mean that in the BEC there are multiple fermions occupying the same state, thereby violating the Pauli exclusion principle?
The fallacy is in the assumption that if two hydrogen atoms are in the same state then the electron associated with one atom must be in the same state as the electron of the other atom. In quantum mechanics, a system of more than one particle is in a joint state. It is often the case that given a de.nite state of several particles we are unable to write down individual states corresponding to each particle in the system. The problem is not simply that we do not know the single-particle states, but that the particles in the system do not have an individual state. Their existence is correlated to the other particles. Any measurement on a given particle will change the state not only of that particle, but also of every particle with which it shares this correlation. This phenomenon is called entanglement and is uniquely quantum mechanical: entanglement has no analog in classical systems.
In an entangled state between particles A and B, many single-particle states of A are represented. Consider for example the following two-particle state:
|.).|f1)A|.1)B + ... + |fd)A|.d)B
where the |f)A’s are basis states for particle A and the |.)B’s are basis states for particle B. Notice that even though we have an exact answer to “what is the state of the two-particle system,” we are unable to answer the question “what is the state of particle A?” A’s state is inextricably linked to B’s state. If we made a measurement on A, we would immediately know the outcome of a measurement on B. That is, if we found A to be in the state |fi)A then we would know that B is in the state |.i)B without making any measurement on B.
Entanglement saves the day in the above situation of a hydrogen BEC. Each hydrogen atom is in the same state as the other atoms in the BEC, which means that every proton-electron pair is in the same state as the other proton-electron pairs. However, neither the proton nor the electron has a de.nite state of its own. Condensation forces all the atoms into the same single-pair state, but does not put the electrons into a single-particle state. The fermions in the hydrogen BEC do not violate the Pauli exclusion principle.
The hydrogen BEC is an example of a system in which pairs of fermions form composite particles that act at least in some respects like bosons. Other examples include Cooper pairs, excitons, and polaritons. Cooper pairs are bound electron pairs at low temperature that are responsible for superconductivity in metals [2]. An exciton is a bound state of an electron and a hole. In the low density limit, excitons obey Bose-Einstein statistics [13]. Polaritons are coupled excitons and photons (each polariton consists of two fermions and a boson). A recent paper by Nelsen et al reports observing super.uid and BEC properties in systems of polaritons with long lifetimes [7]. There are many other examples in nature of pairs (or other even numbers) of fermions whose cumulative behavior is bosonic. However, it is certainly not a requirement that systems of fermion pairs have bosonic properties! An electron and a proton in a plasma, for example, cannot be treated as a boson.
Our discussion so far has dealt with why bosonic behavior of fermion pairs is not a contradiction. We now turn our attention to why it should happen at all. What is special about the above examples that allows the constituent fermions within a composite particle to ignore each other (thus allowing the composite particle to avoid the Pauli exclusion principle)? More concretely, when can we classify a pair of fermions as a boson?
In 2005 C.K. Law hypothesized that the entanglement of a pair of fermions gives rise to the bosonic behavior exhibited by the pair [5]. The following argument motivates why Law’s hypothesis is reasonable.
Consider a pair of fermions in a de.nite pair state. If the fermions are not entangled, then each fermion has one single-particle state available to it. Such a pair cannot occupy the same pair state as another pair as it would force identical fermions into the same state. Now consider an entangled pair state such that each fermion in the pair could be in one of two states. We could conceivably force two pairs into this state. A measurement on one of the fermions in this state would guarantee the opposite outcome for the identical fermion. However, if we try to force a third pair into the state, we will fail. If we were to measure the three identical fermions, we would need the outcome for two of them to be the same, violating the Pauli exclusion principle. To put three pairs into the same state we would need a more highly entangled pair state.
Let bosonic character refer to the ratio of fermion pairs we are able to condense into a single pair state divided by the number of pairs in the system:
fermion pairs in the same pair state
bosonic character = .
fermion pairs in the system
We will make this de.nition precise in the next chapter. Bosonic character of zero
corresponds to no fermion pairs in the same state. In this case the system acts like a
system of ideal fermions (the composite particles obey the Pauli exclusion principle). Bosonic character of one corresponds to every fermion pair in the same state. The pairs are acting like ideal bosons.
This thesis presents evidence supporting and extending Law’s hypothesis. We consider the relationship between bosonic character and entanglement for a system of N pairs of fermions, each pair composed of one A-type and one B-type fermion.
We begin by giving the history of the problem and explaining the relevant mathematical tools in Chapter 2. We de.ne two measures of bosonic character, both based on the above idea of the number of pairs we can put into the same state. Chapter 3 presents evidence in support of Law’s hypothesis by proving two inequalities of the form:
quantity involving entanglement and number of pairs = bosonic character of the pairs.
We can read these inequalities as, “entanglement implies bosonic character.” In Chapter 4 we reverse directions to examine the converse of Law’s hypothesis: bosonic character of the fermion pairs forces high entanglement of the pair. We introduce a third inequality of the form:
bosonic character of the pairs = quantity involving entanglement and number of pairs.
We prove special cases of this inequality and discuss its implications. Chapter 5 continues our exploration of the converse of Law’s hypothesis by analyzing numerical results. We look at more general cases of the inequality presented in Chapter 4 by plotting the two sides of the inequality for randomly chosen states of the system. We also consider several other relationships between entanglement and bosonic character. In Chapter 6 we summarize our results and discuss several of the remaining open questions.
Chapter 2
Background
2.1 History
Entanglement was .rst connected to composite bosons in a paper by Law published in 2005 [5]. Law hypothesized that a composite particle with an even number of highly entangled fermions will act like an ideal boson. His system consisted of N composite C particles, with each C consisting of two distinguishable particles, A and
B. A and B are either both fermions or both bosons. Law measured the bosonic character of C by comparing the annihilation operator of C, denoted c, with that of an ideal boson. The state of a number N of C particles can be written as:
†N
- 1 c
|N) = . 2 v|0)
N
N!where .N is a normalizing factor. Law was able to show that as .N+1 . 1, c
.N
approaches the behavior of an ideal bosonic annihilation operator.
By considering an example wave-function (that of a harmonic oscillator), Law showed that in the limit of high entanglement between A and B, .N+1 approaches
.N
one. He measured entanglement using the inverse of the purity, P , discussed in the next section. From this example, Law reached his hypothesis that the high entanglement of constituent particles gives rise to the bosonic behavior of the composite particle.
In 2010, Chudzicki et al supported Law’s hypothesis by using entanglement to place upper and lower bounds on the bosonic character of a composite particle [1]. Chudzicki et al considered a general wavefunction for a system of composite C particles, with constituent particles A and B both fermions. They measured entanglement using the purity, and used the same measure of bosonic character as Law. By using the Schmidt decomposition, explained in the next section, they were able to prove
.N+1
1 - NP == 1 - P.
.N
The fraction 1 can be thought of as the number of available single-particle states,
P
while N is the number of composite particles in the same pair state. The lower bound
5
tells us that in order to exhibit bosonic behavior, the number of single particles states available to a composite particle must be much larger than the number of composite particles in the system. Intuitively this makes sense: the constituent fermions will not run into the Pauli exclusion principle as long as there is a low probability of two of them occupying the same state. We can therefore ignore the internal structure of the composite particle and treat it as an elementary boson. The lower bound is the strongest possible using the purity. It is achievable whenever the number of composite particles is less than the number of orthogonal states available to each of the two fermions. These orthogonal states are called Schmidt modes (see subsection 2.2.3). The upper bound is weaker as it is not achievable for general N.
A 2011 paper by Ramanathan et al argued against Law’s hypothesis [8]. Ramanathan et al claimed that while entanglement is necessary for composite bosons, it is not su.cient. Their argument proceeds as follows
Ideal bosons can condense using only local operations and classical communication (LOCC). A system of composite bosons should also be able to condense using LOCC. In Law’s model of composite C particles, LOCC means that no quantum interactions are allowed between constituent particles A and B. If there exists a candidate wavefunction for C which is highly entangled, but which requires quantum interactions in order to condense, then this wavefunction proves that entanglement does not guarantee bosonic behavior.
Ramanathan et al are able to .nd such a wavefunction. They conclude that entanglement is not a su.cient condition for bosonic behavior.
The subtle issue in Ramanathan et al’s logic is that they have forbidden quantum interactions within the composite particle. This is not analogous to LOCC in the case of ideal bosons because there are no internal interactions for ideal bosons. In the case of composite bosons, the C’s should be able to condense without communicating with each other, but it might be too strict a restriction to exclude quantum communication among the A’s and B’s. Perhaps it is necessary for the constituent particles to interact in order to preserve the entanglement of the composite particle while undergoing condensation. Unfortunately, entanglement of identical particles is not yet a well-de.ned concept so it is di.cult to address this issue. For our present purposes it is enough to note that Ramanathan et al’s paper does not disprove Law’s hypothesis.
Law’s hypothesis was further supported by Tichy et al in 2012 [11]. Tichy et al also considered composite particles made of two fermions. Using the same measures of bosonic character and entanglement as Chudzicki et al, they were able to tighten the upper bound on bosonic character. They did this by considering the distribution of coe.cients of the Schmidt modes that optimizes the ratio ..NN +1
.N+1 (max)
for .xed P . Letdenote the ratio of normalization factors for the
.N
.N+1 (max)
maximizing distribution. They were able to show that.Nfor a given P grows with increasing number of non-vanishing Schmidt modes. Therefore, in the
(max)
.N+1 .N+1
limit of in.nite number of Schmidt modes, is an upper bound on
.N .N
for a general distribution. By following this line of reasoning they improved the inequality presented in Chudzicki et al to
1 - NP = .N+1 = 1 - P N v
.N 1 + (N - 1) P
2.2 Mathematical Tools
We outline the use of mathematical tools including creation and annihilation operators, Schmidt decomposition, density matrices, and the purity. For a more in depth discussion see [6].
2.2.1 Creation and Annihilation Operators
Creation and annihilation operators represent the creation and destruction of particles. The operator always refers to some .xed quantum state, a wavefunction, for the particle. Let c denote the annihilation operator associated with the particle C in a particular state. The creation operator, c†, acting on the vaccum, |0), creates one C particle in that state:
c †|0) = |1)1 .
The annihilation operator, c, acting on one C particle in that state, |1), returns the vacuum state:
c|1) = |0).
The key idea behind all of the measures of bosonic behavior mentioned in the previous section is the creation and annihilation operator rules for identical bosons and fermions. Operator rules di.er for bosons and fermions because of the symmetries of their associated wavefunctions. Fermions have antisymmetric wavefunctions under interchange of identical particles. Suppose two identical fermions are in two distinct states corresponding to wavefunctions .A and .B. We write the wavefunction of the two-particle state as
1
v (.A(x).B(y) - .A(y).B(x)).
2
This wavefunction is normalized when .A and .B are orthogonal. Interchanging the fermions returns the negative of the initial wavefunction. The two fermions can be distinguished by their states, but they have no intrinsic identities. An equivalent statement in terms of annihilation operators is if aj,ak correspond to two di.erent states of the same type of fermion, then
ajak = -akaj.
1We use the Fock representation |N) to denote N particles in the same state.
Bosons have symmetric wavefunctions under interchange of identical particles. If two identical bosons are in two distinct states, the two-particle wavefunction is
1
v (.A(x).B(y)+ .A(y).B(x)).
2
We see the wavefunction is invariant under interchange of the two particles. The equivalent statement in terms of annihilation operators is that if fj,fk correspond to two di.erent states of the same type of boson, then
fjfk = fkfj.
Identical fermions obey the Pauli exclusion principle and therefore cannot occupy the same quantum state. This physical property is enforced by the antisymmetric wavefunction. Suppose we try to put two identical fermions in the same state. The wavefunction would have to be
1
v (.A(x).A(y) - .A(y).A(x)) = 0,
2
†† ††
which is an impossible wavefunction. In terms of creation operators, aja = -aja
jj
is only satis.ed when ajaj = 0. This tells us that if we try to put two fermions in the same state we will get the number zero:
a †|1) =0.
As any fermion state can contain at most one fermion, the only situations we need to consider for the fermion annihilation operator a are its application to the state of one fermion and to the vacuum state. In these cases we have
a|1) = |0) and a|0) =0.
Identical bosons can occupy the same quantum state. Again, this fact is apparent from the symmetric wavefunction. If two identical bosons are in the same state, the system has wavefunction proportional to
.A(x).A(y)+ .A(y).A(x).
Let f† be the creation operator associated with a boson F in a given state. If we apply f† to a state already containing a boson F in the state associated with f†, we get two F particles in the same state: f†|1) = µ1|2), where µ1 is a normalization factor. There is no limit to the number of bosons we can put in the same state, for any integer N we have:
f†|N) = µN |N +1).
Similarly when we apply the annihilation operator f we have:
f|N) = .N |N - 1),
where by convention we take µN and .N to be real and positive.
We solve for µN and .N by considering the quantum harmonic oscillator. In terms of the harmonic oscillator, f† and f are the raising and lowering operators respectively and |N) denotes an oscillator in the Nth energy state. Applying the raising operator to a state raises the oscillator up one energy level and applying the lowering operator to a state brings the oscillator down an energy level. That is, f†|N).|N +1) and f|N).|N - 1). Raising and lowering operators are de.ned in terms of the position and momentum operators (see [3]). From their de.nitions it is possible to rewrite the Hamiltonian for the harmonic oscillator as
= ..(f†f +1
H ).
2
Of course, the energy eigenvalue equation still applies:
H|N) = EN |N)
11
..(f†f +)|N) = ..(N +)|N)
22
from which we see f†f|N) = N|N).
The operator f†f is called the number operator because its eigenvalue is the energy level of the harmonic oscillator.
Returning to creation and annihilation operators, we still have the number operator f†f, but we now interpret its eigenvalue to be the number of particles in the state corresponding to operator f. The state |N) must be normalized such that (N|N) = 1. If f|N) = .N |N - 1) then normalization requires
(N - 1|.2
.2
(N|f†f|N) = N |N - 1)
N(N|N) = N (N - 1|N - 1)
N = .2
N
v
N = .N
Therefore
v f|N) = N|N - 1).
We can .nd the normalization factor for the creation operator using what we know about the number operator and the annihilation operator:
N|N) = f†f|N)
v = Nf†|N - 1)
v = NµN-1|N)v
Therefore µN-1 = N and
v f†|N) = N +1|N +1).
as an in.nite-dimensional
One can think of the bosonic creation operator f† 10
.
.
.
.
matrix. Let
...
0
0
...
represent the vacuum state,
...
1
0
...
represent the state containing
... ... one F boson in the state associated with f†, and so on. Then:
.
.
f†
=
.......
00 0 0 ... 10 0 0 ...
v
0 200 ...
v
0 0 3 0v ...
0 0 0 4 ...
... ... ... ... ...
.......
.
We see
..
.
..
.
00 0 0 ... 10
f†|0) =
.......
10 0 0
v
.......
.......
0
0
0
0
.......
=
.......
1
0
0
0
.......
= |1),
...
0 200
v
...
00 30 ...
v
000 4 ...
... ... ... ... ... ... ...
just as we expect. The annihilation operator, f is the adjoint of f†:
.
.
.....
010 0 ...
v
00 20 ...
v
000 3
...
000 0 ...
... ... ... ... ...
.....
f
=
and the number operator is
.
.
.....
0000 ...
0100 ...
0020
...
0003 ...
... ... ... ... ...
.....
f†f
=
.
The operator rules for a single fermion or boson can be summarized by the following commuator and anticommuntator relations:
fermions: {ak,al} =0
†
{ak,a l } = dkl bosons: [fk,fl]=0
†
[fk,fl ]= dkl Here {x, y} = xy + yx and [x, y]= xy - yx.
Finally we consider states made of more than one type of particle. Let F and G be two distinct kinds of boson, and let the state |NF ,NG) denote a number NF of F bosons in a given state and a number NG of G bosons in a separate state. If f, g are the annihilation operators associated with these states of F and G bosons, then the following are true:
f|NF ,NG) =NF |NF - 1,NG)
g †|NF ,NG) =NG +1|NF ,NG +1)
f† g|NF ,NG) =(NF + 1)NG|NF +1,NG - 1)
f|0,N2) =0
The last line tells us that if we apply an annihilation operator f on a state that does not contain a particle in the state associated with f, we get the number zero regardless of how many other particles were in the initial state.
Let A, B be two distinct fermions associated with annihilation operators a, b respectively. The following hold:
a|1, 1) = |0, 1)
b†|1, 0) = |1, 1)
a †|1, 1) =0
a|0, 1) =0
The second to last line ensures that two fermions cannot occupy the same state. The last line is the same as for bosons: the annihilation operator for a particle acting on a state that does not contain that particle returns the number zero.
2.2.2 Density Matrices
A pure state is a state to which we can assign a de.nite wavefunction and a mixed state is a state to which we cannot assign a de.nite wavefunction. A mixed state arises both when there does not exist a de.nite wavefunction for the system, and when there exists a de.nite wavefunction but we do not have enough information to determine its form. For an example of the .rst case, consider a system of two particles, A and B, with wavefunction .AB(xA,xB). It is often the case that the composite wavefunction cannot be written in product form as a wavefunction for particle A and a wavefunction for particle B:
= .A(xA).B(xB).
.AB(xA,xB) .
While the two-particle system is in a pure state (it has de.nite wavefunction .AB(xA,xB)), particle A is in a mixed state (as is particle B).
It is important to note that a mixed state is di.erent from a superposition of states. A superposition is a way of combining states by adding vectors. A mixed state combines states in a di.erent way discussed below. A superposition of states corresponds to a de.nite wavefunction, a mixed state does not.
When we have a mixed state, the density matrix . provides the most complete description of the system possible. In some sense, . serves in place of the wave-function. For instance, given a pure state |.) and observable X, we calculate the expectation value of X to be
(X) = (.|X|.). For a mixed state we instead have
(X) = Tr(.X).
The way we calculate a system’s density matrix depends on the information we are given. Suppose we have incomplete information about the preparation of the state of the system, but we know that there is a probability P. of preparing the state |.). The set of possible states, {|.)} may or may not be orthogonal. We can write the density matrix as
. = |.)P.(.|.
.
The eigenvalues of the density matrix constitute a probability distribution (the distribution of outcomes were we to measure the state in a speci.c orthonormal basis of the available Hilbert space). The trace of the matrix is the sum of its eigenvalues, therefore a density matrix always has trace equal to one:
Tr . =1.
A normalized pure state |.) has density matrix . = |.)(.| (because it has probability 1 of being found in state |.)). Consider again a pure state of two particles A and B with normalized wavefunction |.AB). The density matrix of the system is given by
.AB = |.AB)(.AB|.
A closely related concept is the reduced density matrix. In the above situation of a system of particles A and B, the reduced density matrix gives the most complete description of particle A without referencing particle B. One way to compute the reduced density matrix of particle A is to imagine a measurement being performed on particle B. .A describes the possible states available to A without knowing the outcome of the measurement on B. That is, if the measurement on B yields the outcome |.B) with probability P., and |.A) is the corresponding state of particle A for this outcome, then
.A |.A)P.(.A|.
=
.
In general, we calculate the reduced density matrix by taking the partial trace:
.A = TrB .AB = (kB|.AB|kB)k
where the |kB)’s are a basis of the Hilbert space for particle B. We give an example of such a calculation in the following section.
2.2.3 Schmidt Decomposition
The Schmidt decomposition is a way of writing the wavefunction of a two-particle system that reveals the quantum correlation between constituent particles. According to Schmidt’s theorem, any two-particle wavefunction .(xA,xB) can be uniquely written in the form
.(xA,xB)= .nfA(xA)fB(xB),
nn n
where the .’s are the eigenvalues of the reduced density matrix of either particle, and fA,fB are orthonormal bases of the Hilbert space available to particles A and B
nn
respectively. We see the Schmidt decomposition explicitly shows the pairing structure: if particle A is found to have wavefunction fA, then we know with certainty
n
that particle B has wavefunction fB .
n
An equivalent way of writing the two-particle state is
††
|.) =
.na
b
|0),
nn
n
where a larly for b
†
denotes the creation operator for an A particle in the nth state and simi
n
†
n.
From this form we can conclude that the creation operator associated
with the two-particle state, c†, can be written as
†
c =
.na
††
b
.
nn
n
.
.
To better understand the Schmidt decomposition let us work through two ex
.. .. .. ..
...
...
1
2
amples. Consider two spin-
particles represented by the basis
, where the
..
.rst arrow gives the spin of the .rst particle and the second arrow gives the spin 1 0 0 0
...
...
of the second particle. In this basis, the vector
represents the state |..), two
spin-up particles.
.
.
...
1 0 0 1
...
1v 2
Example 1. Write the Schmidt decomposition of the state |.) =
.
We can write this state asform.
1v 2
(|..)+ |..)), which is already in Schmidt
.
.
1
0
Example 2. Write the Schmidt decomposition of the state |.) =
1v 3
...
...
.
1
1
Let’s .rst rewrite the state in terms of |.) and |.) for each particle.
..
1 10
..
..
|.) = v
..
1 1 1
3
= v (|..)+ |..)+ |..))
31
= v (|.). |.)+ |.) . (|.)+ |.))
3
Notice that this state is not yet written in Schmidt form because the second particle is not written in an orthonormal basis: |.) and |.)+ |.)are not orthogonal. To write this state in Schmidt form, we .rst need to .nd the Schmidt bases for particles A and B. These correspond to the eigenvectors of the reduced density matrices.
The vector |.) represents a pure state, therefore our two-particle system has density matrix
..
1011
..
.AB 1 . 0000
.
= |.)(.| = .
..
3 1011 1011
The reduced density matrix for particle A is .A = TrB .AB . To calculate this matrix, begin by separating .AB into four 2 × 2 matrices:
. .
10 11
. .
1 0000
. . .
. .
3 10 11 1011
We get .A by taking the trace of each smaller matrix:
1 11
.A
= .
3 12
v
5
This matrix has eigenvalues 3± that correspond to eigenvectors:
6
. v .. v .
-1+ 5 -1- 5
vv
vv .. v ... . .
10-2 5 10+2 5
,
2
5+ 5 v 10 5+5
To .nd the reduced density matrix for particle B, .B = TrA .AB , we add the matrices sitting along the diagonal:
110 111 121
.B =+= .
300311311
v
5
This matrix has eigenvalues 3± that correspond to eigenvectors:
6
. v .. v .
1+ 5 1- 5
vv
vv
10+2 5 10-25
. .. .
,.
v
2
v 5+ 5 5+5 10
We can now write the Schmidt decomposition of |.) 2:
. .
vvv v
3+ 5 -1+ 5 5+ 5 1+ 5 2
. v |.) +|.). . v |.) +v |.)
6 10 5+ 5
10 - 2 5 10+2 5
. .
vv vv
3 - 5 -1 - 52 1 - 5 5+ 5
. .
+ v |.) +v|.) . v|.) +|.)
6 5+ 5 10
10+2 5 10 - 25
1
Expanding this expression returns the form |.) =(|..)+ |..)+ |..)).
3
The distribution of the .n’s gives a measure of the entanglement of the system.
s
Remembering that Tr . = 1, we know that .n = 1. If there is only one non-zero
n eigenvalue, it must be equal to one and .AB(xA,xB)= .A(xA).B(xB). In this case the particles are not entangled: each particle is in a pure state. Typically, we work in an in.nite dimensional Hilbert space and correspondingly there are in.nitely many .n’s. However, in certain situations it is simpler or more instructive to work in a .nite dimensional Hilbert space. A particle in a .nite dimensional Hilbert space will have .nitely many .n’s. If all the .n’s are equal, then each single-particle state is as far from a pure state as possible; we say that A and B are maximally entangled. One way to quantify entanglement is the purity:
P = .2 .
nn
P = 1 corresponds to a pure state. In .nite dimension, if d is the number of states available to a system then P = 1 corresponds to maximal entanglement. Smaller
d
purity means higher entanglement. To illustrate this concept, let us work through the two examples we considered earlier.
. .
1
. .
0
v1 ..
Example 3. Calculate the purity of the state |.) =
..
2 . 0 1
1
We know the Schmidt decomposition of |.): |.) = v (|..)+ |..)).
2
2The procedure we used in this example does not completely determine the Schmidt decomposition because there could be a phase factor between the two basis states. In our example, though, there is no phase factor.
1
The eigenvalues of the reduced density matrix of each particle are 12 , 2 .
1
Therefore the purity is P =(1 )2 +(1 )2 = 2 . In this example, there are
22
only two states available to each particle, |.) or |.), and the purity is
1
2 , so the two particles are maximally entangled.
.
.
1 0 1 1
...
...
v
1
v 3
From example 2 we know the eigenvalues of the reduced density matrices
v
3± 5 55
are . Therefore the purity of this state is P =(3+ )2 +(3- )2 =
6 66
7 Notice that the purity is greater for this example than in the previ
..
v
Example 4. Calculate the purity of the state |.) =
.
9 .
1
0
0
2
1v
...
...
is more entangled than the state
ous, and therefore the state
1
.
.
...
1
0
1
...
3
1v
.
1
2.2.4 Entanglement of Identical Particles
As mentioned before, entanglement of identical particles is not a well-de.ned concept. The di.culty is partly due to the symmetry of the wavefunctions. Consider the case discussed earlier where two identical fermions or bosons are in distinct states. Let .f denote the two-fermion wavefunction and .b denote the two-boson wavefunction. As discussed before,
1
.f = v (.A(x).B(y) - .A(y).B(x))
2
1
.b = v (.A(x).B(y)+ .A(y).B(x)).
2Clearly neither .f nor .b is factorable. Because the wavefunction cannot be written in product form it has the appearance of being entangled. Now let us switch to the Fock representation and let |NA,NB) denote the state where NA of the identical particles are in state A and NB of the identical particles are in state B. In this representation the state is simply
|1, 1) = |1).|1).
which appears not to be entangled.
Evidently, when we are dealing with identical particles, the way we represent a system a.ects how entangled we view the state. This choice of notation cannot have any physical signi.cance. We see then that the symmetry of identical boson and fermion wavefunctions gives the appearance of entanglement without any physical relevance. As of now there is no universally-accepted way of measuring entanglement of identical particles and fortunately we will avoid this issue in this thesis.
2.3 Setup
2.3.1 Our System
We consider the same system as Chudzicki et al [1]: a number N of composite C particles, with each C particle composed of two distinguishable fermions A and B. From the Schmidt decomposition we can write that the creation operator associated with a C particle is given by
†† b†
c = ana,
nnn where the an’s are the square roots of the eigenvalues of the reduced density matrix for A (or B). The creation operator an † acting on the vacuum creates one A particle in the nth state (similarly for the B’s). Let |O) denote a general state of N number of A, B pairs and |F) denote the N-pair state of the special form:
†† ††
|F) = c †N |0) = aj1 ...ajN aj1 bj1 ...ajN bjN |0)j1,...,jN
(|F) is not normalized to unit length). Note that any terms for which ji is equal to jk do not contribute to the sum by the operator rules described in section 2.2.1.
2.3.2 Measures of Bosonic Character
We measure bosonic character by looking at how many C particles we can put into the same state. The ability to have many particles in the same state is a bosonic property. We want to use bosonic character to place upper and lower bounds on the entanglement of an A, B pair. We introduce two measures of bosonic character. Both of our measures are similar to that of Law, Chudzicki et al, and Tichy et al in that we compare the operator c to an ideal bosonic operator. Our measures di.er in that we are more interested in a comparison of c†c to the number operator rather than the behavior of c itself.
Before we introduce our measures of bosonic character, we .rst recall an important property of the number operator. Let f be an operator associated with an ideal boson. As mentioned before, the state
1
v f†N |0) = |N)N!
is the normalized state containing N bosons in the same state. Remembering f†f is the number operator, we have
(N|f†f|N) = N.
v If we de.ne the unnormalized state |.) as |.) = f†N |0) = N!|N), then we have
(.|f†f|.) N!(N|f†f|N)
== N.
(.|.) N!(N|N)
Consider the ratio
†
(F|cc|F)
.
(F|F)
If c were an ideal bosonic operator, this ratio would be equal to N. As c is not an ideal bosonic operator, we know it will instead always be less than N. 3 Our measures of bosonic character look at the best we can do: how close can we bring this ratio to N?
Let d be an annihilation operator corresponding to some state of our composite particle:
d = ßm,nambn, where |ßm,n|2 =1. m,n m,n
We consider the following two quantities:
(F|d†d|F)
M(F) = max
d (F|F)
(O|d†d|O)
M ' (d) = max .
|O\ (O|O)
The .rst quantity, M(F), .xes |F) and looks at which d maximizes the ratio. That is, we choose a special kind of N-pair state, analogous to the state |.) in the case of ideal bosons, and look for the single-pair state that maximizes the expectation value of d†d.
A simpli.ed way of interpreting how we calculate M(F) is that we choose an N-pair state and look at which single-pair state is most highly occupied. M(F) denotes the level of occupation of this maximally occupied single-pair state. This interpretation is not rigorous because d†d is not a proper number operator and therefore we cannot accurately talk about occupation numbers. However, the more bosonic the fermion pairs are, the more accurate this interpretation is. If we were dealing with bosons, the maximally occupied single-pair state would be the state associated with c, the operator we used to create |F). In this case we would have d = c. That is, the most occupied state would be the state we used to form |F). Interestingly, for our composite particle C, d = c in general.
F|cc|F
3To see this, note that the ratio F† |F can be written as
a2 a2 ...a2 )...(1-djN-1,jN )
m,j1,...,jN
mj1 jN (1-dm,j1
N (the full details can be worked out using
a2 ...a2
r1 rN
r1,...,rN
the de.nitions of c and |F) and the operator rules for identical fermions). The quantitiy
a2 a2
mj1 ...aj2 N (1-dm,j1 )...(1-djN-1,jN )m,j1,...,jN F|cc|F
†
is less than one, thereforeis less than N.
a2 ...a2 (1-dr1,r2 )...(1-drN-1,rN ) F|F
r1 rN
r1,...,rN
The second quantity, M ' (d), .xes d and maximizes the ratio over all |O). Essentially, we choose a particular single-pair state (that associated with d), and then .nd how many composite particles we can cram into that state. M ' (d) addresses the question of how occupied a given single-pair state can be. For the same reason as before, this interpretation is not rigorous, but becomes more so in the limit of high bosonic character. When .nding M ' (d) we consider all N-pair states, not only those restricted to the special form of |F). If we were working with ideal bosons, we would .nd the |O) that maximizes the ratio would be d†N |0) and M ' (d)= N.
Our two measures of bosonic character are the ratios:
M(F) - 1 M ' (d) - 1
and .
N - 1 N - 1
Both ratios range between zero and one. When either of them is equal to zero, we know we can never have more than one composite particle in a given state. In this case our composite particle is acting like a fermion. When either of the above ratios equals one (M(F) or M ' (d) equals N), then c has the same behavior as a bosonic operator and our composite particle acts like a boson, at least in this respect. We subtract one from the numerator and denominator to remove the trivial case of N = 1.
2.3.3 Measures of Entanglement
We measure the entanglement of a single-pair state using the purity, P , of the state of either of the two particles in the pair. Thus, given any single-pair annihilation operator d, we can compute an associated purity. It is important to note that d is di.erent depending on which measure of bosonic character we use.
First consider the d used to calculate M(F). Remember that when .nding M(F) we .rst .x |F) and then maximize over d. In this case, P is the purity of the “winning” d; the purity of the single pair state that is maximally occupied by our N-pair state.
If instead we use M ' (d) to measure bosonic character, then P is the purity of the single-pair state we .x. We know P before we maximize over |O).
As discussed in 2.2.3, high purity corresponds to low entanglement and low purity corresponds to high entanglement.
2.3.4 Single-Pair State Matrix
.F|d†d|F\
We now focus on the problem of .nding the d that maximizes the ratio . To
.F|F\
s
address this problem, let us look in more detail at M(F). Recall d = ßm,nambn.
m,n
Then:
(F|d†d|F)
M(F) = max
d (F|F)
††
(F|ß* b.. am. bn. |F)
m,nnamßm,n
= max
|ß\ (F|F)
m,n,m,n
††
(F|bnamam. bn. |F)
= max ß * ßm.,n.
m,n
|ß\ (F|F)
m,n,m,n
We introduce the single-pair state matrix . as
††
(F|bnamam. bn. |F)
4
.mn,m.n. = . (2.1)
(F|F)
We can rewrite the above expression as
ß *
M(F) = max m,nßm.,n. .mn,m..
n
|ß\
m,n,m,n
= max(ß|.|ß),
|ß\
where |ß) is a column vector with components ßm,n, and the maximum is over all vectors of unit length.
Finally, recall the following fact from linear algebra:
If A is a Hermitian matrix and |f) is any vector of appropriate dimen
sion, then max|f\(f|A|f) is equal to the largest eigenvalue of A. The
maximum is achieved by the corresponding eigenvector of A.
The matrix . is Hermitian, therefore the problem of .nding M(F) is equivalent to .nding the largest eigenvalue of the single-pair state matrix .. The corresponding eigenvector, |ß) is the vector representing the maximally occupied single-pair state (the state associated with d).
It turns out, because of the restriction on the form of the N-pair state |F) 5 we can further reduce this problem to .nding the largest eigenvalue of the reduced matrix
††
(F|bmamanbn|F)
.m,n = (2.2)
(F|F)
(for the full details see A.1). Note that in making this change we are resticting our
s
attention to d’s of the form d = ßkakbk. That is, we are only considering d’s with
k
the same Schmidt modes as c.
s
5Remember |F)= c†N |0) where c = ajaj bj .
j
Using the de.nition of . we can derive the explicit form of our single-pair state matrix. Our N-pair state has the form
††
††
|F) =
|0).
aj1 ...ajN b
...b
a
a
j1 j1 jN jN
j1,...,jN
Following the de.nition of the single-pair state matrix we have
s
††
maman
††
††
(0|akN
bkN ...ak1 bk1 b
|0)
aj1 ....ajN ak1 ...akN bnb
...b
a
a
j1 j1 jN jN
j1,...,jN ,k1,...,kN
.m,n = s
.
††
††
(0|asN
bsN ...as1 bs1 b
|0)
ar1 ...arN as1 ...asN
...b
a
a
r1 r1 rN rN
r1,...,rN ,s1,...sN
We can simplify this expression to:
s
2
j1
2
jN-1
(1 - dmj1 )...(1 - dnjN-1 )(1 - dj1j2 )...(1 - djN-2jN-1
Naman a
...a
)
j1,..,jN-1
.m,n =
(2.3)
s
22
(1 - dr1,r2 )...(1 - dr1,rN )...(1 - drN-1,rN
)
a
...a
r1 rN
2
r1,...,rN
(for the details of this derivation see A.2).
Lastly, we use the vector |ß) to .nd the purity of the maximally occupied single-pair state. The density matrix for |ß) (a pure state), is just |ß)(ß|. The eigenvalues
of the reduced density matrix of particle A (or B) are ß
i .
The purity of a state is
the sum of the squares of the eigenvalues of the reduced density matrix, therefore the purity of the maximally occupied single-pair state is
4
i .
P = ß
i
Chapter 3
Entanglement Implies Bosonic Character
We present further evidence in support of Law’s hypothesis. We prove two inequalities:
M(F) - 1
1 - NPc = , (3.1)
N - 1 M ' (d) - 1
1 - NPd = . (3.2)
N - 1
We interpret all inequalities in this thesis as “left implies right” and therefore read both of the above as “entanglement implies bosonic character.” The next two paragraphs discuss the justi.cation for this interpretation.
The .rst section of this chapter proves inequality 3.1. The left side of inequality
3.1 contains the information about the entanglement of the single-pair state used to create our N-pair state |F). For a given N, the more highly entangled our initial state is, the larger the left side becomes. The right side of inequality 3.1 is just one of our measures of bosonic character. Recall M(F) is calculated by considering the occupation number of the maximally occupied single-pair state. The larger the right side the more fermion pairs we can put into the maximally occupied single-pair state. Higher occupation number means the fermion pairs are acting more bosonic. Therefore inequality 3.1 tells us for .xed N greater entanglement of the initial state sets a higher minimum level for the bosonic character of the system.
Inequality 3.2 is di.erent from inequality 3.1 in both its measure of entanglement and its measure of bosonic character. The left side contains the information about the entanglement of a given single-pair state (unlike before, this single-pair state does not create the N-pair state). Again, the more entangled this state, the larger the left side of the inequality. The right side is our other measure of bosonic character. Recall we .nd M ' (d) by .xing a single-pair state and looking at its maximum occupation number over all N-pair states. In this case, the entanglement of the system and the bosonic character of the system are calculated from the same
23
single-pair state. According to inequality 3.2, after .xing N the entanglement of a given single-pair state sets a lower bound on the maximum occupation number of that state. Therefore inequality 3.2 tells us that for any given state high entanglement forces high bosonic character. Inequality 3.2 is an extension of Chudzicki et al’s result to a di.erent measure of bosonic character.
A subtlety about both inequalities is the dependence on the number of pairs in the system. Although N appears explicitly on both sides of inequalities 3.1 and 3.2, the N dependence is actually entirely contained on the left side of both. To understand why, recall that if the fermion pairs were in fact ideal bosons, both M(F) and M ' (d) would be equal to N. By taking the ratios
M(F) - 1 M ' (d) - 1
and
N - 1 N - 1 we calculate a value between zero and one, where zero corresponds to ideal fermion behavior and one corresponds to ideal boson behavior. The explicit N appearance on the right side of the two inequalities compensates for an implicit N-dependence in M(F) and M ' (d). On the left side of the inequalities, Pd and Pc contain all the information about the entanglement of the system. The N appearance in this case tells us that for a given amount of entanglement, we have a higher minimum level of bosonic character when there are less pairs in the system (the left side decreases with N). A simplistic way of interpreting the two inequalities is:
f(entanglement, number of pairs) = g(bosonic character),
where g is a monotonically increasing function and f grows with increasing entanglement and decreases with increasing N.
= M(F)-1
3.1 1 - NPc N-1
In this section we prove our .rst inequality. Our approach is to .rst prove a relationship between entanglement and bosonic character of the initial single-pair state. We then use this relationship to prove inequality 3.1.
Let Pc denote the purity of the single-pair state used to generate the N-pair state |F). That is, Pc is the purity of the state associated with annihilation operator c.
Recall
c = j ajajbj and |F) = c †N |0). (3.3)
From Section 2.3.4 we have
Pc = a4 j . (3.4)
j
We begin by considering the quantity
Mc = (F|c†c|F)(F|F) . (3.5)
Note that Mc is similar to M(F) (refer to the equation page at the front for the de.nition) except rather than considering the bosonic character of the maximally occupied single-pair state (the state associated with d) we consider the bosonic character of the initial single-pair state c. From the de.nitions of c and |F) we have
Mc = (a|.|a)
s
Na2 a2 a2 ...a2 (1 - dmj1 )...(1 - dnj1 )...(1 - dj1j2 )...(1 - djN-2jN-1 )
mnj1...jN-1 m n j1 jN-1
= s a2 r1 ...a2 rN (1 - dr1r2 )...(1 - drN-1rN ) .
r1...rN
(3.6)
= Mc-1
Theorem 3.1.1. 1 - NPcN-1
Proof. We begin by rewriting our inequality. Mc - 1
1 - NPc =
N - 1 (N - 1)(1 - NPc) =Mc - 1 N(1 - NPc + Pc) =Mc N(1 - (N - 1)Pc) =Mc
By substituting the expression for Mc in terms of the a’s and canceling the factor of N on both sides we .nd
s
a2 a2 a2 ...a2 (1 - dmj1 )...(1 - dnj1 )...(1 - dj1,j2 )...(1 - djN-2jN-1 )
mnj1 jN-1
mnj1...jN-1
1 - (N - 1)Pc = s .
a2 ...a2 (1 - dr1r2 )...(1 - drN-1rN )
r1 rN
r...rN
(3.7)
Multiplying by the denominator gives: a2 ...a2 (1 - dr1,r2 )...(1 - drN-1rN )(1 - (N - 1)Pc) (3.8)
r1 rN r1...rN
a2 a2 a2
= mnj1 ...aj2 N-1 (1 - dmj1 )...(1 - dnj1 )...(1 - dj1j2 )...(1 - djN-2jN-1 ) mnj1...jN-1
(3.9)
Now consider the .rst term in 3.8 a2 ...a2 (1 - dr1r2 )...(1 - drN-1rN ), (3.10)
r1 rN r1...rN
the following algebra shows that 3.10 contains 3.9:
(3.10) = a2 j ar2 1 ...a2 rN (1 - dr1r2 )...(1 - drN-1rN ) (3.11) jr1...rN
= aj 2a2 ...a2 (1 - dr1r2 )...(1 - drN-1rN ) (3.12)
r1 rN
jr1...rN
= a2 j ak2a2 ...a2 (1 - dkr1 )...(1 - dkrN )(1 - dr1r2 )...(1 - drN-1rN )
r1 rN-1
jkr1...rN-1
(3.13)
= a2 j ak2a2 ...a2 (1 - djr1 )...(1 - dkr1 )...(1 - dr1r2 )...(1 - drN-1rN )
r1 rN-1
jkr1...rN-1
(3.14)
+(N - 1) a4 j ak2a2 ...a2 (1 - djk)(1 - djr1 )...(1 - djrN-2 )(1 - dr1r2 )...(1 - drN-2rN-1 )
r1 rN-2
jkr1...rN-2
= (3.9) + (N - 1) aj 4a2 ...a2 (1 - djr1 )...(1 - dr1r2 )...(1 - drN-2rN-1 )
r1 rN-1
jr1...rN-1
(3.15)
s
Lines 3.11 and 3.12 follow by multiplying 3.10 by one in the form aj 2 . In 3.13 we
j relabel index rN as k. In 3.14 we split the sum into components of j = ri and j = ri for some i .{1, ..., N - 1} and relabel indices so that i = N - 1. In 3.15 we relabel index k as N - 1 and recognize the .rst term as 3.9. Substituting line 3.15 for the .rst term of 3.8 and canceling the term 3.9 on each side we can write our inequality as:
0 = (N - 1) aj 4a2 ...a2 (1 - djr1 )...(1 - dr1r2 )...(1 - drN-2rN-1 ) (3.16)
r1 rN-1
jr1...rN-1
+(N - 1) a2 ...a2 (1 - dr1r2 )...(1 - drN-1rN )Pc.
r1 rN
r1...rN
s
Plug in a4 for Pc and divide both sides by (N - 1) to .nd:
k
k
a4 j a2 ...a2 (1 - djr1 )...(1 - dr1r2 )...(1 - drN-2rN-1 )+ (3.17)
r1 rN-1
jr1...rN-1
a4 ka2 ...a2 (1 - dr1r2 )...(1 - drN-1rN ) = 0.
r1 rN
kr1...rN
Let A = a2 ...a2 (1 - dr1r2 )...(1 - drN-1rN-2 ). We rewrite the left side of 3.17:
r1 rN-1
..
A . aj 4(1 - djr1 )...(1 - djrN-1 ) - ak4a2 (1 - dr1rN )...(1 - drN-1rN ). .
rN r1...rN-1 j krN
(3.18)
s
Now multiply by 1 = a2 and relabel rN as j to get
k
k
..
A . a4 j a2 k(1 - djr1 )...(1 - djrN-1 ) - a4 kaj 2(1 - djr1 )...(1 - djrN-1 ). r1...rN-1 jk jk
(3.19)
= Aaj 2ak2(1 - djr1 )...(1 - djrN-1 )(aj 2 - a2 k) (3.20) r1...rN-1 jk
Until this point we have simply been rewriting our inequality. We have shown that
Mc - 1 1 - NPc =
N - 1
is equivalent to
Aaj 2ak2(1 - djr1 )...(1 - djrN-1 )(aj 2 - ak2) = 0. (3.21) r1...rN-1 jk
We now prove inequality 3.21. The index k ranges from 1 through d, whereas j cannot be equal to r1, ..., rN-1. Therefore the term
aj 2ak2(1 - djr1 )...(1 - djrN-1 )(a2 j - a2 k) jk
is negative. It immediately follows that inequality 3.20 is true, which completes the proof.
Now we prove entanglement of the single-pair state associated with c is a lower bound to the bosonic character as measured by M(F).
= M(F)-1
Theorem 3.1.2. 1 - NPcN-1
= Mc-1
Proof. From theorem 3.1 we know 1 - NPcN-1 . By de.nition
†
(F|cc|F)(F|d†d|F)
Mc == max = M(F).
(F|F) d (F|F)
Therefore M(F) - 1
1 - NPc = . (3.22)
N - 1
3.2 1 - NPd = M.(d)-1
N-1
In this section we prove inequality 3.2. That is, we show that for a given single-pair state and .xed N, the entanglement of the state sets a lower bound on the bosonic character as measured by M ' (d). Note that Pd is the purity of the single-pair state we .x.
The following proof relies heavily on the notation and results in Law’s and Chudzicki et al’s papers [5], [1].
M.(d)-1
Theorem 3.2.1. 1 - NPd = N-1
Proof. In his paper, Law de.nes the normalized N pair state - 1 1
|N)= . 2 v d†N |0).
N
N! From the de.nition it follows that (O|d†d|O)(N|d†d|N)
M ' (d) = max = = (N|d†d|N). (3.23)
|O\ (O|O)(N|N) From Law’s paper we also have:
.N+1
(N|[d, d†]|N) = (N|dd†|N)-(N|d†d|N) =2 - 1. (3.24)
.N Consider the term (N|dd†|N). By the de.nition of |N): 1
(N|dd†|N) = (0|dN+1d†(N+1)|0). (3.25)
N!.N Furthermore,
(0|dN+1d†(N+1)|0) =(N + 1)!.N+1(N +1|N +1) =(N + 1)!.N+1. (3.26)
Therefore (N + 1)!.N+1 .N+1
(N|dd†|N) = =(N + 1) . (3.27)
N!.N .N Substituting (N + 1).N+1 for (N|dd†|N) we have:
.N .N+1
(N|dd†|N)-(N|d†d|N) =2 - 1 (3.28)
.N .N+1 .N+1
(N + 1) -(N|d†d|N) =2 - 1 (3.29)
.N .N .N+1
(N - 1) = (N|d†d|N)- 1 (3.30)
.N
.N+1 (N|d†d|N)- 1
= . (3.31)
.N N - 1
Combining the last expression with our earlier result (N|d†d|N)= M ' (d), we have
.N+1 M ' (d) - 1
= . (3.32)
.N N - 1
Finally, we use the result of Chudzicki et al:
.N+1
1 - NPd = , (3.33)
.N
from which we conclude M ' (d) - 1
1 - NPd = . (3.34)
N - 1
3.3 Remarks
When we de.ne our two measures of bosonic character
M(F) - 1 M ' (d) - 1
and
N - 1 N - 1
we examine an important bosonic property: the ability to condense identical particles into the same state. Essentially, the measure used in the proof of theorem
M(F)-1
3.1.2, N-1 , looks at the occupation number of the maximally occupied state for a given N-pair state. The measure of bosonic character in the proof of theorem
M.(d)-1
3.2.1, N-1 looks at how occupied a given pair state can be.
For both measures of bosonic character we .nd a relationship to the purity of a single-pair state and the number of fermion pairs in the system. Note that for both inequalities 3.1 and 3.2 we can raise the minimum amount of bosonic character by decreasing the number of pairs in our system (N). That is, highly entangled fermion pairs (small P ) must act bosonic when we only have a few such pairs in our system. To ensure an equally bosonic system with more fermion pairs we must use more entangled single-pair states. This trend makes sense when you think about the limiting case of N = 1. In this case both an ideal boson and a fermion pair will have occupation number one. Physically, we see this result for atomic BECs: atoms are only able to condense in the low-density limit. A 2002 paper by Rombouts et al .nds the maximum occupation number for di.erent systems of composite bosons, including excitons and trapped atoms [9].
In the following chapters we will show the amazing fact that, while 1 - NPc is a lower bound for bosonic character when using M(F), the similar quantity 1 - NP appears to be an upper bound. That is, we believe:
M(F) - 1
1 - NPc == 1 - NP.
N - 1
Chapter 4
Bosonic Character Implies Entanglement
The previous chapter proved two inequalities supporting Law’s hypothesis. In this chapter we are interested in the converse statement “bosonic character implies entanglement.” In particular we consider the inequality
M(F) - 1 N
= 1 - NP for M(F) = . (4.1)
N - 12
As mentioned in the previous chapter, we read such an inequality as “left implies right.” On the left side of the inequality 4.1 we have the bosonic character of a system of N pairs of distinguishable fermions. Recall that M(F) measures bosonic character by looking at the occupation number of the maximally occupied single-pair state. The right side depends on both the purity and the number of pairs in the system. Fixing N and increasing entanglement (decreasing purity) makes the right side larger. Thus if inequality 4.1 is true, for a given N bosonic character sets a lower bound on the entanglement of the system or “bosonic character implies entanglement.” Proving inequality 4.1 would be evidence in support of a strengthened version of Law’s hypothesis: “a pair of fermions acts like a boson if and only if the pair is highly entangled.” Unlike inequalities 3.1 and 3.2 from the previous chapter, we do not expect inequality 4.1 to hold in general. We have added the restriction
N
that M(F) = because of our numerical results. We are most interested in the
2
region of high entanglement and high bosonic character, which is part of the region for which we expect inequality 4.1 to be satis.ed. As of yet we do not have a physical explanation for why the inequality needs this restriction.
As with inequalities 3.1 and 3.2 the N-dependence is entirely contained on the same side as the entanglement. For a given amount of bosonic character (a .xed value for the left side) a system containing more pairs must have a more highly-entangled maximally occupied single-pair state (larger N must be balanced by smaller P ).
Numerical results (discussed in the next chapter) suggest inequality 4.1 is satis.ed for all N provided the system is in a state such that M(F) = N 2 . We expect
31
inequality 4.1 to hold true for all possible M(F) when our system contains just two pairs. To see why, note that it is always possible to put one fermion pair in our initial single-pair state and therefore M(F) = 1. The two pair system trivially satis.es the condition M(F) = N 2 . We prove this result in the .rst section.
In the second section we return to the case of arbitrary N and consider initial single-pair states of a special form. We can describe the initial single-pair state used to create the N-pair state |F) by its Schmidt coe.cients. Recall that we create |F)by acting the operator c† on the vacuum N times. We de.ne the operator c in the Schmidt basis:
c = ajajbj j
The initial single-pair state associated with c can be described in terms of the Schmidt coe.cients. That is, we .x c by .xing the vector |a) =(a1,a2, ..., ad). In the latter half of this chapter we consider initial single-pair states of the special form |a) =(a, a, ..., a, e) (our interest in these particular states derives from numerical results discussed in the next chapter). We prove the following:
M(F) - 1 =1 - NP in the limit e . 0 (4.2)
N - 1
M(F) - 1 N - 2 N - 2
= (1 - NP ) + in the limit a . 0. (4.3)
N - 1 NN(N - 1)
4.1 System of Two Pairs
We prove
M(F) - 1
= 1 - 2P
2 - 1 (inequalty 4.1 for N = 2). That is, in a system containing two fermion pairs bosonic character sets a lower bound on the entanglement of the system.
Theorem 4.1.1. For the case of two pairs, M(F) - 1 = 1 - 2P
Proof. We .rst rewrite our inequality as
M(F) = 2(1 - P ).
We know that M(F) is the largest eigenvalue of our single-pair state matrix .. If |ß) is an eigenvector of . corresponding to M(F), then (ß|.|ß) = M(F).
For N =2
.mn = s 2 aman ak2(1 - dmk)(1 - dnk). (4.4)
a2 ra2 s(1 - drs) k
rs
De.ne the matrix A:
Aij = aiaj(1 - dij). (4.5)
It follows
(A2)ik = AijAjk (4.6) j
= aiaj(1 - dij)ajak(1 - djk) (4.7) j
= aiak aj 2(1 - dij)(1 - djk), (4.8) j
from which . and M(F) can be rewritten as follows: 2
(A2)mn
.mn
(4.9)
s
=
a2a2(1 - drs)
rs
rs
2
(ß|A2|ß).
M(F) = (ß|.|ß) =
(4.10)
s
a2 ra2 s(1 - drs)
rs
A is a Hermitian matrix. If .i is the ith eigenvalue of ., then in some basis
.
.
.
.
.2
1
.1 0 ... 0
0 ... 0
...
...
and A2
=
...
...
.2
2
...
0
0 .2 ... 0
0
A =
.
... ... ... ...
... ... ... ...
00 ... .d 00 ... .d 2 We assume without loss of generality that .21 is the largest eigenvalue of A2 . Then
.
.
1
0
|x) =
...
...
...
0
is clearly an eigenvector of A2 corresponding to the maximum eigenvalue. Note that |x) is also an eigenvector of A. Let |ß) denote the eigenvector |x) in our original basis. As |ß) is an eigenvector of both A and A2 we can rewrite equation 4.10 as
v
2
s
(ß|A|ß) = M(F). (4.11)
a2a2 )
rs(1 - drs
rs
Notice that we can rewrite the denominator of equation 4.11
v
ai 2aj 2(1 - dij)= (A2)ii = Tr A2 . (4.12) ij i
Furthermore, we can rewrite part of the numerator by de.ning the matrix Bij = ßißj(1 - dij):
(ß|A|ß) = aiajßißj(1 - dij)= (AB)ii = Tr AB. (4.13) ij i
Substituting equations 4.12 and 4.13 into equation 4.11 we have
v
2
M(F) = v Tr AB. (4.14) Tr A2
Recall from section 2.3.4 that the purity associated with the state |ß) is Pß =
ss
ßj 4 . Normalization requires ß2 = 1. Using these two conditions we rewrite the
j
jj
quantity 1 - Pß as
1 - Pß =1 - ß4
j j
=( ßj 2)2 - ß4
j jj
= ßi 2ßj 2(1 - dij) ij
= ßißj(1 - dij)ßjßi(1 - dji) ij
= Tr B2 .
As A and B are both symmetric real matrices it follows that (Tr AB)2 = Tr A2 Tr B2 . Therefore 2 (Tr AB)2 = 2 Tr B2 ,
Tr A2 which implies M(F) = 2(1 - Pß).
Finally, in section 2.3.3 we de.ned the purity P of our system to be the purity of the maximally occupied single-pair state. The maximally occupied single-pair state, |.) is the eigenvector of . which corresponds to the largest eigenvalue. In the case that multiple such eigenvectors exist we choose the one with minimum purity. When M(F) is non-degenerate |ß) = |.). When M(F) is degenerate, Pß = P and 2(1 - Pß) = 2(1 - P ). Therefore
M(F) = 2(1 - P ),
which says that inequality 4.1 is satis.ed for N = 2.
All of the sums in the above proof converge in the limit that d goes to in.nity.
ss
To see this, note that a2 = 1 and a4 = 1 regardless of whether the sum is
jj jj
.nite or in.nite. The same is true of the sums involving the ß’s. Whenever the sum involves delta functions, we are simply subtracting positive terms from one of the above quantities that are always less than or equal to one. Furthermore, each sum is bounded from below by zero (the sums are always positive). Our sums are bounded from above and below and thus must converge. Therefore, theorem 4.1.1 holds in in.nite dimension.
4.2 Special Initial Single-Pair States
We create our N-pair state |F) by making N copies of an initial single-pair state:
|F)= c †N |0)
† = ††
cakabk.
kk
The vector of coe.cients |a) =(a1,a2, ..., ad) (where d is the dimension or number of orthogonal states for each fermion) describes the initial single-pair state. In the
M(F)-1
next chapter we plot N-1 on the y-axis and 1 - NP on the x-axis for randomly chosen |a)’s. We see that |a) states of the particular form |a) =(a, a, ..., a, e) appear to form an upper boundary on such plots. These special states correspond to the “J-shapes” discussed in the next chapter. A better understanding of these boundary states could lead to a proof of inequality 4.1.
Of the |a) states of this special form, there are two cases of particular interest:
|a) =(a, a, ..., a, 0) and |a) =(e, e, ..., e, a),e . 0.
Again, the justi.cation for our interest in these two cases must wait until the next chapter when we can relate them to our numerical plots. Our purpose now is to
M(F)-1
show that in each case for .xed N and varying dimension d the quantities N-1 and 1 - NP satisfy a linear relationship.
The density matrix for the state |a) =(a, a, ..., a, 0) is:
.
d-1-Nd-1-N
1 ... 0
d-2 d-2
d-1-Nd-1-N
1 ... 0
d-2 d-2
d-1-Nd-1-N
d-2 d-2
1 ... 0
.
.....
.....
. =
N
d - 1
,
(4.15)
... 0 ... 0 ... 0 ... ... ... 0
which has largest eigenvalue M(F) = N(d - N) d - 1 . (4.16)
The density matrix for the state |a) = (e, e, ..., a) is:
.
.
. =
1
d - 1
.......
(N-1)(d-N)(N-1)(d-N)
(N - 1) ... (d - N) e
(d-2) (d-2) a (N-1)(d-N)(N-1)(d-N)
(N - 1) ... (d - N) e
(d-2) (d-2) a
(N-1)(d-N)(N-1)(d-N)
(d-2) (d-2)
(N - 1) ... (d - N) e
a
... ... ... ... ... (d - N) e (d - N) e (d - N) e ... 1
aaa
.......
,
(4.17)
with largest eigenvalue
(N - 1)(d - N + 1)
M(F) = . (4.18)
d - 1
Both density matrices are easily derived from the de.ntion of the single-pair state matrix. For the details of these derivations see Appendix B.
v 1
The corresponding eigenvector (to zeroeth order in e) for both matrices is (1, 1, 1, ..., 0),
d-1
which has purity 1
P = . (4.19)
d - 1 Simple algebra shows that for a given N and d, when |a) =(e, e, ..., a),e . 0,
M(F) - 1 N - 2 N - 2
= (1 - NP ) + (4.20)
N - 1 NN(N - 1)
M(F)-1 N-2 N-2
(i.e. the point on the plot of 1-NP vs. lies on the line y = x+ ).
N-1 NN(N-1)
When |a) =(a, a, ..., 0),
M(F) - 1 = (1 - NP ) (4.21)
N - 1
M(F)-1
(i.e. the point on the plot of 1 - NP vs. N-1 lies on the line y = x).
From numerical results, we believe inequality 4.1 begins taking e.ect at the intersection of these two lines (equivalently when M(F) = N ).
2
Chapter 5
Numerical Work
In this chapter we discuss numerical results in suport of the claim “bosonic character implies entanglement.” Our main focus is on inequality 4.1 discussed in the previous chapter
M(F) - 1 N
= 1 - NP for M(F) = .
N - 12 The .rst section focuses on the inequality 4.1. We use random initial states |a)to generate values for M(F) and P , then plot the left side of inequality 4.1 on the y-axis and the right side on the x-axis. We .nd boundary lines for these plots and justify the claim in 4.2 that special forms of |a) form a boundary for our plots. In the second section we compare the purity of our initial single-pair state, Pc, to the purity of the maximally occupied single-pair state, P . It appears to be the case that Pc = P : the maximally occupied single-pair state is more entangled than the intial single-pair state.
M(F)-1
5.1 = 1 - NP
N-1
The correlation between the system’s bosonic character as measured by M(F) and the system’s entanglement as measured by the purity is divided into three regions (see .gure 5.1). In each region, entanglement bounds bosonic character from above by a distinct inequality. In this section we .nd these linear bounds and look more closely at the structure of the inequalities.
5.1.1 Broken-Line Bound
M(F)-1
Figure 5.1 plots bosonic character (measured by N-1 ) against entanglement (measured by 1 - NP ) for N =2, 3, 4, and 5. For all choices of N, entanglement is an upper bound on the bosonic character. Every point on the graph corresponds to a di.erent intial choice for the single-pair state associated with annihilation operator c (the operator used to generate our N-pair state |F)). We are interested in an inequality relating M(F) and P , therefore we are interested in the boundary of the
37
(a) N = 2 (b) N =3
(c) N = 4 (d) N =5
Figure 5.1: Broken line plots for N =2, 3, 4, and 5. Each plot has three regions:
111 1
= P , = P = , and P =
N 2N-2 N 2N-2
plot rather than the distribution of points. For this reason we do not worry about creating a uniform distribution of intial states.1
The plots show there are three regions, each with a di.erent linear upper bound. The three regions are divided by the the purity of the maximally occupied single-pair states:
1
Region One: = P (5.1)
N
11
Region Two: = P = (5.2)
2(N - 1) N
1
Region Three: P = (5.3)
2N - 2
s
1We choose an initial state in the following way: recall c = j aj aj bj. Using Mathematica’s built-in RandomReal program we assign values to all aj and then normalize them
s
such that a2 = 1. Fixing the aj determines ., M(F), and P .
j
j
For simplicity of notation let
M(F) - 1
x =1 - NP and y = .
N - 1
In region one we see
2 N - 2
y = x + (5.4)
N(N - 1) N(N - 1)
and in region two
N - 2 N - 2
y = x + . (5.5)
NN(N - 1)
Our primary focus is on the limit of high entanglement and high bosonic character, thus we are most interested in region three. States of the system in region three appear to satisfy inequality 4.1, equivalently
y = x. (5.6)
For the case of N = 2 shown in Figure 5.1a the boundary lines for regions one, two and three are all the same. When the system contains two pairs of distinguishable fermions we have
M(F) - 1 = 1 - 2P for all choices of |a). (5.7)
N - 1
We proved this result in the Section 4.1.
For N = 3 (shown in Figure 5.1b) the boundary lines for the .rst and second regions are the same. For N = 4 regions one, two, and three all have distinct bounds.
5.1.2 Dimension of Initial Single-Pair State
To generate .gures 5.1 we .rst create a plot for .xed number of fermion pairs N and .xed dimension d (number of orthogonal states available to each fermion). For any
d
d, P = 1 plots with the same N and di.erent d’s. For example, Figure 5.2a plots points for N =3,d = 4, Figure 5.2b plots points for N =3,d = 5, and Figure 5.2c overlays Figures 5.2a and 5.2b, thereby giving us a better understanding of the boundary for N = 3.
N
The largest possible x-coordinate for a given d is x =1 -
We overlay
.
.
d
(a) N =3,d = 4 (b) N =3,d =5
(c) N =3,d =4 or 5
Figure 5.2: Figure 5.2a shows the plot for three pairs with four modes available to each pair, while .gure 5.2b shows the plot for three pairs with .ve modes available to each pair. In .gure 5.2c we overlay .gures 5.2a and 5.2b to get a better sense of the boundary. By overlaying plots for more and more choices of d we reach a better understanding of the boundary of the plot for three pairs with an arbitrary number of modes available to each pair.
5.1.3 Boundary States
We can better understand the boundary of the plot for any given N and d by considering the special case when all but one of the states available to an A, B pair are equally probable. This physical restriction is mathematically equivalent to setting a1 = a2 = ... = ad-1. Let |a) =(a1,a2, ..., ad) denote the vector of all ai’s. The boundary case here considers |a)’s of the form |a) =(a, a, ..., a, e). Figure 5.3a shows this boundary case for N =4,d = 5. Figure 5.3b overlays these boundary cases for N = 4 and d ranging from 5 to 10. We refer to these plots as “J-shapes” because they roughly resemble a J.
The switch from region two to region three occurs when M(F) = N , equivalently
2
when d =2N - 2. Let e