Substituting Quantum Entanglement for Communication

For decades, quantum entanglement has been the focus of much work in the foundations of quantum mechanics. Beyond this fundamental aspect, creating and manipulating of entangled states are essential for quantum information applications. Hence, quantum entanglement [1] has been viewed as an essential resource for quantum information processing. In recent years, much attention has been paid to the presence of entanglement in condensed-matter systems at finite temperatures. The state of a typical condensed-matter system at thermal equilibrium (temperature T ) is described by the density operator ρ = e−βH /Z where H is the Hamiltonian, Z = tre−βH is the partition function, and β = 1/kT where k is Boltzmanns constant. The entanglement associated with the thermal state ρ is referred to as thermal entanglement [2]. Superconducting Josephson devices (SJD’s) are simple but realistic and extensively studied solid systems. In fact, they may be thought of as two-state systems realizing the elementary unit of quantum information, known as a qubit. Moreover, Josephson devices [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] can be scaled up to a large number of qubits and their dynamics may be controlled by externally applied voltages and magnetic fluxes. Superconducting devices such as Cooper pair boxes, Josephson junctions, or superconducting quantum interference devices (SQUID’s) have been thus proposed and used as basic elements for the practical realization of quantum gates. It is an interesting topic to study thermal entanglement in systems involving SJD’s, such as SQUID’s. As we know, so far no study on thermal entanglement of Josephson qubit systems has been reported. The purpose of this paper is to investigate thermal entanglement of such a Josephson qubit system which consists of a flux

Circuit Theory and Computational ModelsG ILSON A NTONIO G IRALDI11LNCC–National Laboratory for Scientific Computing-Av.Getulio Vargas,333,25651-070,Petropolis,RJ,Brazil{gilson}@lncc.brAbstract.Concepts in Circuit Theory and Computational Paradigms1IntroductionIn this chapter we develop basic elements of another model of computation,the circuit model,which is equivalent to the recursive partial functions(RPF)model in terms of computational power,but is more convenient for many applications.Both models can be used to address the problem of characterizing the set of computable functions.However,sometimes the circuit models is more intuitive then the RPF model.Besides,for some applications the viewpoint of breaking the computation in a graph of gates is much more natural and convenient then functional operations,like composition and recursion.That is the case of quantum computing[4,6]and neural computation[1]paradigms,which are briefly discussed in sections3.2and3.1,respectively.In this Chapter,we summarize the basic elements of circuit theory and its application for the men-tioned computational paradigms.Section2will development elements of circuit theory.Then,quantum computing and neural networks are briefly discussed and their relationships with circuit theory pre-sented.The equivalence between RPF and circuit framework is proposed as an exercise(section4).2Circuit TheoryIn this section we start with a practical development found in[4],page129.So,we may say that a circuit is made up wires and gates.The wires carry information around and the gates(functions)perform simple computational tasks.The simplest gates are the logic operators presented on a previous Chapter and the corresponding functions are defined by their truth Tables.The Figure1shows the elementary gates and their graphical representation.Also,we can add to the set of elementary gates the identity(just a wire) and the FANOUT which replace a bit with two copies of itself.In Figure2it is pictured an example of a circuit that outputs the result of the Boolean expression x⊕y together with a carry bit set to1if x and y are both1,or0otherwise(check it!).Formally,let G be a set of gates that map several bits to one bit.For each n≥0,a circuit C n over the set G is a directed,acyclic graph with a list of input nodes(with no incoming edges),a list of output nodes(with no outgoing edges),and a gate in G labeling each non-input/non-output node.Given a binary input string(x1,x2,...,x n)∈{0,1}n,we label each input node x i or∼x i(NOT(x i)).For every other node v with m predecessors(y1,y2,...,y n),we recursively assign a value g(y1,y2,...,y n), where g is the gate that labels node v.The circuit C n outputs the value given by the list of output nodes.The size of a circuit C n is its number of nodes and the depth of C n is the length of the longest path from any input node to an output node.Figure1:Basic Gates.Figure2:Circuit example.A circuit family is a list of circuits C=(C1,C2,...,C n,...)where C i has i binary inputs.The family C computes a family of Boolean functions(f1,f2,...,f n,...),where f i is the boolean function computed by circuit C i.We say that the family C has size complexity s(n)and depth complexity d(n)if for all n≥0circuit C n has size at most s(n)and depth at most d(n).Size and depth are important complexity descriptions of circuits that respectively characterize the computational resources and the number of steps needed to compute a family of boolean functions.Important classes of circuits are the following ones.2.1Threshold CircuitsA weighted threshold function of weight bound w and threshold∆∈Z(set of integers)is a boolean function denoted by T n,∆w1,w2,...,w nand defined as follows[7]:T h n,∆w1,w2,...,w n:{0,1}n→{0,1},T h n,∆w1,w2,...,w n (x1,x2,...,x n)=1,ifni=1w i x i≥∆,T h n,∆w1,w2,...,w n(x1,x2,...,x n)=0,otherwise,where(w1,w2,...,w n)∈Z n,|w i|≤w for all i.If we restricted to the case of w i=1,i=1,2,..,n,we just denote T h n,∆:{0,1}n→{0,1}and call T h n,∆a threshold function with threshold∆.A threshold circuit is a circuit over the set of threshold gates(functions).Let T C(s(n),d(n))denote the collection of threshold circuits of size s(n)and depth d(n).The class of functions that can be computable by these circuits will be denoted as T C.We have analogous definitions for the collection of weighted threshold circuits and the class of functions that are computable by them,denoted by W T C(s(n),d(n))and W T C,respectively.An important theorem about threshold circuits is the following one:Theorem:Suppose an analytic function f(x)has a convergent Taylor series expansion f(x)= ∞n=0c n(x−x0)n over an interval|x−x0|<ε,where0<ε<1,and the coefficients are rational numbers c n=a n/b n,where a n,b n are integers with magnitude at most2n O(1).Then,f(x)can be efficiently computed over this interval,within accuracy2−n c,for any constant c≥1,by threshold circuits of polynomial size and simultaneous constant depth[7].2.2Equality Threshold CircuitsA weighted equality threshold function of weight bound w≥0is a boolean function defined by:Et n w1,...,w n:{0,1}n→{0,1},Et n w1,...,w n (x1,x2,...,x n)=0,ifni=1w i x i=0,Et n w1,...,w n(x1,x2,...,x n)=1,otherwise,for(w1,w2,...,w n)∈Z n,|w i|≤w for all i.An equality threshold circuit of weight bound w≥0is a circuit over the set of equality threshold gates such that all the weights are absolutely bounded by w.Let EC(s(n);d(n))denote the collection of equality circuits of size s(n)and depth d(n).EC will also be used to denote the class of functions computable by these circuits.It is possible to show that the weights of a EC circuit can be encoded appropriately in unitary matrices,and quantum entanglement and interference can be used to compute n i=1w i x i in one step. Thus,EC provide a good model to help in the combination of ideas from quantum computing and threshold circuits[3].3Circuit Theory and Computational ParadigmsIt is important to observe how the theory summarized above helps the development of non usual com-putational paradigms.Specifically,quantum computing and neural computing are two such frameworks for which the theory of circuits are fundamental.Artificial Neural Networks(ANNs)have been stud-ied and applied infields like cognitive science,pattern recognition and classification[1].On the other hand,in the last two decades we observed a growing interest in Quantum Computation and Quantum Information due to the possibility to solve efficiently hard problem for conventional computer science paradigms.Quantum computation and quantum information encompasses processing and transmission of data stored in quantum states(see[5]and references therein).Recently,several works have been done in artificial neural networks(ANNs)based on quantum theoretical concepts and techniques due to cognitive science and computer science aspects.The so called Quantum Neural Networks(QNNs)is an interesting area of research in thefield of quantum computation and quantum information[2].However,a key question about QNNs is what such an architecture will look like as an implementation on quantum hardware.Unfortunately,up to now,there is no a complete solution for the implementation of QNNs.Next,we show some details of both quantum computing and ANNs to explain the application of circuit theory in thesefields.3.1Classical Neural NetworksThefirst logical neuron was developed by W.S.McCulloch and W.A.Pitts in1943[1].It describes the fundamentals functions and structures of a neural cell reporting that a neuron willfire an impulse only if a threshold value is exceeded.Figure3:McCulloch-Pitts neuron model.Figure3shows the basic elements of McCulloch-Pitts model:x is the input vector,w are weights input associates,y is output,R is number of elements in input and f is the activation function thatdetermine the value in output.A simple choice for f is the signal function sgn(.).In this case,if the sum,across all the inputs with its respective weights exceeds the threshold b the output is1else the value of y is−1,that is:y=sgn(Ri=1w i x i−b).(1)But the McCulloch-Pitts neuron did not have a mechanisms for learning.Based on biological evidences,D.O.Hebb suggested a rule to adapt the weights input,which is interpreted as learning rule for the system[1].This biological inspired procedure can be expressed in the following manner:w newi=w old i+∆w i;∆w i=η(y desired−y)x i,(2) where w new and w old are adapted weights and initials weights respectively,ηis a real parameter to control the rate of learning and y desired is the desired(know)output.This learning rule plus the elements of Figure3is called the perceptron model for a neuron.Then,the learning typically occurs for example through training,or exposure to a know set of input/output data.The training algorithm iteratively adjusts the connection weights{w i}analogous to synapses in biological nervous.These connection weights store the knowledge necessary to solve specific problems.3.2Quantum ComputationIn practice,the most useful model for quantum computation is the Quantum Computational Network also called Deutsch’s model[3,6].The basic information unit in this model is a qubit[4],which can be considered a superposition of two independent states|0 and|1 ,denoted by|ψ =α0|0 +α1|1 , whereα0,α1are complex numbers such that|α0|2+|α1|2=1.A composed system with n qubits is described using N=2n independent states obtained through the tensor product of the Hilbert Space associated with each qubit.Thus,the resulting space has a natural basis that can be denoted by:{|i0i1...i n−1 ;i j∈{0,1}}.(3) This set can be indexed by|i ;i=0,1,...,N−1.Following the Quantum Mechanics Postulates, the state system|ψ ,in any time t,can be expanded as a superposition of the basis states:|ψ =N−1i=0αi|i ;N−1i=0|αi|2=1.(4)Entanglement is another important concept for quantum computation with no classical counterpart. To understand it,a simple example is worthwhile..Let us suppose that we have a composed system with two qubits.According to the above explana-tion,the resulting Hilbert Space has N=22independent states.Let the Hilbert Space associated with thefirst qubit(indexed by1)denoted by H1and the Hilbert Space associated with the second qubit(indexed by2)denoted by H2.The computational basis for these spaces are given by:{|0 1,|1 1}and{|0 2,|1 2},respectively.If qubit1is in the state|ψ 1=a10|0 1+a11|1 1and qubit2in the state|ψ 2=a20|0 2+a21|1 2,then the composed system is in the state:|ψ =|ψ 1⊗|ψ 2,explicitly given by:a1i a2j|i 1⊗|j 2.(5)|ψ =i,j∈{0,1}Every state that can be represented by a tensor product|ψ 1⊗|ψ 2belongs to the tensor product space H1⊗H2.However,there are some states in H1⊗H2that can not be represented in the form |ψ 1⊗|ψ 2.They are called entangled states.The Bell state(or EPR pair)presented next is a very known example:|ψ =12(|0 1⊗|0 2+|1 1⊗|1 2).(6) Trying to represent this state as a tensor product|ψ 1⊗|ψ 2,with|ψ 1∈H1and|ψ 2∈H2, produces an inconsistent linear system without solution.Entangled states are fundamental for teleportation[4].In recent years,there has been tremendous efforts trying to better understand the properties of entanglement,not only as a fundamental resource for the Nature,but also for quantum computation and information.The computation unit in Deutsch’s model consists of quantum gates which are unitary operators that evolves an initial state performing the necessary computation to get the desired result.A quantum computing algorithm can be summarized in three steps:(1)Prepare the initial state;(2)A sequence of (universal)quantum gates to evolve the system;(3)Quantum measurements.From quantum mechanics theory,the last stage performs a collapse and only what we know in advance is the probability distribution associated to the measurement operation.So,it is possible that the result obtained by measuring the system should be post-processed to achieve the target(quantum factoring(Chapter6of[6])is a nice example).The most used computational model in quantum computing is the circuit one because itfits very well with the Deutsch’s model.The Figure4shows a representation of a quantum circuit.Figure4:Example of quantum circuit with generic quantum gates denoted by U and D.4Exercises1.Show that any functions f:{0,1}n→{0,1}can be computed by a circuit.2.Show that the circuit model and the Partial Recursive Functions(PRF)model are equivalent;that is,show that any function f:{0,1}n→{0,1}can be computed by both the PRF and circuit model.3.Consider the weighted threshold circuits of section2.1.What kind of function f:{0,1}n→{0,1}can be computed by this model?Now,discuss the question:What kind of function f:{0,1}n→{0,1}can be computed by the McCulloch-Pitts(neuron)model of section3.1?4.Develop a circuit that computes the operation x−y(subtraction),for x,y∈{0,1}.The circuit must have x,y as inputs and two outputs,one of them used to send the signal of the operation,say,0for ”+”and1for”−”.5.Develop circuits to implement the following operations:a)x⊕y,for x,y∈{0,1}.b)Product:x.y,where x,y∈{0,1}.c)x.y⊕z.w,x,y,z,w∈{0,1}.d)Sum of two natural numbers m and n.e)Given a pair(x,y)∈{0,1}2,gives as output the pair(y,x).6.Simplify the following expression and give the corresponding circuit:∼((a∧∼b)∨c)∧((a∨c)∧∼c).7.The computation units in quantum computing are unitary operators.The whole computation, forget measurements now,can be seen as a sequence A1·A2...·A n(x),where A i are unitary matrices, the operation”·”is the usual matrix product,and x∈C n is a vector representing the initial state(C is the set of complex numbers).Therefore,answer the questions:(a)Are there functions f:{0,1}n→{0,1}that can not be computed by unitary gates?(b)Which kind of functions can be computed by unitary gates?(c)Based on items(a)-(b)discuss the power of quantum computing paradigm if compared with the usual one based on logic(electronic)circuits.References[1]R.Beale and T.Jackson.Neural Computing.MIT Press,1994.[2]Gilson A.Giraldi and Jean Faber.Quantum models for artificial neural networks.Technical report,virtual01.lncc.br/˜g iraldi/TechReport/QNN-Review.pdf.gz,2002.[3]S.Gupta and R.Zia.Quantum neural networks.Technical report,/PS。

Quantum entanglementMaciej LewensteinMaciej Lewenstein has obtained his degree in Physics from Warsaw University. From 1980 he worked at the Center for Theoretical Physics of the Polish Academy of Sciences. He received his doctoral degree in 1983 at the University of Essen and habilitation in 1986 in Warsaw. He became a full Professor in Poland in 1993. In 1995 he joined “Service de Photones, Atomes et Molecules” of CEA in Saclay. In 1998 he became a full professor and a head of the quantum optics theory group at the University of Hannover. In 2005 he started a new theory group at the “Insitut de Ciencias Fotoniques” in Barcelona. His research interests include: quantum optics, quantum information and statistical physics.Chiara MacchiavelloChiara Macchiavello finished her degree in Physics in 1991 and her PhD in 1995 at the University of Pavia. She held a post-doctoral for two years at the University of Oxford. Since 1998 she has been an Assistant Professor at the University of Pavia.Her research interests include quantum information processing and quantum optics.Dagmar BrussSince 2003 Dagmar Bruss is a professor at the Institute of Theoretical Physics at the University of Duesseldorf, Germany. Her research interests include the foundations of quantum information theory, classification of entanglement and quantum optical implementations of quantum computation.AbstractEntanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essential ingredient in some quantum cryptographic protocols and in quantum algorithms. We give a brief overview of the concept of entanglement in quantum mechanics, and discuss the major results and open problems related to the recent scientific progress in this field.IntroductionEntanglement is a property of the states of quantum systems that are composed of many parties, nowadays frequently called Alice, Bob, Charles etc. Entanglement expresses particularly strong correlations between these parties, persistent even in the case of large separations among the parties, and going beyond simple intuition.Historically, the concept of entanglement goes back to the famous Einstein-Podolski-Rosen (EPR) “paradox”. Einstein, who discovered relativity theory and the modern meaning of causality, was never really happy with quantum mechanics. In his opinion every reasonable physical theory should exhibit a so called local realism.Suppose that we consider two particles, one of which is sent to Alice and one to Bob, and we perform independent local measurements of “reasonable” physical observables on these particles. Of course, the results might be correlated, because the particles come from the same source. But Einstein wanted really to restrict the correlations for “reasonable” physical observables to the ones that result from statistical distributions of some hidden (i.e. unknown to us and not controlled by us) variables that characterize the source of the particles. Since quantum mechanics did not seem to produce correlations consistent with a local hidden variable (LHV) model, Einstein concluded that quantum mechanics is not a complete theory. Erwin Schrödinger, in answer to Einstein’s doubts, introduced in 1935 the term “Verschränkung” (in English “entanglement”) in order to describe these particularly strong quantum mechanical correlations.Entanglement was since then a subject of intense discussions among experts in the foundations of quantum mechanics and philosophers of science (and not only science). It took, however, nearly 30 years until John Bell was able to set the framework for experimental investigations on the question of local realism. Bell formulated his famous inequalities, which have to be fulfilled in any multiparty system described by a LHV model. Alain Aspect and coworkers in Paris have demonstrated in their seminal experiment in 1981 that quantum mechanical states violate these inequalities. Recent very precise experiments of Anton Zeilinger’s group in Vienna confirmed fully Aspect’s demonstrations. All these experiments indicate the correctness of quantum mechanics, and despite various loopholes, they exclude the possibility of LHV models describing properly the physics of the considered systems.Entanglement has become again the subject of cover pages news in the 90’s, when quantum information was born. It was very quickly realized that entanglement is one of the most important resources for quantum information processing. Entanglement is a necessary ingredient for quantum cryptography, quantum teleportation, quantum densecoding, and if not necessary, then at least a much desired ingredient for quantum computing.At the same time the theory of entanglement is related to some of the open questions of mathematics, or more precisely linear algebra and functional analysis. A solution of the entanglement problem could help to characterize the so called positive linear maps, i.e. linear transformations of positive definite operators (or physically speaking quantum mechanical density matrices, see below) into positive definite operators.Entanglement of pure statesIn quantum mechanics (QM) a state of a quantum system corresponds to a vector |Psi> in some vector space, called Hilbert space. Such states are called pure states. One of the most important properties of QM is that linear superpositions of state-vectors are also legitimate state-vectors. This superposition principle lies at the heart of the matter-wave dualism and of quantum interference phenomena.Entanglement is also a result of superposition, but in the composite space of the involved parties. Let us for the moment focus on two parties, Alice and Bob. It is then easy to define states which are not entangled. Such states are product states of the form |Φ>= |a>|b>, i.e. Alice has at her disposal |a>, while Bob has |b>. Product states obviously carry no correlations between Alice and Bob. Entangled pure states may be now defined as those which are superpositions of at least two product states, such as|Φ> = α1|a1>|b1> + α2|a2>|b2> + etc.but cannot be written as a single product state in any other basis. All entangled pure states contain strong quantum mechanical correlations, and do not admit LHV models.Entanglement of mixed states and the separability problemVerify whether a given state-vector is a product state or not is a relatively easy task. In practice, however, we often either do not have full information about the system, or are not able to prepare a desired state perfectly. In effect in everyday situations we deal practically always with statistical mixtures of pure states. There exists a very convenient way to represent such mixtures as so called density operators, or matrices. A density matrix rho corresponding to a pure state-vector |Φ> is a projector onto this state. More general density matrices can be represented as sums of projectors onto pure state-vectors weighted by the corresponding probabilities.The definition of entangled mixed states for composite systems has been formulated by Reinhard Werner from Braunschweig in 1989. In fact, this definition determines which states are not entangled. Non-entangled states, called separable states, are mixtures of pure product states, i.e. convex sums of projectors onto product vectors:ρ = Σι pi|ai>|bi><ai|<bi|, (*)where 0 ≤ pi ≤ 1 are probabilities, i.e. Σιpi= 1. The physical interpretation of thisdefinition is simple: a separable state can be prepared by Alice and Bob by using local operations and classical communication. Checking whether a given state is separable or not is a notoriously difficult task, since one has to check whether the decomposition (*) exists or not. This difficult problem is known under the name of “separability or entanglement problem”, and has been a subject of intensive studies in the recent years.Simple entanglement criteriaThe difficulty of the separability problem comes from the fact that rho admits in general an infinite number of decompositions into a mixture of some states, and one has to check whether among them there exists at least one of the form (*). One of the most powerful necessary conditions for separability has been found by one of the fathers of quantum information, the late Asher Peres. Peres (Technion, Haifa) observed that since Alice and Bob may prepare separable states using local operations, Alice may safely reverse the time arrow in her system, which will change the state, but will not produce something unphysical. In general, such a partial time reversal is not a physical operation, and can transform a density operator (which is positive definite) into an operator that is no more positive definite. In fact this is what happens with all pure entangled states. Mathematically speaking partial time reversal corresponds to partial transposition of the density matrix (only on Alice's side). We arrive in this way at the Peres criterion: If a stateρis separable then its partial transposition has to be positive definite.This criterion is usually called positive partial transpose condition, or shortly PPT condition. Amazingly, the PPT condition is not only necessary for separability, but it is also a sufficient condition for low dimensional systems such as two qubits (dimension 2x2)and a system composed of one qubit and one qutrit (dimension 2x3). In higher dimensions, starting from 2x4 and 3x3, this is no longer true: there exist entangled states with positive partial transpose, which are called PPT entangled states.There exist several other necessary or sufficient separability criteria which have been established and frequently discussed in recent years. For example, states that are close to the completely chaotic state (whose density operator is equal to the normalized identity) are necessarily separable. There exist also other criteria that employ entropic inequalities, uncertainty relations, or an appropriate reordering of the density matrix (so called realignment criterion) etc. There exists, however, no general simple operational criterion of separability that would work in systems of arbitrary dimension.Entanglement witnessesThe set of all states P is obviously compact and convex. If ρ1 and ρ2are legitimate states,so is their convex mixture. The set of separable states S is also compact and convex (seeFigure 1). From the theory of convex sets and Hahn-Banach theorem we conclude that for any entangled state there exists a hyperplane in the space of operators separating rhofrom S. Such a hyperplane defines uniquely a Hermitian operator W (observable) which has the following properties: The expectation value of W on all separable states, <W> ≥ 0, whereas its expectation value on ρ is negative, i.e. <W>ρ< 0.Figure 1Such an observable is for obvious reasons called entanglement witness, since it “detects” the entanglement of ρ. Every entangled state has its witnesses; the problem obviously is to find appropriate witnesses for a given state. To find out whether a given state is separable one should check whether its expectation value is non-negative for all witnesses. Obviously this is a necessary and sufficient separability criterion, but unfortunately it is not operational, in the sense that there is no simple procedure to test for all witnesses.Nevertheless, witnesses provide a very useful tool to study entanglement, especially if one has some knowledge about the state in question. They provide a sufficient entanglement condition, and may be obviously optimized (see Figure 2) by shifting the hyperplane in a parallel way towards S.Figure 2Bell inequalitiesAfter introducing the concept of separability and entanglement for mixed states, it is legitimate to ask what is the relation of mixed state entanglement and the existence of a LHV model, which requires that the state cannot violate any of the Bell-like inequalities. Let us discuss an example of such inequalities, the so called Clauser-Horne-Shimony- Holt inequality for two qubits. Let us assume that Alice and Bob measure two binary observables each, namely A 1, A 2, and B 1, B 2. The observables are random variables taking the values +1 or − 1, correlated possibly through some dependence on local hidden variables. It is easy to see that in the classical world, if B 1 + B 2 is zero, then B 1 − B 2 is either +2 or −2, and vice versa. Therefore if we define s = A 1(B 1 + B 2 ) + A 2 (B 1 − B 2 ) , we obtain that 2 ≥ s ≥ −2. This inequality holds also after averaging over various realizations. On the other hand, it can be shown that by taking suitable sets of observables for Alice and Bob we can find pure and even mixed quantum states that violate this inequality.Are Bell-like inequalities similar in this respect to witnesses, i.e. for a given entangled state can one always find a Bell-like inequality that “detects” it? The answer to this question is no, and has been already given by R. Werner in 1989. Even for two qubits there exist entangled states that admit an LHV model, i.e. cannot violate any Bell-like inequality.This observation indicates already that there is more structure in the “eggs” of Figure 1 and Figure 2. Separable states are evidently inside the PPT egg, according to the Peres condition. They admit an LHV model, i.e. they are also inside the LHV egg. But what about PPT entangled states? Do they violate some Bell-like inequality? Peres has formulated a conjecture that this not the case, and there is a lot of evidence that this conjecture is correct, although a rigorous proof is still missing.The distillability problem and bound entanglement Above we have classified quantum states according to the property of being either separable or entangled. An alternative classification approach is based on the possibility of distilling the entanglement of a given state. In a distillation protocol the entanglement of a given state is increased by performing local operations and classical communication on a set of identically prepared copies. In this way one obtains fewer, but “more entangled”, copies. This kind of technique was originally proposed in 1996 by Bennett and coworkers in the context of quantum teleportation, in order to achieve faithful transmission of quantum states over noisy channels. It also has applications in quantum cryptography as a method for quantum privacy amplification in entanglement based protocols in the presence of noise, as pointed out by David Deutsch and coworkers from Oxford.The distillability problem poses the question whether a given quantum state can be distilled or not. A separable state can never be distilled because the average entanglement of a set of states cannot be increased by local operations. Furthermore, the positivity of the partial transpose ensures that no distillation is possible. Thus, a given PPT entangled state is not distillable, and is therefore called bound entangled. There mayeven exist undistillable entangled states which do not have the PPT property. However, this conjecture is not proved at the moment.The first example of a PPT entangled state has been found by Pawel Horodecki from Gdansk in 1997. These states are so called edge states, which means that they cannot be written as a mixture of a separable state and a PPT entangled state. Particularly simple families of states have been suggested by Charles Bennett and coworkers at IBM, New York. They have found the so called unextendible product bases (UPB), i.e. sets of orthogonal product state-vectors, with the property that the space orthogonal to this set does not contain any product vector. It turns out that the projector onto this space is a PPT state, which obviously has to be entangled since it does not contain any product vector in its range (note that all state-vectors in the decomposition of a separable state ρinto a mixture of product states belong automatically to the range of ρ).The existence of bound entanglement is a mysterious invention of Nature. It is an interesting question to ask whether bound entanglement is a useful resource to perform quantum information processing tasks. It was shown so far that this is not the case for communication protocols such as quantum teleportation and quantum dense coding (i.e.a protocol that allows to enhance the transmission of classical information, using entanglement). However, surprisingly, it is possible to distill a secret key in quantum cryptography, starting from certain bound entangled states.Entanglement detectionAs discussed above, entanglement is a precious resource in quantum information processing. Typically in a real world experiment noise is always present and it leads to a decrease of entanglement in general. Thus, it is of fundamental interest for experimental applications to be able to test the entanglement properties of the generated states. A traditional method to this aim is represented by the Bell inequalities, a violation of which indicates the presence of entanglement. However, as mentioned above, not every entangled state violates a Bell inequality. So, not all entangled states can be detected by using this method.Another possibility is to perform complete state tomography, which allows determining all the elements of the density matrix. This is a useful method to get a complete knowledge of the density operator of a quantum system, but to detect entanglement it is an expensive process as it requires an unnecessary large number of measurements. If one has certain knowledge about the state the most appropriate technique is the measurement of the witness observable, which can be achieved by few local measurements. A negative expectation value clearly indicates the presence of entanglement.All these methods have been successfully implemented in various experiments. Recently another method for the detection of entanglement was suggested based on the physical approximation of the partial transpose. It remains a challenge to implement this idea in the laboratory because it requires the implementation of non local measurements.Entanglement measuresWhen classifying a quantum state as being entangled, a natural question is to quantify the amount of entanglement it contains. For pure quantum states there exists a well defined entanglement measure, namely the von Neumann entropy of the density operator of a subsystem of the composite state. For mixed states the situation is more complicated. There are several different possibilities to define an entanglement measure. The so called entanglement cost describes the amount of entanglement one needs in order to generate a given state. An alternative measure is the entanglement of formation, which is a more abstract definition. A further possibility to quantify entanglement is given by the minimum distance to separable states. Finally, motivated by physical applications, one can introduce the distillable entanglement which quantifies the extractable amount of entanglement.Unfortunately all of these quantities are very difficult to compute in general. For example, in order to determine the entanglement of formation one has to find the decomposition of the state that leads to the minimum average von Neumann entropy of a subsystem and this is a very challenging task. So far a complete analytical formula for the entanglement of formation only exists for composite systems of two qubits.Entanglement in multipartite systemsSo far, we have restricted ourselves to the case of composite systems with two subsystems, so called bipartite systems. When considering more than two parties, i.e multipartite systems, the situation becomes much more complex. For example, for the most simple tripartite case of three qubits, a pure state can be either completely separable, or biseparable (i.e. one of the three parties is not entangled with the other two), or genuinely entangled among all three parties. The latter class again consists of inequivalent subclasses, the so called GHZ and W states. This concept can be generalized to mixed states. For more than three parties it is easy to imagine that the number of subclasses grows fast.In recent years there has been much progress in the creation of multipartite entangled states in the laboratory. The existence of genuine multipartite entanglement has also been demonstrated experimentally by using the concept of witness operators.Even if the full classification of multipartite entanglement is a formidable task, certain classes of states, the so called graph states, have been completely characterized and shown to be useful both for quantum computational and quantum error correction protocols. Moreover, a deeper understanding of entanglement has proved to be very fruitful in connection with statistical properties of physical systems. All of these problems are discussed in more details in other sections of this publication.References[1] Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935)[2] J.S. Bell, Physics 1, 195 (1964)[3] P. Horodecki, Phys. Lett. A 232, 333 (1997)[4] M. Lewenstein et al., J. Mod. Opt. 47, 2481 (2000)[5] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)[6] E. Schrödinger, Naturwissenschaften 23, 807 (1935)[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989) Contact information of the author of this article Maciej LewensteinInstitut de Ciènces Fotòniques (ICFO)C/Jordi Girona 29, Nexus 2908034 BarcelonaSpainEmail: maciej.lewenstein@icfo.esChiara MacchiavelloIstituto Nazionale di Fisicadella Materia, Unita' di Pavia Dipartimento di Fisica "A. Volta"via Bassi 6I-27100 PaviaItalyEmail: chiara@unipv.itProf. Dr. Dagmar BrussInst. fuer Theoretische Physik IIIHeinrich-Heine-Universitaet Duesseldorf Universitaetsstr. 1, Geb. 25.32D-40225 Duesseldorf,GermanyEmail: bruss@thphy.uni-duesseldorf.de。

Journ al of Sha ngha i Univ er sity (En glish Edition ),2006,10(3):215-218Article ID :1007 6417(2006)03 0215 04Entanglement of atoms in Tavis Cummings modelDONG Chuan hua (董传华), ZHANG Ya li (张亚利)Depar tm ent of Phy sics ,College of Scien ces ,Shan ghai Un iver sity ,Shan gha i 200444,P .R .ChinaReceived Sep.24,2004; Revised Ja n.7,2005DO NG Chuan hua,Prof.,E mail:chdong@ma Abstract Entanglem ent is used to measure correlation between separated subsystems.von Neumann entropy is used to study evolu tions of entanglement of atoms in processes of interaction between atoms with the field prepared in coherent state.The effects of field intensity and detuning on entanglement a re investiga ted.It is shown that the entanglement exhibited osc illations in its evolutions,their amplitudes and mean values decrease with inc reasing field intensity.Oscillation frequenc ies increase with detuning,but the maximum values a re almost independent of detuning.Key words entanglem ent,entangled state,Jaynes Cummings model.PACS 2001 42.50. P1 IntroductionSuperposition principle is one of the fundamentals in quantum m echanics.A two level atom may be in the ground state |0 ,the excited state |1 o r the superpo sition states C 0|0 +C 1|1 .The coefficients C 0and C 1are complex probability amplitudes of |0 and |1 respectively.For a system consisting of two atoms,the state of total system may be in the direct p roduct states |0,1 or |1,0 .Acco rding to the superposition principle,the total system may also be in the superpo sition states 1 2(C 0|0,1 +C 1|1,0 ).This kind of superposition states of multi particle is called entan gled state.Recently,people have devoted much attention to the applications of entangled states in quantum information,such as quantum teleporting [1,2],quantumcopying and cloning [3],quantum cryptography andquantum secret sharing [4],and quantum computation [5].Various schemes of preparing entangledstates have been proposed [6].The concept of entan gling has been generalized to the states of multi parti cle system,i .e .,G reenberger Ho rne Zeilinger states.Various methods of preparing G H Z states have alsobeen proposed [7,8].Entangling can be realized with non local correlation and entanglement is defined to measure the degree ofnon local correlation.Some definitions of entanglement have been given [9,10].Zuo and Xia have studied the evolution properties of three body entanglement of two identical two level atoms in the presence of fieldin vacuum [11].Rendell and Rajagopal have studied the entanglement of initially entangled mixed state in damped Jaynes Cummings model and found the timeevolutions of the concurrence lower bound [12].In this paper,we will use the partial entropy to study evolu tions of entanglement for two atoms in the presence of the field in coherent state with Tavis Cummings model.Effects of light intensity and the detunning on evolutions of entanglement are investigated.2 Definition of entanglementConsidering a system consisting of two atoms A and B,its entanglement (partial entropy entanglement)E is defined with von Neumann entropy,i .e .E =S ( A )=S ( B ),(1)where A and B are reduced density operators of at oms A and B.von Neumann entropy is a generalization of Shannon entropy in classic system to quantum sys tem,which describes the quantum co rrelation betweenthe sub systems [13].von Neumann entropy is defined asS ( )=-tr( log 2 ).(2)With Schmidt decomposition,the density operato rs of atoms A and B can be written in the fo rmA=tr B( A B)=mj=12j|j A A!j|,B=tr A( A B)=mj=1 2j|j B B!j|,(3)where m∀min{dim H A,dim H B},H A and H B are the Hilbert subspaces of atoms A and B,respectively. It follows that A and B have m non zero eigenvalues j(j=1-m).The relation between E and j isE=-mj=1j log2 j.(4)For two atoms in the entangled states| AB=C1|1,1 +C2|1,0 +C3|0,1+C4|0,0 ,(5) the eigenvalues of A(or B)arej=121-(-1)j1-4|C1C4-C2C3|2,j=1or2.(6)The partial entropy entanglement can be calculated as follows.E=-( 1log2 1+ 2log2 2).(7) 3 Entanglement of two atoms in TavisCummings modelConsidering a system consisting of two level atoms A and B,the ground state and excited state are denot ed|0 and|1 respectively.The light field is initially prepared in the coherent state|! =nf n|n ,where f n=exp(- n 2)!n n!and n is mean photon num ber.Two atoms are in the entangled state|(0) ato m=1 2(|0,1 +|1,0 ).(8) This entangled state is one of Bell bases,which is maximally entangled state with entanglement E=1. The state evolves from the initial state to|(t) at time t>0|(t) =C1(t)|1,1;n-1 +C2(t)|1,0;n+C3(t)|0,1;n +C4(t)|0,0;n+1 ,(9) where the coefficients C j(t)(j=1-4)are to be de termined.In the interaction picture,the Hamiltonian of the system consisting of two atoms and a field in Tavis Cummings model is,H I=g(a+S-e i∀t+S+a e-i∀t),(10)where∀=#-#0,#0is the transiting frequency of at oms and#is the f requency of field,S+=S(A)++S(B)+, S-=S(A)-+S(B)-,and S(j)#(j=A or B)are the transit ing operato rs of atom A(or B).Substituting the inter action Hamiltonian Eq.(10)and the state vector Eq.(9)into the Schr dinger equation and solving it using a standard p rocedure with the initial condition Eq.(8), the coefficients C j(t),(j=1∃4)in Eq.(9)can be obtained as follows.C1(t)=-2n gm1∃1-∀[ei(∃1-∀)t-1]+m2∃2-∀[ei(∃2-∀)t-1]+m3∃3-∀[ei(∃3-∀)t-1],C2(t)=C3(t)=m1e i∃1t+m2e i∃2t+m3e i∃3t,C4(t)=-2n+1gm1∃1+∀[ei(∃1+∀)t-1]+m2∃2+∀[ei(∃2+∀)t-1]+m3∃3+∀[ei(∃3+∀)t-1],(11)wherem1=f n2∃2∃3+2(2n+1)g2(∃1-∃2)(∃1-∃3),m2=f n2∃1∃3+2(2n+1)g2(∃2-∃1)(∃2-∃3),m3=f n2∃1∃2+2(2n+1)g2(∃3-∃1)(∃3-∃2)(12)and∃1=2r1 3cos%,∃2=2r1 3cos(%+2& 3),∃3=2r1 3cos(%+4& 3),(13) r=[2(2n+1)g2+∀2]3 27,%=1 3arccos(-∀g2 r).(14) In the case of∀=0,C j(t)can be reduced toC1(t)=-i2nf n g sin(∋n t) ∋n,C2(t)=C3(t)=2-1 2f n cos(∋n t),C4(t)=-i2(n+1)f n g sin(∋n t) ∋n,(15) where Rabi frequency is2∋n and.(16)216J our nal of Shan gha i Univer sityThe reduced density operator of atom A can beobtainedA = n(|C 1|2+|C 2|2)|1 !1|+(|C 3|2+|C 4|2)|0 !0|+(C *2C 4+C *1C 3)|0 !1| +(C 2C *4+C 1C *3)|1 !0|,(17)its eigenvalues are =0.5(1#1-(),(18)where(=4[ n (|C 1|2+|C 2|2) n(|C 3|2+|C 4|2)- n(C 1C *3+C 2C *4)2].(19)Using Eq.(7),the evolutions of entanglement can be obtained,which are shown in Fig.1for the case of resonance.The atoms A and B are prepared initially in Bell base |1,0 +|0,1 ,so the initial entanglement is 1.In the processes of interaction between the atoms and the field,the state of the system consisting of at oms A and B will deviate from this Bell base.In case of resonance,the states will evolve to the state | (t ) at a time t >0,which can be rewritten as | (t ) =1 2 nf n {cos (∋n t )(|1,0;n +|0,1;n ) -i sin (∋n t )[1-1 (2n +1)|1,1n -1+1+1 (2n +1)|0,0;n +1 ]}=1 2 n f n {cos (∋n t )(|1,0;n +|0,1;n ) -i sin (∋n t )[(|1,1;n -1 +|0,0;n +1 )-1 (4n +2)(|1,1;n -1 -|0,0;n +1 )+%]}.(20)It can be seen from Eq.(20)that the other two Bellbases |1,1 #|0,0 appear in atom field interaction.These Bell bases evolve according to cosine and sine with frequency ∋n that is different for various pho ton numbers n .Superposing these Bell bases form the os cillations of entanglement in evolution. In a weaker field ( n <4),the valley values of oscil lations rise obviously with increasing field intensity so that the mean values of entanglements in the oscil lations increase (see Fig.1(a)-(c)).In a stronger field ( n >4),the mean values of entanglements de crease rapidly (see Fig.1(d)and (e)),and the am plitudes shrink sharply with enhancing the field.When n =15,entanglement is maintained roughly constant inevolution.Thus,enhancing the field makes the atomsdetangle.This can be explained by the fact that thequantum co rrelation is weakened due to enhancing the interaction of two atoms with the field respectively in a stronger field.Fig.1 Entanglement of two atoms in the case of resonanc e((a) n =0.2,(b) n =1.0,(c) n =4.0,(d) n =9.0,(e) n =15.0)Evolution of entanglement in off resonance are shown in Fig.2(for weaker field, n =0.2)and Fig.3(for stronger field, n =9).From these figures,it can be seen that oscillation frequencies increase and the amplitudes shrink with increasing detuning in a weaker field.Except these oscillations,there are slow varia tions with large amplitudes in evolution for a stronger field and large detuning.The minimum values rise in a weaker field (see Fig.2)and drop in a stronger field (see Fig.3)with increasing the detuning.The maxi mum values of entanglement in the evolution are inde pendent of detuning almost,while dependent only on the field intensity.In summary,in interactions between atoms and the field,entanglements of two atoms will oscillate due to the Rabi oscillations.The field intensity and the detun ing affect evolutions of atom entanglement.Entanglements of two atoms depend on the field in tensity.The mean entanglem ents increase slightly in a weak field and decrease obviously in a strong field217Vol.10 No.3 Jun.2006DONG C H,et al .: Entanglem ent of atom s in Tavis Cummings modelFig.2 Entanglement of two atoms in the case of off resonanc efor n =0.2((a)∀=0.5g ,(b)∀=2.0g ,(c)∀=4.0g ,(d)∀=10.0g ,(e)∀=20.5g)Fig.3 Entanglement of two atoms in the case of off resonanc efor n =9.0((a)∀=0,(b)∀=0.5g ,(c )∀=1.0g ,(d)∀=4.0g ,(e)∀=10g )with increasing field intensity.It follows that a strong field makes the atoms detangle.The amplitudes of oscillations of entanglements shrink sharply with in creasing field intensity.The maximum values of entan glement are independent of detuning but dependent on the field intensity.The oscillation frequencies increase and the amplitudes shrink in a weak field with the increasing detuning.References[1] Bennett C H,Wiesner S munication v ia one andtwo particle operators on Einstein Podolsky Rosen states [J].Phys .Rev .Lett .,1992,69(20):2881-2884.[2] Bennett C H,Brassard G,Cr peau C,et al .Teleportingan unknown quantum state via dual classical and Ein stein Podolsky Rosen c hannels [J].Phy s .Rev .Lett .,1993,70(13):1895-1898.[3] Buzek V,H illery M.Quantum copying:Beyond the nocloning theorem[J].Phys .Rev .,1996,A54(3):1844-1852.[4] Hillery M,Buzek V,Berthiaume A.Quantum secret sharing[J].Phys .Rev .,1999,A59(3):1829-1834.[5] Anders S rensen,Klaus M lme r.Entanglement and quantum computation with ions in the therm al motion [J ].Phys .Rev .,2000,A 62(2):022311 1-022311 11.[6] Cirace J I,Zoller P.Preparation of macroscopic superposition in many atom systems [J ].Phy s .Rev .,1994,A50(4):R2799-R2802.[7] Yao C M,Guo G C.Generation of spin type G HZ statesof the cavity field in squeezed coherent states[J].Acta Phy sica Sinica ,2001,50(1):59-62(in Chinese).[8] Liu X,Li H C.Prepa ration of multi atom GHZ states v iathe Raman interaction of V type three level atoms and one cavity field[J].Acta Phy sica Sin ica ,2001,50(9):1689-1692(in Chinese).[9] Vedral V,Plenio M B,Rippin M A,et al .Quantifyingentanglement[J].Phys .Rev .Lett .,1997,78(12):2275-2279.[10] Vedral V,Plenio M B.Entanglement measures and purification procedures[J].Phy s .Rev .,1998,A57:1619-1633.[11] Zhou Z C,Xia Y J.The evolution property of three bodyentanglement measure in Tavis Cumm ings model [J ].Acta Physica Sinica ,2003,52(11):2687-2693(in Chinese).[12] Rendell R W,Rajagopal A K.Revivals and entanglementfrom initially entangled mixed sta tes of a damped Jaynes Cumm ings model [J].Phys .Rev .,2003,A67(6):062110 1-062110 11.[13] von Neumann.Mathem atical F oundat ion s of Qua ntumMechan ics [M].Princeton University Press,Princeton,NJ,1955.(Editor YAO Yu e yuan )218J our nal of Shan gha i Univer sity。