Entanglement: Concept, Measures and Open Problems

Project Work: Entanglement: concept, measures and open problems

Daniel Karlsson and Miroslav Hopjan

June 10, 2013

Abstract In this short report, we have presented a review about entanglement research and entanglement measures. By giving an introduction to entanglement, and to the different measures possible, we have tried to give a glimpse of how entanglement research can look like, and wherein the difficulties lie. In the report we have discussed different entanglement measures, such as the entanglement cost, the entanglement distillation and the entanglement entropy and we have also discussed several important states in entanglement research, such as the Bell states, the GHZ states and the W states.

1 1 Introduction

Entanglement is one of the most important concepts of quantum mechanics. It appears to be a necessary implication of quantum mechanics and plays the key role for a confirmation of its validity. Roughly speaking, the quantum entanglement, or quantum interconnection, is a state of a system in which the states of its subsystems are mutually correlated. As an example, the wave function of two entangled particles A (Alice) and B (Bob) can be written in bracket formalism as |Ψi = |1Ai|0Bi + |0Ai|1Bi. (1) Note that there is no way how to write the state |Ψi as a direct product of functions N of two subsystems, |Ψi= 6 (|Ψi)A (|Ψi)B. In general, entangled states can contain superpositions of more than two states, and can link more than two particles. The entanglement has interesting consequences. Suppose particle A is measured to be in the state |1i. This means that the wave function collapses into the left branch and the particles B is in state |0i. An analogous argument applies to the measurement of particle A in state |0i. Such a sudden action is long-distance, and called non-local. The non-local character of the collapse of the wave function during measurements was only partially admitted by physicist in the early days. The so-called EPR paradox and subsequent discussions between defenders of quantum mechanics, and its opponents calling for some background theory (with hidden parameters) beyond quantum mechanics, lead to the first attempts to quantify the measure of entanglement. The Bell inequalities served as a tool to distinguish between correlations induced by classical physics and purely quantum mechanical correlations. The quantum correlations are very counter-intuitive. However, such phenomena has been confirmed by experiments and therefore one have to accept them as facts. It the past it was hardly imaginable that such correlations could be created in well controlled environments between distinct quantum systems. During the last decades, the development of experimental methods and the technological progress has allowed now-days to coherently prepare individual quantum systems, manipulate them and perform measurements, as well as create controllable quantum correlations between such systems. Once the quantum correlation became controllable they started to be an important resource for quantum computational science. Compared to a classical bit, a quantum state can store much more information. Thus, instead of the classical bits 0 and 1 we can use qubits, represented by α|0i + β|1i. The possible operations made with qubits, where quantum correlations are important, could efficiently solve problems which are extremely inefficient using classical information science. Motivated by the development of quantum information and the rising importance of quantum correlations, physicists try to establish a rigorous theory of entanglement and somehow characterize, manipulate and even measure the entanglement of quantum systems. In the next sections we would like to give a brief overview of the current status of entanglement research.

2 Introduction to Entanglement

We need start with a discussion of what entanglement actually means. For this purpose we choose a highly operational point of view. It is motivated both by technological and fundamental aspects and naturally appears within quantum communication framework and a teleportation over long distances.

2 We will consider the motivation in detail. The main aim in quantum communication is to be able to transport quantum particles to the laboratories separated by macroscopic distances. Ideally, we want to distribute entanglement without any losses. If the transport of qubits can be done without coherence losses the entanglement will be distributed perfectly. One can reverse the idea. If one can distribute entanglement perfectly, then it is also possible to transfer qubits perfectly. This is actually the heart of quantum teleportation experiments. Noise will make us lose information about the system, which means that we cannot perform the transfer perfectly. One can overcome this problem, using noisy quantum channels for distributions of particles, by eliminating the influence of noise a posteriori. The post-process can be done using local quantum operations with high accuracy in separated laboratories. One has a chance that such post-processes could be done without any influence of noise because they can be performed in well controlled environments locally. One can then use classical communication to coordinate the quantum post-processing in the laboratories. The classical communication is, for these purposes, regarded as perfect. Note that exactly this scheme is implemented in promising quantum teleportation protocols. These ideas bring us to the concept of Local quantum Operations with Classical Communication, the ”LOCC” paradigm. The fundamental motivations of the LOCC paradigm are even more important than the technological motivation mentioned above. The LOCC paradigm help us to distinguish between classical and quantum correlation and a way of characterising entanglement. The entanglement is naively defined as quantum correlations that appear in many- party quantum states. In the framework of LOCC we can define the quantum correlation as those correlations which we are not able to create via only LOCC operations. Thus, additional correlations which were created by LOCC operations are classical correlations. For illustration, we consider quantum states from a noisy channel, and after that we performed the LOCC operations. We thus obtain states which can be used for some task in quantum information. So, if the task which we perform cannot be simulated by classical correlations, i.e. they do not satisfy the Bell inequalities, we can not attribute such correlations to LOCC operations. These correlations are quantum and they are something which is present from the initial states. This is very important. Thus, we can see the LOCC framework as a tool for characterising entanglement.

3 Quantum Operations

In Quantum Information, there is a slightly different definition of quantum measurement compared to standard quantum mechanics, where the measurement process is defined as a non-unitary operation, namely as a projection measurement to one of the eigenstates of the observable we are interested in. In Quantum Information, we use a so-called generalized measurement. The original system is evolved while adding another system which interacts with it. The setup consists of three steps. The first step is the addition of ancilla subsystems, then we perform a unitary evolution of the whole system, and then perform a measurement on both the ancilla system and the original system. The subsystems which are not important after such an evolution are discarded. The whole system where the ancilla particles are uncorrelated with the rest of the system is usually described with K raus operators. The measurement outcomes will be

3 † distributed according to probabilities pi = tr(AiρinAi ) as † AiρinAi ρi = † (2) tr(AiρinAi )

where ρin is the initial state and the Ai are Kraus operators, which satisfy the normaliza- P † tion condition i Ai Ai = 1. In some situations all or part of the measurement outcomes might not be accessible. If we are in the extreme case where the ancilla subsystems were traced out, the map is given by X † σ = AiρinAi , (3) i sometimes referred to as a trace preserving quantum operation. The operations where measurement outcomes are retained are referred to as measurement operators. P † Conversely, for any set of linear operators Ai with conditions i Ai Ai = 1 we can ﬁnd a process, where we add ancilla bits, provide unitary evolution and von Neumann † AiρinAi measurement which give us ρi = † . tr(AiρinAi ) The reason for generalising the quantum operations in terms of Kraus operators is to use them to determine which operations are implementable by the LOCC constraint. The LOCC is complicated, because Alice and Bob can communicate classically before or after any given round of local actions, and these actions are dependent on outcomes of previous ones. There is no simple characterization of LOCC operations. One need to take broader class of operations which still retain the element of LOCCality. Such a class is easy to characterize and is represented by separable operations. In Kraus representation: † † Ak ⊗ BkρinAk ⊗ Bk ρk = † † (4) tr(Ak ⊗ BkρinAk ⊗ Bk) P † † which satisfy k AkAk ⊗ BkBk = 1. Any LOCC operation can be cast into the form of these separable operations, as the local Kraus operator, corresponding to the individual actions of Bob and Alice, can be jointed to the product of Kraus operators. The converse is not true.

4 Basic properties of entanglement

The basic properties of entanglement are determined by the LOCC class of operations. They will be used as a guide for the quantiﬁcation of quantum entanglement later on. • Separable states contain no entanglement - A state is separable when we can write it as a product of states of its parts

X i i i ρABC = piρA ⊗ ρB ⊗ ρC (5) i

where pi is probability that we ﬁnd the system in the state i. In other words, the density matrix of the system is the tensor product of the density matrices of its parts. These states are easily created by LOCC operations. Alice samples from the distribution, and tell the other parties about the outcome i. Each party X thus i locally create state ρX and discards informations about outcome i. These states created by LOCC transformation satisfy local hidden variables models and all their correlations can be described classically. We conclude that separable states contain no entanglement, and thus serve as a lower bound for a measure of entanglement.

4 • All non-separable states allow some task to be achieved better than by LOCC alone, so all non-separable states are entangled -The entangled state cannot be created by LOCC action only. This statement push the entanglement to the position of ”negative” meaning. A quantum state can be perfectly generated by LOCC operations if and only if it is separable. For any non-separable state ρ we can ﬁnd a state σ whose teleportation ﬁdelity can be enhanced if ρ is also present. This gives to the non-separable state the ”positive” meaning, as the non-separable state posses a useful resource that is not present in separable states. This allow us to identify the terms non-separable and entangled as synonymous.

• Entanglement does not increase under LOCC operations - Starting from separable states we are not able to create more entanglement in the system using only LOCC operations. This is a consequence of the fact that LOCC can only create separable states. If we start from the entangled or non-separable state we cannot add any entanglement by LOCC. Thus the entanglement cannot increase under LOCC. The reverse is not true, it is very much possible to decrease entanglement by LOCC.

• The entanglement does not change under local unitary evolutions, without classical communication. - If we imagine only local unitary transformation we can reverse the evolution. Both are in the same time LOCC. The property above can be applied for both operation and thus the entanglement stays unchanged by local unitary transformations.

• There are maximally entangled states - We saw that minimally entangled states are separable and we can characterize that one state is more entangled than the other, we would like to ﬁnd also upper boundary or maximally entangled state. Can we ﬁnd such states? In the two-party system with d-dimensional sub-system such states exist and can be represented upon local unitary transformation as (generalized Bell state)

|00i + |11i + ··· + |ddi |ψ i = √ . (6) d d This is well justiﬁed. Any pure or mixed state of two d-dimensional subsystems can be prepared in this state with certainty using only LOCC transformation. As we said before we can only lose entanglement during LOCC evolution. In a many-particle system it is harder to ﬁnd such states, making the characterisation of entanglement even harder.

The main idea of LOCC is that it can give a prescription to judge whether one system is more entangled than the other. We say that one state ρ is more entangled than σ if we can create σ from ρ using only LOCC operations. This give us an ordering, but we do not know if the ordering is partial or total. We even don’t know if we can always do transformation from ρ to σ. We look at this in next section.

5 5 Local manipulation of single bipartite system - manipulation of single bi-partite states

In this section we will discuss bi-partite system where, as we showed previously, there exists maximally entangled states - the states from which any other state can be created, under LOCC. Let us do this explicitly in an example, so we will ﬁnd the Kraus operators for a system of two qubits. We start with the maximally entangled state - the Bell state.

|0, 0i + |1, 1i |ψ i = √ . (7) 2 2 or its unitary similar state. This state can be transformed to any other state, pure or mixed. For this purpose we start with the state

|φi = α|0, 0i + β|1, 1i. (8) which is a general state in Schmidt decomposition. Now we want to ﬁnd the Kraus operators for process |ψ2i → |φi.

A0 = (α|0, 0i + β|1, 1i) ⊗ 1 (9) A1 = (α|1, 0i + β|0, 1i) ⊗ (|1, 0i + |0, 1i)

† † 2 2 † † where A0A0 + A1A = 1 ⊗ 1 and |φihφ| = p0|φihφ| + p1|φihφ| = A0|ψihψ|A0 + A1|ψihψ|A1. Indeed we can implement such a transformation by LOCC only. We add an ancilla bit |0i to Alice’s bit |0, 0i |0i + |0, 1i |1i A B√ A B . (10) 2 and then we perform local unitary operations on Alice qubits |00i → |α00i + |α11i and |01i → |α01i + |α10i so we end up with

|0i (|0, 0i + |1, 1i ) + |1i (|0, 1i + |1, 0i ) A AB AB √ A AB AB . (11) 2 Depending on the result of a measurement on the ancilla bit Alice can instruct, by classical communication, which local operation Bob should do on his side. Bob will do nothing if the outcome from the ancilla qubit measurement is |0i, and will perform σx if the result is |1i. P For mixed state, the above can be generalised. The desired state is ρ = j pj|φjihφj| = P † † j pj(Uj ⊗ Vj)(αj|00i + βj|11i)(αjh00| + βjh11|)(Vj ⊗ Uj ). Thus, the state is the sum of locally unitary transformed states (αj|00i + βj|11i) which can be prepared by the prescription above. This can be done if we start with a maximally entangled state. A natural question is then: what happens if we start from a general state? Can we ﬁnd conditions when we can transform one general pure state to another using LOCC? The answer to this question was given by the theory of majorization. The result of this theory are conditions (necessary and suﬃcient) for the possibility of the LOCC interconversion between two pure states. The conditions can be seen by using the Schmidt decomposition

N X √ |ψi = UA ⊗ UB αi|iiA|iiB (12) i=1

6 where the positive numbers αi are the Schmidt coefficients of the state |φi. The local unitary transformation does not change the entanglement, so we can always prepare the initial and final states in the Schmidt bases N N q X √ X 0 0 0 |ψ1i = αi|iiA|iiB |ψ2i = αi|i iA|i iB (13) i=1 i=1 where the sum is up to n (dimension of each quantum system). Without loss of generality 0 0 0 we assume that α1 ≤ α2 ≤ · · · ≤ αn and α1 ≤ α2 ≤ · · · ≤ αn. For states with the same Schmidt coefficients the interconversion should be straightforward by using local unitary operations. If it so we can decide about the interconvertibility of state under LOCC. The state |ψ1 can be converted to state |ψ2 with certainty if the set of coefficients αi is 0 Pl Pl 0 Pn Pn 0 majored by the set αi i.e. i=1 αi ≤ i=1 αi for 1 ≤ l < n and i=1 αi = i=1 αi for n non-zero Schmidt coefficients. For states where we can not perform such a transformation, neither by going from the first to second, or the opposite way, with certainty, one must conclude that such states are incomparable: We can not say which one is more entangled. One way to compare these state is via entanglement catalysis. This means that, even if the transformation |ψi → |φi is not possible, the transformation |ψi|ηi → |φi|ηi is possible with certainty. One problem with majorization condition are discontinuities. Suppose two transformations (|0, 0i + |1, 1i) √ → (0.8|0, 0i + 0.6|1, 1i) 2 (14) (|0, 0i + |1, 1i) (0.8|0, 0i + 0.6|1, 1i + |2, 2i) √ → √ . 2 1 + 2 Maximal probability of success for the LOCC transformation from the first state in the first row to the second one is one. For the second row such transformation success is always zero. We can conclude that the number of Schmidt coefficients cannot increase by using a LOCC protocol, even probabilistically. Physically we would like to admit some elevation from desired state and we would not need to be exact. This follow to non-zero probability of the transformation from the second row. Thus, by allowing for imprecision, the discontinuity is removed. However, the success probability now depends on the size of the imprecision. In the next section we will discuss this further.

6 Local manipulation of single bipartite system - asymptotic manipulation of single bi-partite states

The exact LOCC can only induce a partial order on the set of quantum states. For mixed states the situation is even worse, because we can in general not decide if it is possible to transform each pair of states. These problems can be solved if we look in the asymptotic regime, where we look for the transformation of many copies of states ρ⊗n → σ⊗m instead of single exact transformations ρ → σ. The largest ration m/n which one can achieve is our new indicator for relative entanglement content of these two states. The question is: What do we actually want to have as an output? We require either exact or asymptotically exact transformation. They are not the same and may leads to diﬀerent results.

7 In the exact transformation we allow no errors. We are asking if it is possible to perform the transformation ρ⊗n → σ⊗m with certainty for given values m and n. The supreme of all such achievable rates r = m/n is rexact(ρ → ω). The exact rate is a measure of the exact LOCC ”exchange rate” between the states ρ and σ. Asymptotically exact transformations are physically more reasonable. Here, we allow asymptotically vanishing imperfections, leading to asymptotically small changes in the ⊗n bounded observables. We consider transformation ρ → σm where σm is approximation to state σ⊗m for some large m. Asymptotically means that we take n → ∞ with the same constant rate r = m/n, which means m → ∞ also. If in the asymptotic regime the state ⊗m ρm is arbitrarily close to the state ρ we will call the rate r achievable. The supremal achievable rate rapprox is then related to the relative entanglement content of ρ and σ in the asymptotic regime. The asymptotic rate rapprox is in general diﬀerent from the exact one rexact. The asymptotic approach gives a total ordering on bipartite pure system, leading to a natural measure, which is in the same time one of the most important ones, the entanglement cost ⊗n rn EC = inf r : lim inf D ρ − Ψ(Φ(2 )) . n→∞ ψ (15) = inf r : lim inf Dρ⊗n − Ψ(Φ(2)⊗rn) . n→∞ ψ

For a given state ρ this measure quantifies the maximal rate r at which blocks of 2- qubit maximally entangled states can be converted into output states that approximate many copies of ρ in the asymptotically sense explained above. The distance between this two states is usually measured by trace D(σ, η) = tr|σ − η|, but the results seem to be independent of the choice of measure. The evolution Φ(K) of a general state K are trace preserving. The entanglement cost is conceptually important, but very hard to compute practi- cally. It classifies the whole sale exchange rate for converting maximally entangled states to ρ using only LOCC transformations. Thus, we can think about the maximally entangled states as a standard measure. There is also a connection to another measure which is called entanglement of formation, EF (ρ), described below. We can also consider the reverse process: In what rate can one obtain maximally entangled states from an initial state ρ? This defines the entanglement of distillation (or entanglement of concentration for pure states). We define the Distillable entanglement ⊗n rn ED = sup r : lim inf D(Ψ ρ − Φ(2) ) s (16) n→∞ ψ

ED tell us the rate at which noisy mixed states can be be transformed back to the ’gold standard singlet state’ using only LOCC. It turn out that the trace preserving operations are not important for definition of Distillable entanglement. Using the singlet state is not necessary either, but it is convenient. Because of the large similarity, we can ask if, or when, ED = EC . It turns out that for pure states, the equality is indeed satisfied. Moreover, in this case the entanglement measures are equal to the entropy of entanglement, defined as: EC (|φihφ|) = ED(|φihφ|) = E(|φihφ|) = S trA|φihφ| = S trB|φihφ| = −tr(trB|φihφ| log2(trB|φihφ|)) (17)

7 Postulates for axiomatic entanglement measures

It is quite difficult to come up with a definition of an entanglement measure; to state what entanglement really is. A possible way to move forward is to define a set of rules, postulates, which an entanglement measure should follow. These postulates are then more or less In order to have physically realizable measurements of entanglement, one can list possible postulates, which the measures should obey. We here consider a composite system AB.

• An entanglement measure E is a map from density matrices to a positive real number. The maximally entangled (Bell state for qubits) are taken as the maximum entanglement, with E(|BihB|) = log d.

• If the state is separable, then E(ρAB) = 0. • LOCC operations cannot increase E, on average.

• For a pure state, the entanglement measure on ρAB is given by the von Neumann entropy of one of the subsystems. According to the Schmidt decomposition, the entropy of the subsystems is the same. E(ρAB) = S(ρA) = S(ρB) if ρAB is pure. Equivalently, E((|ψihψ|) = S(T rB{|ψihψ})

The ﬁrst and second postulate introduces a scale. Since a separable state is classical, it should not contain any entanglement, thus giving a lower bound to the entanglement, which is 0. The ﬁrst postulate also introduces an upper bound, that no state can be more entangled than the maximally entangled state. This gives the bounds 0 ≤ E(ρ) ≤ log d. The third postulate states that we cannot increase the entanglement of a state using only local operations. Aside from these postulates, there exists several other requirements, which are useful, but not all measures follow them. P P • Convexity, meaning E( i piρi) ≤ i piE(ρi). • Additivity, meaning E(σ⊗n) = nE(σ).

However, even though the entanglement measures are set according to a scale, different entanglement measures will give different entanglement for the same states. More importantly, the entanglement ordering from one measure will in general not coincide with another measure. This seem to suggest that there is no unique definition of what entanglement really is, and that it depends what we want to use the entangled states for.

9 7.1 von Neumann entropy Since the previous section contained the entanglement entropy, the von Neumann entropy, we here shortly discuss the properties of this measure. To discuss the concept of entanglement entropy, we first start by giving a definition of classical entropy regarding information, the Shannon entropy X H(X) = − pi log(pi) (18) i where pi is the classical probability distribution. The Shannon entropy is zero when we have only one outcome (a die which can only be 3, for example), and is maximal when the 1 1 probability distribution is totally random (for a 6-sided die, pi = 6 , H(X) = − log( 6 ) = log(d), where d is the number of outcomes). Thus the entropy has lower and upper bounds, which can be very useful in describing systems. In the quantum mechanical case, the entanglement entropy is defined so that it reduces to the classical one in the classical limit, that is, for a completely mixed state. The von Neumann entanglement entropy is defined as X S(ρ) = −tr(ρ log(ρ)) = − λi log(λi) (19) i where in the last equality, a unitary transformation has been used to transform the density matrix to a diagonal basis. This definition reduces to the Shannon entropy for completely mixed states, since then ρ = I/d, where d is the dimension of the space, and the entanglement entropy becomes

S(ρ) = −tr(I/d log(I/d)) = log(d). (20)

Analogous to the case of classical entropy, if the state is pure, then the von Neumann entropy is zero. Also, if the entanglement entropy is zero, then the state is pure. Thus means that also this kind of entropy has a natural lower and upper bound, since it can be shown that the entropy can never exceed log d in a d-dimensional system. Several other properties follow from the deﬁnition of von Neumann entropy:

• 0 ≤ S(ρ) ≤ log d

• For a pure state ρAB, S(ρA) = S(ρB). P P • S ( i piρi) = H(p) + i piS(ρi)

8 A survey of entanglement measures

There exists several different measures of entanglement. In a sense, they all describe how correlated a state is, but there are many ways of measuring this. However, in order to get a well defined measure of entanglement, and to have physical insight into it, several different properties has to be fulfilled in order for it to be called entanglement. The properties was discussed in the previous section.

Entanglement cost However, even if the requirements are met, it is not obvious how to deﬁne an entanglement measure. A deﬁnition should be physically appealing, and also possible to measure in some way. Mathematically, it would be good if it was easily

10 calculated, both for pure and for mixed states. One concept is to use the maximally entangled states, the Bell states, as a measure. Since this is the maximally entangled state, we can use these states to create other states of lesser entanglement using LOCC. How easy it is to create such states is called the entanglement cost. One could think of: given m Bell states, how many states n of a given state ρ can I create? The ratio, called rate, is defined then as r = m/n. However, there are some problems for finite n, m, which are related to the fact that the conversion is not exact. Instead, we are more interested in the asymptotical limit. Thus, the definition of the entanglement cost is: ⊗n ⊗rn EC = inf r : lim inf T r ρ − Ψ(Φ(2) ) . (21) n→∞ ψ where as a distance between states, the trace distance is used as a measure. ψ is a LOCC operation. Since the entanglement cost is defined as a rate, this means that, in the asymptotic limit, if we have m Bell states, we can obtain n = m/r = m/EC states of our choice. This gives a physical motivation of the entanglement cost as an entanglement measure. This measure is not easy to calculate, however, for pure states it is equal to the entropy of entanglement. Fortunately, bounds can be made on the entanglement cost, by using other measures of entanglement, in particular the entanglement of formation, discussed below. Also, the conditional entropy, C(A|B) = S(ρAB) − S(ρB) yields a lower bound for the entanglement cost, and the lower bound is −C(A|B).

Distillable entanglement Distillable entanglement is the opposite of entanglement cost: how many n states ρ does it take to produce m Bell states? This is the distillable entanglement, and is defined in a similar way as the entanglement cost: ⊗n rn ED = sup r : lim inf D Ψ ρ − Φ(2) . (22) n→∞ ψ Also in this case, we find that the entanglement distillation is a rate, which means that, given n states, whichever they might be, we can create m = nr = nED Bell states. As for the entanglement cost, it is not easy to evaluate, and it is only known for pure states (equal to entropy of entanglement, postulate 4) and for a few more specific states. Because of the similarity between the entanglement cost and the entanglement distillation, one can ask if they will give the same values. For pure states, this is the case, and then both measures will give the same result, the entropy of entanglement, as in postulate 4 above. As in the case of the entanglement cost, the conditional entropy C(A|B) can be used to gain a lower bound for the distillable entanglement, as ED(ρAB) ≥ max(−C(A|B), 0)

Entanglement of formation The entanglement of formation for a mixed state ρ = P i pi|ψiihψi| is deﬁned as ( ) X EF (ρ) = inf piE (|ψiihψi|) (23) i where E is the entropy of entanglement. This quantity is related to the entanglement cost, and it can be shown that ⊗n EF (ρ ) lim = EC (ρ) (24) n→∞ n

11 However, the asymptotic entanglement of formation is also quite hard to obtain. In fact, one of the major open problems in the ﬁeld is the question if this measure is additive. ⊗n EF (ρ ) This would mean that EF (ρ) = limn→∞ n , and EF is a much easier quantity to compute. Thus, if it was true that the entanglement of formation is additive, it would immediately make it possible to compute the entanglement cost. For two qubit states, one can calculate the entanglement of formation explicitly, using a quantity called the concurrence C:

C(ρ) = max (0, λ1 − λ2 − λ3 − λ4) (25)

The entanglement of formation is then given by ! 1 + p1 − C2(ρ) E (ρ) = s (26) F 2 where

s(x) = −x log x − (1 − x) log(1 − x) (27)

This is however only true for a two-qubit system. For more particles, the concurrence does not have a unique deﬁnition.

9 Entanglement for speciﬁc states

To show examples of entropy measures for specific states, we first define important states in quantum information: The Bell states, The GW states and the Z-states.

Bell states The Bell states, famous for playing a large role in the EPR thought experiments) are defined as 1 |B1i = √ (|0iA ⊗ |0iB + |1iA ⊗ |1iB) (28) 2 1 |B2i = √ (|0iA ⊗ |0iB − |1iA ⊗ |1iB) (29) 2 1 |B3i = √ (|0iA ⊗ |1iB + |1iA ⊗ |0iB) (30) 2 1 |B4i = √ (|0iA ⊗ |1iB − |1iA ⊗ |0iB) (31) 2 In the composite system AB, this is a pure state, which is non-separable. They are the maximally entangled states, can can thus be used as a relative scale, from which entanglement of other states are measured. If we consider the von Neumann entanglement entropy for AB, we find S(ρAB) = 0, since it is a pure state. However, tracing out A or B, we find that S(ρA) = S(ρB) = log 2. This is the entropy of a completely mixed state, which is what we get if we take the partial trace. However, from the postulates of entanglement measures, E(Bi) = log 2. This is because of postulate 4, E(|ψihψ|) = (S(T rB(|ψihψ|))). These states has played a large role in the early studies of entanglement, and are still used as resources for quantum computation. These maximally entangled states are also the basis for the entanglement measures entanglement cost and entanglement distillation.

12 GHZ states The Greenberger–Horne–Zeilinger (GHZ) state [3] is an example of a state which is not bipartite. It is composed of three or more subsystems, and is given by 1 |GHZi = √ (|0i⊗n + |1i⊗n) (32) 2 and in the speciﬁc case of n = 3, we get 1 |GHZi = √ (|000i + |111i) (33) 2 . This can then be seen as a generalization of the Bell states to more than two subsystems. In fact, there are some entanglement measures which deﬁnes the GHZ state to be maximally entangled, but the situation is more involved when dealing with multiple subsystems. By performing a local measurement in one of the subsystems, the state will collapse in the other subsystems, meaning that if A measures |0i, then the state of the system will be |000i, and all the subsystems will measure |0i. Thus this is a three-way entangled state.

W states The Werner (W) state 1 |W i = √ (|001i + |010i + |100i) (34) 3 is yet another example of a state consisting of three subsystems. This state is diﬀerent than the GHZ state though, in the sense that if A measures |1i, then the system collapses into |100i, but if A measures |0i, then the system collapses into a linear combination, |001i + |010i. Thus, this state cannot really be said to be a maximally entangled state, like the GHZ states.

10 Summary and outlook

The field of entanglement research is definitively a rich and ongoing field, with important proofs and theorems proved in recent times. There are certainly several big open questions in the field, many of them related to what is actually meant by entanglement, and how to quantify it in a useful way. Especially for multipartite systems there is a need for further study. One of the major open problems is to find a good calculatable measure for the entanglement cost, and here the additivity of the entanglement of formation is a very interesting open question. In this plethora of different entanglement measures, one can start to wonder if there will be a final theory of entanglement. This, in our opinion, is the case, that it probably never will be such a theory. Since entanglement is a resource for doing quantum com- puting, different algorithms need different definitions of what is meant by entanglement. One could hazard a guess, that when the size of experimental quantum computers will grow, then the quantum teleportation algorithms will talk about one measure of entanglement, while the groups doing quantum cryptography will talk about another measure. Indeed, there is nothing strange with this, since the most important characterisation of entanglement should be, in our opinion, what kind of algorithms can be done with it. This could suggest that a general theory of entanglement and entanglement measures is not critical for quantum information. Then most important property for entanglement

13 measure is, in our opinion, is that it is measurable in some way, and that it provides an ordering of entanglement, and that the same ordering can be used within similar quantum algorithms. In the end, the most important role of an entanglement measure is to help researchers quantify which states are better than others for speciﬁc algorithms.

References

[1] Quantum Computation and Quantum Information, Michael A. Nielsen, Isaac L. Chuang, 9th edition (2007)

[2] An introduction to entanglement measures, Martin B. Plenio and Shashank, Quant.Inf.Comput. 7, 1-51 (2007)

[3] Three qubits can be entangled in two inequivalent ways, W. D¨ur,G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000)