Author: Seán Gray
Supervisor: Joseph A. Minahan Subject Reader: Ulf Lindström
Department of Physics and Astronomy Division of Theoretical Physics
Uppsala University
June 24, 2014
Abstract
In this paper the features of quantum systems which lay the foundation of quantum entanglement are studied. General properties of entangled states are discussed, in- cluding their entropy and relation to Bell’s inequality. Applications of entanglement, namely quantum teleportation and quantum cryptography, are also considered.
J o e S u s a n n e Family &
F r i e nds
1 Introduction 7
1.1 Background . . . 7
2 Objective 8 2.1 Method . . . 8
3 Quantum Systems 9 3.1 Pure States . . . 9
3.2 Mixed States . . . 10
3.2.1 Density Matrix . . . 10
4 Entropy 12 4.1 Shannon Entropy . . . 12
4.2 von Neumann Entropy . . . 13
5 Quantum Entanglement 15 5.1 Definition . . . 15
5.1.1 Pure State Entanglement . . . 15
5.1.2 Mixed State Entanglement . . . 16
5.2 The Bell States . . . 16
5.3 Entropy of Entangled States . . . 18
5.3.1 Density Matrix of Entangled Subsystems . . . 18
5.3.2 Entropy of Entangled Subsystems . . . 19
5.3.3 Entropic Inequalities . . . 19
5.4 Monogamy of Entanglement . . . 20
6 Bell’s Inequality 21 6.1 Violation of Bell’s Inequality by Entangled State . . . 22
6.2 CHSH Inequality . . . 24
7 Quantum Teleportation 24 7.1 No Cloning Theorem . . . 25
7.2 Quantum Teleportation Protocol . . . 25
7.3 Generalisation . . . 27
8 Quantum Cryptography 28 8.1 Asymmetrical and Symmetrical Cryptosystems . . . 28
8.1.1 Asymmetrical Cryptosystems . . . 29
8.2.1 BB84 Protocol . . . 30
8.2.2 E91 Protocol . . . 31
8.2.3 Teleporting Keys . . . 33
9 Discussion 33 9.1 Conslusion . . . 34
Bibliography 35 A Appendix 36 A.1 Derivations and Proofs . . . 36
A.1.1 Completeness Releation . . . 36
A.1.2 Trace of Density Matrix . . . 36
A.1.3 Probability for Projection Operator . . . 37
A.1.4 Unitary Transformation . . . 37
A.1.5 Correlation Coefficient for Spin Singlet . . . 37
1 Introduction
One may likely assume, given the nature of human interpersonal relations, that sensitive information has been shared for as long as there has been a language capable of faithfully communicating a message.1 In order to exclude any unwanted third party from taking part of the shared secrets, many techniques have been developed in order to scramble messages in such a way that only the concerned peers can understand their contents. Such scrambling is known as encryption, which is a part of the subject of cryptography.
In our contemporary, technology dependent society, vast amounts of delicate information, both private and governmental, is transferred using encrypted channels. The security of such channels are to a large extent based upon computational complexity [1], meaning that unscrambling a message by brute force would be extremely time consuming.
With the advent of quantum computing, the world of cryptography is facing a paradigm shift. Once quantum computation is used on a tangible scale, the complexity of the present cryptosystems will be impaired [2]. To this end, cryptography must be taken into the realm of quantum mechanics.
Quantum cryptography incorporates many aspects of quantum mechanics to obtain, in theory, completely secure communications. One especially noteworthy possibility is quantum teleportation, which enables secure transmission of quantum states to between two specially separated parties without having to disclose any locations.
This paper will introduce the quantum mechanical concepts that are of most importance in quantum cryptography, with emphasis on quantum entanglement and its implications, including quantum information theory. Quantum teleportation will also be presented and studied. The final part of this paper is dedicated to presenting the elegant and beautiful protocols used in quantum cryptography, using the principles presented throughout the paper.
1.1 Background
Quantum entanglement is a property of compound quantum systems which gives rise to correlations between subsystems which enable completely accurate predictions of measure- ments that ordinarily would be of a statistical nature, even if the subsystems are separated by an arbitrary distance. Such “actions at a distance” were first recognised by Schrödinger [3], and Einstein, Podolsky and Rosen2, who in their 1935 paper refuted such extraordinary correlations [4]. More specifically, EPR argued that the quantum mechanical description of reality was incomplete, referring to the existence of local hidden variables. In essence, this
1Indeed, informational fidelity may for some not always be of highest priority.
2Einstein, Podolsky and Rosen are often referred to as EPR.
meant that all quantum mechanical systems would be governed by an underlying classical, deterministic framework.
Unknown parameters gave rise to alternate models, called local hidden variable models.
However, such models did not provide any unique experimentally verifiable predictions, which threatened to place the discussion in the domain of metaphysics. Thirty years after the original papers, John S. Bell presented a theorem which must be obeyed by any classical system, whether it is a local hidden variable model or not [5]. It was simultaneously shown that the correlations between entangled states violate this theorem, meaning no classical model would be sufficient at explaining the quantum phenomenon.
Bell’s theorem revived the study of quantum entanglement, but it is only recently that utilisations have been considered. The two applications that are treated in this paper are quantum teleportation [6], and quantum cryptography [7, 8]. Both of these make use of relatively simple properties of quantum mechanics, together with entanglement, and are striking in their refinement. Perhaps the knowledge of the quantum world needed to mature before it could be used to its full extent.
2 Objective
This work strives to understand the possibilities and limitations of quantum entanglement and, by extension, quantum information theory; along with their function in quantum teleportation and quantum cryptography.
2.1 Method
This paper consists of a literature study which aims to review and study relevant issues concerning the topic of quantum entanglement, quantum information theory, and their specified implementations. The source material has primarily been chosen by relevance and importance to the field, and has been used as guidelines for the work presented.
The paper begins by considering the different representations of quantum systems which are of importance in this context, based upon [9]. Pure and mixed states are discussed, and properties of density matrices are studied.
Section 4 makes use of [10] to study classical Shannon entropy, and quantum von Neumann entropy and their interpretations in information theory. The general expression for the von Neumann entropy is simplified and analysed.
Section 5 discusses quantum entanglement based upon [10] and [11]. The general representations of entangled systems are considered, followed by a more rigorous analysis of bipartite entanglement, focusing on the Bell states. Entropy of entangled states is discussed, followed by the monogamy of entanglement.
Bell’s inequality is presented and proven, following [12], in section 6. It is shown to be violated by an entangled state, and the implications of which are discussed.
In section 7 the protocol for quantum teleportation [6] is analysed, taking classical and quantum information theory into consideration. The protocol is also generalised for arbitrary quantum states.
Finally, in section 8, some important protocols for quantum cryptography are presented and studied, making use of the ideas and concepts presented previously; and the security of said protocols is scrutinised from an eavesdroppers perspective.
3 Quantum Systems
Quantum mechanical systems can behave in many different ways. A system’s behaviour is heavily dependent on its composition; if it is a single known state or a mixture of many distinguishable states. In this section, the mathematical tools used to describe such quantum ensembles will be presented.
3.1 Pure States
A pure state is a quantum state which can be described by a single state vector. For example, in the case of a spin one-half state, a pure state is a particle with a definite spin in either one of the two spin directions, described by the state vectors |↑i and |↓i. A general form of a spin one-half state
|ψi = α |↑i + β |↓i , (3.1)
where |α|2+|β|2 = 1, is also a pure state, since it is a superposition of pure states of the same Hilbert space with known relative phases. A system of pure states is an ensemble of quantum states which are all described by the same state vector, i.e. a system of particles each in the same quantum state.
An example of a pure state is an electron which has passed through a Stern-Gerlach apparatus, a device which sorts atoms depending on their spin by applying a strong magnetic field to an incoming beam of unpolarised states.3 The outgoing beam will then be a pure ensemble of states, since all electrons will have the same spin.
The expectation value for an observable A for any pure state is given by the familiar formula
hAi = hφ | A | φi , (3.2)
where |φi is a pure state.
3An unpolarised beam is in this example a beam containing a mixture of all possible spin states with different probability weights.
3.2 Mixed States
Continuing the example of the Stern-Gerlach apparatus, one may want to describe the unpolarised beam of incoming electrons. Such a system, containing a statistical mixture of subsystems, is known as a mixed state. For the task of describing the total system, the expression (3.1) will by its own not suffice, since it clearly cannot describe a statistical ensemble of spin orientations due to the fact that each state has a definite direction. For this reason, we introduce the more general notion of a density matrix.
3.2.1 Density Matrix
In classical physics, one may for several reasons want to consider the probability density of a system. For example, in classical mechanics, the phase space for a system is a space in which each point corresponds to a position and momenta which the system can obtain.
The probability density for position and momenta is thus called the phase space probability measure.
The quantum mechanical equivalent of knowing the probability density for a system is knowing the density matrix, ρ. As the name suggests, the density matrix is a matrix which contains all the information regarding the probabilities of encountering a subsystem of a mixed state. The density matrix is defined as
ρ =X
i
pi|α(i)i hα(i)| , (3.3)
where |α(i)i are the state vectors for each subsystem, and piare the normalised probabilistic weights for each subsystem, such that
X
i
pi = 1 . (3.4)
From the definition, it is clear that the density matrix is Hermitian.
A mixed state in which the probabilities for each outcome are the same is called maximally mixed, since there is no statistically preferred subsystem; while a mixed state with different statistical weights is called partially mixed, since some state will appear more frequently than others. A pure state is the special case where there is only one non-zero probability weight, hence yielding the density matrix
ρ = |αi hα| , (3.5)
which is known as the projection operator. It is important to note that the states in a density matrix need not be orthogonal.
Example 3.1. The density matrix for the pure spin up state |↑i =
1 0
is given by ρ = |↑i h↑| =
1 0
(1, 0) =
1 0 0 0
.
Example 3.2. The density matrix for the maximally mixed state of spin one-half particles with each state having probability weight 1/2 is given by
ρ = 1
2 |↑i h↑| + |↓i h↓|= 1 2
1 0 0 1
.
From the above examples it is possible to see an important property of the density matrix. Namely, that a density matrix for a mixed state has more than one non-zero eigenvalue λi ∈ R; while a density matrix for a pure state only has a single eigenvalue.
Somewhat more obviously, one may also note that the trace of the above matrices sum to unity4, meaning that the probability is normalised. These two properties are in line with classical probability theory, which is a clear advantage when trying to make sense of these issues.
Let us now find an expression for the expectation value of an observable A in a mixed state. We begin by considering an ensemble of weighted states pi|α(i)i, which yields the expectation value
hAi =X
i
pihα(i)|A|α(i)i , (3.6)
which is the usual expectation value for each subsystem weighted by the probability pi, summed over all states. Using the completeness relation,
X
b0
|b0i hb0| = 1 (3.7)
we can write the above expression in a general basis as follows hAi =X
i
pi
X
b0
X
b00
hα(i)|b0i hb0| A |b00i hb00|α(i)i (3.8)
=X
b0
X
b00
X
i
pihb00|α(i)i hα(i)|b0i
hb0| A |b00i . (3.9) Remembering the expression for the density matrix, we obtain
hAi =X
b0
X
b00
hb00|ρ|b0i hb0| A |b00i . (3.10)
4This is proven in the appendix A.1.2
Using the completeness relation, together with Pb00hb00|A|b00i = Tr(A), the final expression for the expectation value of an observable A will be given by
hAi = Tr(ρA) . (3.11)
Example 3.3. The expectation value of the spin for the maximally mixed state considered in the previous example will be given by
hσii = Tr(ρσi) , where σi are the three Pauli matrices
σ1=
0 1 1 0
, σ2= i
0 −1
1 0
, σ3 =
1 0
0 −1
.
Since the density matrix is proportional to the identity matrix, and the Pauli matrices have trace zero, the expectation value will be
hσii = 0 ,
meaning the two possible outcomes, spin up or down, are equally likely, which is consistent with the amount of mixing.
4 Entropy
Entropy is a fundamental quantity in thermodynamics and statistical physics. The physical interpretation of entropy is the amount of chaos within a system; meaning the number of microstates which have the same macroscopic properties. It increases with the number of microstates in a system, and the smaller the statistical weight is for each microstate.
However, entropy also has an information theoretical interpretation, which will be discussed in this section. This section will begin with the classical formalism, later presenting the quantum analogue.
4.1 Shannon Entropy
The Shannon entropy5 quantifies the uncertainty of an outcome for a discreet variable X associated with the sample space Ω belonging to the classical probability distribution P . It may also be regarded as the amount of information6 needed to convey the outcome that
5The Shannon entropy is named after its discoverer and the father of information theory, Claude Shannon.
6Information may seem abstract and difficult to quantify. The unit used for information depends in which base the measurement of X is made. For the purpose of this paper base two is most convenient, yielding the unit bit.
is measured, or the information gained by measuring the variable. The Shannon entropy is given by
H(X) = −X
i
pilogb(pi) , (4.1)
where pi are the normalised probabilities for each outcome of X, and b the base in which the measurement of the variable is made.
From the expression for the entropy, one can draw several conclusions. Firstly, the entropy of an outcome decreases the more likely the outcome is. Meaning, the greater the probability for an outcome to occur, the less information it provides about the distribution.
Secondly, the entropy is maximum, and thus the ignorance about the system maximum, when X is uniformly disitributed, i.e. pi = 1/N . It then obtains the value
H(X) = −
N
X
i=1
1
N logb(1/N ) (4.2)
= logb(N ) . (4.3)
Further, the entropy is at its minimum when the variable only has a single outcome, in which case the entropy is zero7 and one has complete knowledge of the system.
Example 4.1. Consider flipping an unbiased coin such that Ω = {H, T } and pH = pT = 1/2. The amount of information, in base two, needed to specify heads (H) or tails (T ) will then be given by the Shannon entropy
H(X) = log2(2) = 1 bit
Example 4.2. Now consider the weighted coin such that Ω = {H, T } with the probabilities pH = 2/3 and pT = 1/3. The entropy will then be
H(X) = −2/3 log2(2/3) − 1/3 log2(1/3) = 0.91 bits
This result is less than for the unbiased coin. Reasonably, the amount of information contained in a message carrying the result is less for a biased coin, since one of the results is more expected.
4.2 von Neumann Entropy
The quantum mechanical equivalent to the Shannon entropy is the von Neumann entropy.8 Analogously to its classical counterpart, the von Neumann entropy is a measure of the
7Here we have used that 0 log 0 = 0
8Named after John von Neumann. The unit of the von Neumann entropy is quantum bit, shortened qubit. From here on, the use of entropy will refer to the von Neumann entropy in the quantum case, and Shannon entropy in the classical case.
ignorance concerning a quantum state. Instead of using a probability distribution, the entropy of a quantum system makes use of the density matrix, which behaves similarly.
The von Neumann entropy is given by
S(ρ) = −Tr(ρ ln ρ) . (4.4)
Since the density matrix is Hermitian, it will always be diagonalisable. In which case, the von Neumann entropy for a diagonal matrix can be simplified. We write the diagonal matrix as
ρ0 = P−1ρP , (4.5)
where P is the diagonal matrix of eigenvectors for ρ. The logarithm for the diagonalisable density matrix will be given by
ln(ρ) = P−1ln(ρ0)P , (4.6)
where the logarithm acts on each eigenvalue λi of the diagonal matrix. Thus, the expression for the von Neumann entropy will be
S(ρ) = −Tr(ρ0P−1ln(ρ0)P ) = −Tr(ρ0ln(ρ0)) , (4.7) where we have used that the trace operation is invariant under matrix conjugation, together with P−1P = I. Since the trace will be the sum over the eigenvalues, the final expression for the von Neuman entropy will be
S(ρ) = −X
i
λiln(λi) , (4.8)
which is reminiscent of the Shannon entropy. From the above expression, we can draw the conclusion that the eigenvalues for the density matrix can be thought of as probability weights for each subsystem.
Entropy has the quantum interpretation as the amount of mixing in a quantum ensemble.
The greater the mixture, the less one knows about the outcome of any measurement.
Therefore, zero entropy corresponds to a pure state, where one has maximum knowledge.
In the same way as for the Shannon entropy, the maximum value for the quantum entropy is obtained in the maximally mixed state, given by
S(ρ) = ln(N ) . (4.9)
Example 4.3. The von Neumann entropy for the pure spin state used in example 3.1 is S(ρ) = − ln(1) = 0 qubits
Example 4.4. The entropy of the mixed state considered in example 3.2 is S(ρ) = 1/2 ln(2) + 1/2 ln(2) = ln(2) qubits ,
which is the maximum entropy for a mixed spin one-half state.
5 Quantum Entanglement
Quantum entanglement is one of the most intriguing aspects of quantum mechanics. Two particles that have once interacted continue to, in some sense, share a wave function even after being separated by large distances. In this section, the quantum mechanical description of entanglement and its properties will be studied.
5.1 Definition
5.1.1 Pure State Entanglement
Classically, the space for a system will be represented by a set. This implies that the total space of a multipartite system consisting of n subsystems is the Cartesian product of all subspaces corresponding to each state.9 This means that the total space will aways be a product space of the individual subspaces, hence it will always be separable.
In quantum mechanics, the space for a system will instead be represented by a vector space, called a Hilbert space. This has significance for the total space of a multipartite system. Instead of the Cartesian product, the total Hilbert space for a quantum system will be given by the tensor product10 between the Hilbert spaces for each subsystem
H =
n
O
i=1
Hi, (5.1)
where Hiare the vectorspaces for each subsystem. Together with the superposition principle of quantum mechanics, a total system consisting of n subsystems can be expressed by
|ψi = X
i1,...,in
ci1,...,in|i1i ⊗ |i2i ⊗ · · · ⊗ |ini , (5.2)
where ci1,...,in is a normalisation constant, and |iji the state vectors for each subsystem.
Generally, the universal state cannot be written as a product of subsystems,
|ψi 6= |ψ1i ⊗ |ψ2i ⊗ · · · ⊗ |ψni ; (5.3) i.e. not all multipartite quantum systems can be written as a product of subsystem, meaning they may be inseparable. An important point is, for an inseparable system, even though the total system is in a pure state, it is impossible to assign a single state vector to any of the subsystems. This is because the two state vectors belong to separate Hilbert
9The Cartesian product is a product between two sets, A and B, which returns a product set A × B with cardinality equal to the sum of the cardinalities of A and B.
10The tensor product takes two vector spaces V and W to produce a third vector space V ⊗ W with dimension equal to the product of the dimensions of V and W .
spaces, whereas the global state is in a total Hilbert space. This fundamentally describes entanglement.
In the case of bipartite entanglement, the expression (5.2) reduces to
|ψiAB=
dA−1
X
i=0 dB−1
X
j=0
Aψij|eiAi ⊗ |ejBi , (5.4)
where, |eiAi⊗|ejBi are the orthonormal product basis for the state, dAand dBthe dimensions of each subspace, and Aψij is some matrix coefficient. This state is separable if and only if the matrix coefficient is of rank one.11 This rank is equivalent to the rank of either density matrix for subsystem A or B; meaning that if either one of the subsystems is a pure state, the system will not be entangled.
5.1.2 Mixed State Entanglement
The topic of mixed state entanglement is somewhat more delicate than pure state entan- glement, primarily because a mixed state is more complex and hard to predict than a pure state. However, the definition for an entangled mixed state is similar to that of a pure state, although with some distinctions.
A mixed state ρ is entangled if it cannot be written as a sum of product states ρ 6=X
i
piρ1i ⊗ · · · ⊗ ρni , (5.5) where the superscripts denote the subspaces. Since the density matrix can be thought of as a probability distribution for the quantum system, a mixed state is entangled if its total probability distribution cannot be expressed as a product distribution for other states.12 5.2 The Bell States
For the remainder of this paper pure state bipartite entanglement will be considered since it is sufficient for future purposes; more specifically, the four maximally entangled states of two spin one-half particles. The four states are called the Bell states, named after John S.
Bell, but are also known as EPR-states.13 The four states, which together form a complete basis, are
|Ψ±i = 1
√2 |↑↓i ± |↓↑i, |Φ±i = 1
√2 |↑↑i ± |↓↓i, (5.6)
11Meaning that all columns in the matrix are linearly dependent.
12It is worth noting that classical probability distributions may always be written as mixtures of product distributions. This manifests yet another divergence between classical and quantum formalism.
13In fact, any configuration of two, two level, maximally entangled states are still Bell states. For example, the same configurations may be used for photons where spin is substituted by polarisation.
where the state |ψ−i is called the spin singlet, since it is the single state with total spin zero;
and the three other states are called spin triplets, since they are three states with total spin one. In the above expressions we have used the short notation |αβi = |αi |βi = |αi ⊗ |βi.
For these states, either subsystem has equal probability of measurement. Entanglement comes about since a measurement of either particle immediately reveals the spin direction for the other. For example, a measurement of the singlet will produce anti-correlated results.
Remark 5.1. The expectation value for the spin along any axis is zero for either particle in the entangled Bell states. This will be shown for σ3 acting on the first particle in the singlet state, but is shown analogously for any other configuration.
Since the singlet state is a pure state the expectation value for σ3 will be given by hσ3i = hΨ−|σ3|Ψ−i .
Letting the Pauli matrix act on the state to the right, we get hσ3i = 1
√
2hΨ−| |↑↓i + |↓↑i .
By expanding the bra and multiplying, and discarding orthogonal terms, the expectation value will be
hσ3i = 1
2 h↑↓ | ↑↓i − h↓↑ | ↓↑i= 0 .
However, for any product of spin one-half states, there will always be some axis in which either particle will have a definite spin direction, since both particles are independent.
Remark 5.2. It is important to stress that there is no violation of causality for entangled states. Suppose the two entangled particles are separated by an arbitrary distance; one particle held by Alice, the second held by Bob. At any time, Alice can measure the spin along any axis for her particle, and will have equal probability of measuring either spin up or spin down. Once the measurement is made, Alice immediately knows the spin orientation of Bob’s particle. However, Bob has no knowledge about Alice’s result. For him, the state which he holds is still in a superposition of the two possible spin states, each having equal probability of measurement. It is only after his own measurement that he will have full knowledge about the system. Thus, no information has propagated from Alice to Bob faster that the speed of light. Nevertheless, it is possible for Alice to call Bob and inform him of her result, thus giving Bob full knowledge without him measuring his state; yet such communications are limited by the speed of light.
5.3 Entropy of Entangled States
As mentioned in section 5.1.1, it is impossible to assign a single state vector to any of the subsystems of an entangled state. By definition, this means that each subsystem must be represented by a density matrix, and that the entropy of each subsystem is larger than zero. How to extract the density matrix of a subsystem, and in turn calculate the entropy, will be presented in the next section.
5.3.1 Density Matrix of Entangled Subsystems
To obtain the density matrix for a subsystem of a total multipartite system, known as the reduced density matrix, one needs to take the partial trace of the universal density matrix over the degrees of freedom that do not correspond to the system of interest.
For simplicity, let us consider the pure entangled state which acts in the Hilbert space HA⊗ HB. The state can be written as a sum of orthogonal product basis
|ψiAB=X
i,j
cij|iiA|jiB , (5.7)
which in turn yields the density matrix ρAB= X
i,j,i0,j0
cijc∗i0j0|iiA|jiBhi0|Ahj0|B . (5.8)
Now, to take the partial trace over the degrees of freedom for B, we put the density matrix in brackets which only act on the B vectors; hence
TrB(ρAB) =X
k
hk|ρ|ki
B B . (5.9)
Expanding the above expression we obtain TrB(ρAB) = X
i,j,i0,j0,k
cijc∗i0j0|iiA Bhk|jiB Bhj0|kiBhi0|A (5.10)
= X
i,j,i0,j0
cijc∗i0j0|ii hi0| hj0|ji , (5.11)
where we in the last step use the completeness relation. Due to orthogonality, we must have that j = j0, which leaves the final expression for the partial trace over B to
TrB(ρAB) =X
i
c2i |iiAhi|A , (5.12)
which translates equivalently for trace over any subsystem for any system.
Example 5.1. The total density matrix for the singlet state |Ψ−iAB= √1
2 |↑A↓Bi − |↓A↑Bi will be given by
ρAB= |ΨiABhΨ|AB, and thus the density matrix for subsystem A will be given by
ρA= TrB(ρAB) =X
i
c2i |iiAhi|A= 1
2 |↑i h↑| + |↓i h↓|,
which is the same matrix as for the maximally mixed state in example 3.2. Thus, the density matrix will be
ρA= 1 2
1 0 0 1
.
In this case, the density matrices for both subsystems will be the same.
5.3.2 Entropy of Entangled Subsystems
The von Neumann entropy for a subsystem of an entangled system will be given as usual, the only exception being that the density matrix used to compute the entropy will be that of the subsystem. However, there are some interesting properties of the entropy of entangled states.
The entropies for the subsystems in example 5.1 are S(ρA) = S(ρB) = ln(2). However, the entropy of the entire system, S(ρAB), is zero since it is a pure state. In other words, there is more uncertainty in the subsystems than for the global system. This is a property unique for entangled states, hence the von Neumann entropy is also known as entropy of entanglement. This means, the closer the subsystems in an entangled ensemble are to mixed states, the greater the entanglement of the total state.
5.3.3 Entropic Inequalities
For the classical Shannon entropy, the entropy of two variables X and Y is always greater or equal to the entropy of only one of the two,
H(X, Y ) ≥ H(X) , H(X, Y ) ≥ H(X) . (5.13) The quantum analog of these inequalities uses the von Neumann entropy of a quantum ensemble. For the two particle case, this will be given by
S(ρAB) ≥ S(ρA) , S(ρAB) ≥ S(ρB) . (5.14)
This can in turn be written as
S(ρAB) − S(ρA) ≥ 0 , S(ρAB) − S(ρB) ≥ 0 , (5.15) which can be taken as the positivity of the conditional quantum entropy14,
S(A|B) = S(ρAB) − S(ρA) , S(B|A) = S(ρAB) − S(ρB) . (5.16) Conditional quantum entropy has the interpretation analogous to conditional probability considered in classical probability theory; namely the information carried by a state given some prior information about the configuration.
As we have seen in the previous section, the entropies of the subsystems in an entangled pure state are greater than the entropy of the universal state. Hence, an entangled state will have negative conditional entropy. Classically, this would have no meaning; however, in the context of quantum communication, it does. More precisely, it is of importance in quantum teleportation, where it means that classical information is sufficient to reconstruct a quantum state. Simultaneously, the two communicators, Alice and Bob, end up sharing
−S(A|B) additional EPR-states, which can be used for future communications [13, 14].
This will be evident when considering quantum teleportation in a later section.
5.4 Monogamy of Entanglement
Monogamy of entanglement is a fundamental property of quantum entanglement. In its most basic form, it states that if two systems, A and B, are completely entangled, then neither one of the systems can have any entanglement with a third system, C.15 For example, if particle A is in a singlet state with particle B, it cannot also be in a singlet state with particle C. This can be expressed in the general form
E(A : B1) + E(A : B2) + · · · + E(A : BN) ≤ E(A : B1. . . BN) , (5.17) where E is some measure of entanglement.
For the sake of argument, say that particle A is in a singlet with both particles B and C. One may then measure the spin along the x-axis for particle B, and the spin along the z-axis for particle C; thus gaining information about the spin of particle A in both the x and z direction, which is not possible. Hence, we are left with a contradiction.
However, as the inequality suggests, there may be a trade-off in the amount of en- tanglement between multiple systems. This trade-off is partially what mutes entangled properties in macroscopic compounds, since any entanglement will be shared by a large amount of states.
14The conditional entropy for classical variables is given analogously, using Shannon entropy.
15In contrast, for the classical case there are no limitations for correlations between several variables.
Monogamy of entanglement is of great importance for the security of quantum key distribution, as will become apparent.
Example 5.2. Consider the simplest tripartite entangled state
|Ψi123 = 1
√
2 |↑1↑2↑3i + |↓1↓2↓2i ,
which is known as the GHZ state. As a total state, this is a maximally entangled state of three spin one-half states. Extracting the density matrix for particle one and two, by taking the partial trace over the degrees of freedom for particle three in the complete density matrix for the system, we get
ρ12= 1
2 |↑1↑2i h↑1↑2| + |↓1↓2i h↓1↓2|. If we identify the state vectors with
|↑1↑2i = (1, 0, 0, 0)T
|↓1↓2i = (0, 1, 0, 0)T, the density matrix will be
ρ12= 1 2
1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
.
As we can see, the subsystem is maximally mixed. However, it is separable.16 This means that the removal of one particle causes the entanglement to collapse, even though the global state is maximally entangled. This has the interpretation that one particle is maximally entangled with a two particle state, however the subsystems experience no mutual entanglement.
6 Bell’s Inequality
Bell’s inequality is an inequality which puts restrictions on correlations in classical proba- bility theory; namely, it is an inequality which holds for all classical physical systems. If the inequality is violated, it implies that no classical theory can describe the system at hand.
In this section, we consider the case where the classical system can be regarded as a set Ω such that there are some subsets A, B, C ⊂ Ω.
16Here we use the positive partial transpose criteria [15].
Theorem 1. For any given set Ω, and subsets A, B, C ⊂ Ω, the following relation must hold,
N (A \ B) + N (B \ C) ≥ N (A \ C) where N denotes the cardinality of a set.
Proof. Consider the Venn diagram of Ω, with its subsets A, B and C,
C A
B
5
3
1 2
4 6
7
then we will have
N (A \ B) = N (1) + N (2) N (B \ C) = N (7) + N (4) N (A \ C) = N (1) + N (4) . Using these sets in the Bell inequality, we get
N (1) + N (2) + N (7) + N (4) ≥ N (1) + N (4) , which is trivially the case.
6.1 Violation of Bell’s Inequality by Entangled State
As mentioned earlier, violation of Bell’s inequality implies a system that is not governed by classical physical rules. Hence, this section is dedicated to showing that the four Bell states violate said inequality. This will be shown for the spin singlet state, however it may be shown analogously for any formation.
Proof. First, we take the system to be two particles in the singlet state
|Ψ−i = 1
√
2 |↑↓i − |↓↑i , (6.1)
yet we suppose the system is purely classical. We will now define the subsets of our system, which are generated from the first spin. Due to the properties of the spins in the singlet
state, any negation of a set for the first spin will be the same set for the second spin. We define
A = #1 up along the z-axis.
B = #1 up 45◦ along the z-x-plane.
C = #1 up along the x-axis.
¬A = #2 up along z-axis.
¬B = #2 up 45◦ along the z-x-plane.
¬C = #2 up along the x-axis.
Plugging these sets into the Bell inequality, we will obtain
N (↑, %) + N (%, →) ≥ N (↑, →) . (6.2)
Due to rotational invariance of the singlet state, the first and second term in the above expression are the same; thus, it can be expressed as
2N (↑, %) ≥ N (↑, →) , (6.3)
where each slot corresponds the first or second spin, respectively. By taking a sufficiently large amount of particles, the number of elements in a set can be approximated by the probability for each element. In which case, the inequality will be given by
2P (↑, %) ≥ P (↑, →) . (6.4)
These probabilities can now be computed for the singlet state. Before doing this, let us introduce measurement by projection.
Consider the projection operator Π = |αPi hαP| which projects a state vector onto any desired subspace such that
Π |αi = hαP|αi |αPi , (6.5)
where |αPi is the projected state. The probability for a projected state is thus given by17
P (αP) = hα|Π|αi . (6.6)
For any state, there exists a corresponding projection operator which projects any arbitrary state vector onto the eigenvector of said state. For the spin states in our system, the corresponding projection operators can be written in terms of the Pauli matrices as follows
Π↑,1 = I + σ3
2 , Π%,2= 1
2 I +τ1+ τ3
√ 2
!
, Π→,2= I + τ1
2 , (6.7)
where τi are Pauli matrices acting on the second particle, while σi act on the first. We may now compute the desired probabilities for the singlet state.
17This is shown in appendix A.1.3
The probability will be given by P (↑, %) = hΨ−|1
2
I + σ3
2
!
I +τ1+ τ3
√ 2
!
|Ψ−i = 1 4
√
2 − 1 . (6.8) Analogously, the second probability is given by
P (↑, →) = hΨ−| I + σ3
2
! I + τ1
2
!
|Ψ−i = 1
4. (6.9)
Hence, the inequality will be
1 2(√
2 − 1) 1
4. (6.10)
This result means that the singlet state, and in turn entanglement, cannot be described using any local hidden variable models.
6.2 CHSH Inequality
Out of the plethora of different takes on Bell’s theorem, there is one of special interest. It is called the CHSH inequality, and is experimentally testable for a system which has two outcomes [16]. In a system where Alice can measure a and a0, and Bob can measure b and b0 with values ±1, it is expressed as18
E(a, b) + E(a, b0) + E(a0, b) − E(a0, b0) ≤ 2 , (6.11) where E is the correlation coefficient given by
E(x, y) = Pii(x, y) + Pjj(x, y) − Pij(x, y) − Pji(x, y) , (6.12) where i and j are the different outcomes for each measurement, and P (x, y) is the probability.
In the same way as the previous inequality, the CHSH inequality is satisfied by any system admitting classical models. The maximum violation of the inequality is by a factor√
2.
This inequality is of importance for security in some quantum key distribution protocols, which we shall see in the following.
7 Quantum Teleportation
The word teleportation provokes images of interstellar travel and futuristic laboratories which disintegrate a specimen into light which is then beamed to some distant location.
However, the present state of teleportation is not yet at that level. In this section we will present the protocol for teleporting single states over arbitrary distances using entanglement.
18It may also be expressed in the shorter form |S| ≤ 2.
7.1 No Cloning Theorem
A quantum operator which makes exact copies of any given state would be a nice thing to have. If not only for personal entertainment, one may want to clone some state and send it to a friend while keeping one or more copies for oneself. However, such an operator does not exist [17].
Theorem 2. It is impossible to clone an unknown quantum state by any unitary operation.
Proof. Suppose there exists some unitary operator Ucwhich when it acts on some quantum states makes a copy of said state. Accordingly
Uc|αi |0i = |αi |αi , (7.1)
where |0i is the vacuum state. Now, consider a general state of a spin one-half particle
|ψi = 1
√2 |↑i + |↓i. (7.2)
Attempting to clone the full state, we get Uc|ψi |0i = |ψi |ψi = 1
2 |↑↑i + |↑↓i + |↓↑i + |↓↓i. (7.3) Instead, when attempting to clone the expanded state, we obtain
√1
2Uc |↑i + |↓i|0i , (7.4)
which, using the linearity of quantum operators, can be written as
√1
2 Uc|↑i |0i + Uc|↓i |0i= 1
√2 |↑i |↑i + |↓i |↓i. (7.5) Clearly, the two attempts at cloning the same state yield different results, meaning such efforts are of no use.
7.2 Quantum Teleportation Protocol
Quantum teleportation is a means of transmitting a quantum state from a sender, which we called Alice, to a receiver, Bob. Although its name suggests otherwise, quantum teleportation is not purely quantum. In order for Alice and Bob to successfully send and receive a state, some classical communication between them must take place.
Suppose Alice wants to send the general spin state |φi1 = α |↑i+β |↓i, where|α|2+|β|2= 1, to Bob, who is located at some distant location unknown to Alice. Assume also that Alice and Bob each posses a particle of the same EPR singlet state
|Ψ−i23= 1
√2 |↑i2|↓i3− |↓i2|↑i3, (7.6)
where Alice holds particle two, and Bob holds particle three.19 The complete system for all three particles will then be of the form
|Ψi123 = α
√2 |↑i1|↑i2|↓i3− |↑i1|↓i2|↑i3+ β
√2 |↓i1|↑i2|↓i3− |↓i1|↓i2|↑i3, (7.7) where we have used |Ψi123= |φi1|Ψ−i23. So far there is no entanglement between particle one and two. However, since the Bell states form a complete basis the above expression can be expressed in terms of the Bell basis for particles one and two,
|Ψ±i = 1
√2 |↑i1|↓i2± |↓i1|↑i2, |Ψ±i = 1
√2 |↑i1|↑i2± |↓i1|↓i2. (7.8) Thus, the total state written in the Bell basis is
|Ψi123 = 1 2
h|Ψ−i12 −α |↑i3− β |↓i3+ |Ψ+i12 −α |↑i3+ β |↓i3 + |Φ−i12 α |↓i3+ β |↑i3+ |Φ+i12 −α |↓i3− β |↑i3i
(7.9)
Next, Alice makes a joint measurement of particles one and two in the Bell basis.
Due to the configuration of the total system, she will measure one of four Bell states with equal probability. Once Alice has identified the state which she has measured, she communicates her result to Bob using a classical channel. The amount of information needed to communicate the outcome of the joint measurement will be given by the Shannon entropy for a completely random variable, in this case yielding H(X) = log24 = 2 bits.
When Bob is informed of Alice’s result, he knows that the state which he has at hand will be related to the original particle by some unitary transformation, as is evident from the expression of the total state. Respectively, the state held by Bob will have the form
− |φi3≡ |φi3 ,
−1 0
0 1
|φi3 ,
0 1 1 0
|φi3 ,
0 −1
1 0
|φi3 . (7.10) For Bob to obtain the original state sent by Alice, he performs one of the three possible transformations shown above, since the first state only differs by an overall phase. When the rotation is made and Bob obtains the state intended by Alice, the protocol is fulfilled.
Remark 7.1. The process of teleporting the unknown state |φi1 from Alice to Bob destroys said state for Alice when it is paired together with her original particle in the joint measurement, only to be re-constructed when Bob performs his rotations. Thus, teleportation does not violate the no cloning theorem.
19For simplicity, say the particles have been divided between Alice and Bob before Bob left on his journey, in the intention of future use.
There is no displacement of matter. Rather, the properties of the transmitted particle are re-constructed at the receivers location, i.e. the quantum information about a state is teleported. Furthermore, the negative conditional entropy of the system, discussed in section 5.3.3, enables the original quantum state to be re-constructed given only classical information; while an additional EPR-state is obtained when Alice makes her joint measurement, which can be used for future communications. Thus, potential information is gained.
7.3 Generalisation
Let us now generalise teleportation for particles with N > 2 orthogonal states. In the simplest case, the entangled state shared between Alice and Bob can be given by
|ψiAB=X
j
|ji |ji /√
N , (7.11)
where the kets label the basis vectors for the N-state systems, and j ∈ [0, N − 1].
Let us now construct the eigenvectors for the basis in which the joint measurement of particles one and two is made. In the simple case of a single particle, the ket for the state is given by summing over the basis vectors
|Ψi =X
i
ci|ii , (7.12)
where ci is some complex number containing a relative phase between the basis vectors.
The basis vectors for the joint measurement will be of the form |ji |ki. To find the relative phase between the basis vectors, we consider the complex plane. Each orthonormal basis vector must lay equally spaced on the unit circle in the complex plane. This gives the relative phase e2πijn/N, where j, n ∈ [0, N − 1]. Here, n generates the relative sign between the basis vectors, while j sums over all possible states. In order to include all possible combinations of states in the basis vectors, we can shift the summation of the second state by j + m mod N , where we take the modulo in order to constrict the vectors to the unit circle. Hence, the basis vectors will be of the form
|ji |j + m mod N i (7.13)
Combining the above result, we get the expression for the basis eigenvectors for particles one and two
|Ψnmi12=X
j
e2πijn/N|ji |j + m mod N i /√
N , (7.14)
where n and m are iterated through all possible combinations.
Once Alice has made her measurement and communicated her result to Bob, he uses a unitary transformation given by
Unm =X
k
e2πikn/N|ki hk + m mod N | , (7.15)
where k ∈ [0, N − 1]; which is a matrix that rotates the measured vector by the shift in the measured basis.20 These result are the same as the ones presented in [6].
8 Quantum Cryptography
The purpose of encryption is to systematically scramble a message, known as the plaintext, in such a way that it is rendered unintelligible to any party not meant to take part of the information. The distorted text, called the cryptogram, is produced by algorithms, known as cryptosystems or ciphers, which combine the plaintext with some additional, meaningless information, known as the key. Thus, the reader must in turn have access to the key in order to reverse the algorithm, decrypt the cryptogram, to get access to the plaintext. The process can be expressed as
Ek(P ) = C , Dk(C) = P . (8.1)
A cryptosystem is secure if it is impossible to decipher a cryptogram without having access to the key.
There is one main weakness in classical cryptography. Namely, the possibility for any eavesdropper to collect information unnoticed; for example by listening to a channel used to pass keys. This makes the matter of key distribution somewhat complicated. However, if one were to use a quantum channel, any interference by an eavesdropper must obey the laws of quantum mechanics, in which case any information gained will perturb the system of interest. Such perturbations can then be detected in the output of the quantum channel, and any communications aborted.
This section will begin by presenting some existing cryptosystems, to later discuss methods of key distribution using quantum channels.
8.1 Asymmetrical and Symmetrical Cryptosystems
There exists two kinds of cryptosystems: asymmetrical, also known as public key cryp- tosystems; and symmetrical, or secret key cryptosystems.
20The proof that the transformation is unitary is in the appendix A.1.4.
8.1.1 Asymmetrical Cryptosystems
The most commonly used encryption protocols belong to asymmetrical cryptosystems, which are popular due to their convenience. The principle is that two different keys are used for encryption and decryption, hence the name.
Suppose Alice wants to send classified information to Bob using a public channel. Bob, who is a person of much importance, is used to receiving sensitive information meant for his eyes only. In order to receive messages from several peers without having to use different keys for each sender, he uses a public key protocol. First, Bob generates a key which he keeps for himself. From this private key, he constructs another key which he announces publicly to whomever wants to use it. Alice then uses this public key to encrypt her message, and sends it to Bob using the public channel. Finally, Bob obtains the plaintext by decrypting the cryptogram using his private key.21
In classical asymmetrical protocols, there is nothing stopping a malicious eavesdropper, Eve, from listening to the public channel and copying the encrypted message passed between Alice and Bob. After all, Eve also has access to the public key. However, simply knowing the public key will not provide sufficient information to read the message. Specifically, the public key is constructed in such a way that it acts as a one-way function. Such a function, f (x), is easily computed for any given x; but difficult to inverse and compute x given f (x). The security of public key cryptosystems is therefore based upon computational complexity; the idea being that brute force decryption will be too time consuming. For example, the RSA protocol is based upon prime factorisation of large integers [2].
Unfortunately, the security of one way functions is not yet proven, meaning it is impos- sible to rule out that some computational algorithm may be discovered which efficiently cracks the ciphers. With the rise of quantum computation, and the algorithms which come with it [2], public key cryptosystems may not be safe for much longer; implicating that a move to symmetrical protocols is crucial.
8.1.2 Symmetrical Cryptosystems
A symmetrical cipher uses the same key, shared between Alice and Bob, for both encryption and decryption. The best way of illustrating this is by presenting the Vernam cipher, also known as the one time pad protocol, which is the only cipher proven to be secure.
In the one time pad protocol, Alice and Bob share a common key. For binary operations, the key is a string of randomly generated bits the same length as the converted plaintext.
21An analogy is that Bob distributes boxes with open padlocks to which he holds the master key. Anybody can place a message in a box and lock it, while only Bob can open the box and read its contents.
The cryptogram is produced by adding the key to the plaintext
c = p ⊕ k , (8.2)
where ⊕ is binary addition modulo 2 acting on each bit. Since the key is completely random, the cryptogram will by itself not contain any information. Once the cryptogram is produced, Alice sends her message to Bob choosing any sort of classical channel she pleases.
Bob then decrypts the cryptogram to obtain the message by subtracting the same key
p = c k . (8.3)
It is crucial for the security of the one time pad protocol that the key is used only once.
This is a source of difficulties, since the key must be handed over personally, or distributed by a trusted currier. Such a logistical operation would prove too expensive on a large scale, hence such cryptosystems are used only under exceptional circumstances.
8.2 Quantum Key Distribution Protocols
The primary objective of quantum cryptography is the distribution of keys for use as one time pads. Quantum key distribution protocols consist of two phases: sending of the quantum states and public announcement of the measurement procedure. Due to the nature of quantum mechanics, the security of the protocols are guaranteed even when using public channels, hence such protocols are cost effective. Furthermore, the keys cannot be intercepted during the sending of the states, since the key only comes into existence after the protocols are completed.
8.2.1 BB84 Protocol
The BB84 protocol [7] was the first proposed protocol for quantum key distribution.22 It will be presented using the original idea of polarised photons, but any two level quantum state would be adequate.
Let us begin by defining the technicalities. Vertically and horizontally polarised photons are denoted |li and |↔i, respectively; and they are sent in the rectilinear basis denoted ⊕.
Diagonally polarised photons are denoted | l i and | l i, and are sent in the diagonal basis denoted ⊗. In order to produce binary keys, the polarisations are assigned binary values;
|li and | l i code for 0, while |↔i and | l i code for 1.23
22The protocol is named after its proposers, Charles H. Bennet and Gilles Brassard, who presented the protocol in 1984.
23Quantum states with assigned bit values are called qubits, and have the general form
|ψi = α |0i + β |1i, where the coefficients are complex number which modulus square sum to unity.
