Pawe l HorodeckiFaculty of Applied Physics and Mathematics,

Technical University of Gdansk, 80-952 Gdansk, Poland.

Artur K. EkertCentre for Quantum Computation,

Clarendon Laboratory, University of Oxford,

Parks Road, Oxford OX1 3PU, U.K.

Carolina Moura AlvesCentre for Quantum Computation,

Clarendon Laboratory, University of Oxford,

Parks Road, Oxford OX1 3PU, U.K.

Daniel K. L. OiCentre for Quantum Computation,

Clarendon Laboratory, University of Oxford,

Parks Road, Oxford OX1 3PU, U.K.

L. C. KwekDepartment of Natural Sciences,

National Institute of Education, Nanyang Technological University,

1 Nanyang Walk, Singapore 637616.

Dagomir KaszlikowskiDepartment of Physics, Faculty of Science,

National University of Singapore, Lower Kent Ridge,

Singapore 119260.

Micha l HorodeckiInstitute of Theoretical Physics and Astrophysics,

University of Gdansk, 80-952 Gdansk, Poland.

Direct Detection

of Quantum Entanglement

Received March 31, 2003

After playing a significant role in the development of the foundations of quantum mechan-

ics, entanglement has been recently rediscovered as a new physical resource with potential

commercial applications, ranging from quantum cryptography to very precise frequency stan-

dards. Thus, the detection of quantum entanglement is vital in the experimental context.

We present a direct method of detecting the presence of entanglement and we put it in the

context of quantum information science.

1. Entanglement

Probably the best way to agitate a group of jaded, but philosophically-inclined,physicists is to buy them a bottle of wine and mention interpretations of quantum

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mechanics. Opening Pandora’s box isn’t much different, despite the fact that, asfar as lip-service goes, the orthodoxy established by Niels Bohr over 60 years ago,known as the “Copenhagen interpretation”, still effectively holds sway. It seems thateverybody agrees with the formalism of quantum mechanics, but no one agrees on itsmeaning. Quantum theory, according to the “Copenhagen interpretation”, providesmerely a calculational procedure and does not attempt to describe objective physicalreality. According to Bohr

There is no quantum world. There is only an abstract physical descrip-tion. It is wrong to think that the task of physics is to find out how thenature is. Physics concerns what we can say about nature.1

A very defeatist view indeed, but hardly surprising. The intellectual atmosphereand the philosophy of science at the time were dominated by positivism that emergedin Vienna in the early 1920s. According to this school of thought all statements otherthan those describing or predicting observations are meaningless2.

One of the first who found the pragmatic instrumentalism of Bohr unacceptablewas Albert Einstein. In 1927 during, the fifth Solvay Conference in Brussels, Einsteindirectly challenged Bohr over the meaning of quantum theory. This triggered theBohr-Einstein debate, which lasted almost three decades and which, among manyother things, brought quantum entanglement into the remit of modern physics. Inearly 1935, Einstein, together with Boris Podolsky and Nathan Rosen, published aclassic paper which featured a composite quantum system consisting of two distantparticles with entangled wave function [1]. The entanglement of the wave function assuch was never mentioned in the paper, as the authors focused on the incompletenessof quantum mechanical description of reality. Erwin Schrodinger, at the time a fellowof Magdalen College in Oxford, seems to have been the first to spot it and to realizeits importance. In August 1935, the Cambridge Philosophical Society received apaper written by Schrodinger, and communicated by Max Born, titled “Discussion ofprobability relations between separated system” [2]. The paper was read 28 October1935. Its opening paragraph states

When two systems, of which we know the states by their respective repre-sentatives, enter into temporary physical interaction due to known forcesbetween them, and when after a time of mutual influence the systemsseparate again, then they can no longer be described in the same way asbefore, viz. by endowing each of them with a representative of its own.I would not call it one but rather the characteristic trait of quantummechanics, the one that enforces its entire departure from classical linesof thought. By the interaction the two representatives (or ψ-functions)have become entangled...3

1Quoted after Aage Petersen in Niels Bohr: a centenary volume, eds. A.P. French and

P.J. Kennedy, Harvard University Press, 1985.2It is quite amusing to notice that according to this criterion positivism is itself meaningless.3The emphasized “one” and “the” are due to Schrodinger.

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After playing a significant role in the development of the foundations of quan-tum mechanics [3], entanglement has been recently rediscovered as a new physicalresource with potential commercial applications such as, for example, quantum cryp-tography [4], better frequency standards [5] or quantum-enhanced positioning andclock synchronization [6]. On the mathematical side, the studies of entanglementhave revealed very interesting connections with the theory of positive maps [7, 8].The capacity to generate entangled states is one of the basic requirements for buildingquantum computers. Hence, efficient experimental methods for detection, verificationand estimation of quantum entanglement are of great practical importance. Here, wedescribe an experimentally viable, direct detection of quantum entanglement whichhas been proposed recently by Horodecki and Ekert [10]. It is efficient and does notrequire any a priori knowledge about the quantum state. In the particular case oftwo entangled qubits, it provides an estimation of the amount of entanglement. Wealso discuss it in context of some aspects of quantum information theory.

2. Specification of the problem

Suppose we are given n pairs of particles, all in the same quantum state described bysome density operator %, which is unknown. We need to decide whether the particlesin each pair are entangled or not. From a mathematical point of view we need toassert whether % can be written as a convex sum of product states [12],

% =

k∑

i

pi; |αi〉〈αi| ⊗ βi〉〈βi|, (2.1)

with |αi〉 and |βi〉 pertaining to different particles in the pair and∑

i pi = 1. It isassumed that the Hilbert spaces associated with each particle are of finite dimensiond (taken to be the same for the two particles), so that one can always find k 6 d2. If% were known, we could try either to find the decomposition (1) directly or to use oneof the mathematical separability criteria [8]. For sufficiently large n, we may indeedstart with the quantum state estimation. However, this involves estimating d4 − 1real parameters of %, most of which are irrelevant in the context of the entanglementdetection.

Another possibility is the recourse to a particular class of two-particle observ-ables, called entanglement witnesses, which can detect quantum entanglement is somespecial cases (see [9, 8]). They have positive mean values on all separable states andnegative on some entangled states. Therefore, any individual entanglement witnessleaves many entangled states undetected. When % is unknown, we need to checkinfinitely many witnesses, which effectively reduces this approach to the quantumstate estimation. However, if some information about the state is available, the en-tanglement witnesses approach may be very convenient [11]. In this paper we willfocus on the case of unknown %.

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3. What are positive maps good for?

In the following we describe a direct method of detecting quantum entanglementwithout invoking the state estimation.

We construct a measurement, as powerful in detecting quantum entanglement asthe best mathematical test based on positive maps [7], which can be performed on allcopies of %. The measurement can be viewed as two consecutive physical operations:firstly, we construct a transformation which maps % into an appropriate state %′;secondly, we measure the lowest eigenvalue of %′. This eigenvalue alone serves as aseparability indicator.

A convenient starting point for our construction is the most powerful, albeitpurely mathematical and not directly implementable in a physical way, separabilitycriterion proposed to date. The criterion is based on mathematical properties of linearpositive maps acting on matrices [7]. Let Md be a space of matrices of dimension d;recall that Λ : Md 7→ Md is called positive if X > 0 implies Λ(X) > 0 (expressionX > 0 means that the matrix X has a nonnegative spectrum). If the inducedmap I ⊗ Λ is also positive, then Λ is called completely positive and, as such, itrepresents a physically allowed transformation of density operators (here I denotesthe identity map on an auxiliary system of any dimension). Using this terminology,the separability criterion reads [7]: % is separable iff

[I ⊗ Λ](%) ≥ 0, (3.1)

for all positive but not completely positive maps Λ : Md 7→ Md acting on thesecond particle. In fact, it is sufficient to consider only positive maps Λ suchthat the maximum of trΛ(%), over all %, is equal to unity. Other positive mapsdiffer only by a positive multiplicative factor which does not affect the condi-tion (2).

Furthermore, in some cases, instead of scanning all positive maps, we can choosejust one. For example, all entangled states of two qubits can be detected by choosingΛ to be transposition [13, 7]. The snag is that positive maps Λ, such as an anti-unitarytransposition and the induced maps I ⊗ Λ, cannot be implemented in a laboratory.Thus, the criterion (2) tacitly assumes prior knowledge of %. However, there is a wayto modify it, so that it becomes experimentally viable and without involving anystate estimation.

4. Operational criterium for detecting entanglement

If we mix in an appropriate proportion of [I⊗Λ] with a depolarizing map, that turnsany density matrix into a maximally mixed state, the resulting map can be completelypositive This is because the lowest negative eigenvalues generated by the induced map[(I ⊗ I) ⊗ (I ⊗ Λ)] can be offset by the positive eigenvalues of the maximally mixedstate generated by the depolarizing map. The most negative eigenvalue −λ < 0 isobtained when [(I ⊗ I) ⊗ (I ⊗ Λ)] acts on the maximally entangled state of the form

114 Quantum Computers and Computing, V. 4,����������

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1√

d2

∑d2

i=1 |i〉|i〉, where each state | i〉 pertains to a d2 dimensional subsystem, itself

is composed of two d dimensional parts. Thus, the map

[I ⊗ Λ](%) = pI ⊗ I

d2+ (1 − p)[I ⊗ Λ](%), (4.1)

is completely positive, therefore physically implementable, when the induced map

[(I⊗I)⊗(I ⊗ Λ)] is positive. This happens for p > (d4λ)/(d4λ+1) [14]. By insertingthe threshold value p = (d4λ)/(d4λ+ 1) into (3), we can modify the criterion (2) asfollows: % is separable iff for all positive maps Λ,

[I ⊗ Λ](%) >d2λ

d4λ+ 1, (4.2)

i.e. when the minimal eigenvalue of the transformed state %′ = [I ⊗ Λ](%) is greaterthan (d2λ)/(d4λ + 1). In general, for some maps Λ, the related completely positive

maps I ⊗ Λ are not trace-preserving and require postselections in their physical im-

plementations. Maps such as I ⊗ Λ have been referred to as “structural” physicalapproximations of unphysical maps I ⊗ Λ [14].

For example, if we take Λ to be transposition T (the first positive map used fordetecting entanglement), we obtain

[I ⊗ T ](%) = dd3 + 1

I ⊗ I + 1d3 + 1

[I ⊗ T ](%). (4.3)

In the two qubit case, where the partial transposition is a sharp test for entanglement,we obtain,

[I ⊗ T ](%) = 29I ⊗ I + 1

9[I ⊗ T ](%), (4.4)

which can be represented and implemented as

13Λ1 ⊗ Λ2 + 2

3I ⊗ σxσzΛ1σzσx, (4.5)

with the two channels defined as:

Λ1(%) = 13

∑

i=x,y,z

σi%σi, Λ2(%) = 14

∑

i=o,x,y,z

σi%σi. (4.6)

The map can be implemented by applying selected products of unitary (Pauli)transformations with the prescribed probabilities. Since the map is trace-preserving,any post-selection in experimental realizations is avoided.

Thus, in order to detect entanglement of an arbitrary two-qubit state %, it is

enough to estimate a single parameter, i.e. the minimal eigenvalue of [I ⊗ T ](%). The

state % is separable if this eigenvalue satisfies λmin ≥ 29. Let us also point out an extra

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bonus: λmin gives us −λ′, the most negative eigenvalue of [I ⊗ T ](%), which entersthe expression for the upper and lower bounds for the entanglement of formation,

H

(

1 +√

1 − 4λ′2

2

)

≤ E(%) ≤ H

1 +√

1 − 4(√

2λ′2 + λ′ − λ′)

2

, (4.7)

where H(x) is the Shannon entropy. The above formulae can be derived from theestimations of the concurrence provided by Verstraete et al [15].

5. Spectrum estimation

Suppose for a moment that I ⊗ Λ is a trace-preserving map, e.g. the transpositioncase. The first part of our entanglement detection measurement is accomplished by

applying I ⊗ Λ to each of the n pairs to obtain n copies of %′ = [I ⊗ Λ](%). Then,following the criterion (4), we need to measure the lowest eigenvalue of %′.

This can be viewed as a special case of the spectrum estimation, and possibleapproaches depend a lot on particular physical realizations of %′. Here, we provide twogeneral solutions. The first one, based on quantum interferometry, is conceptuallysimple and relies on estimating d2 − 1 parameters from which the spectrum of %′ canbe calculated (this is a significant gain over the state estimation which involves d4−1parameters). The second solution is a joint measurement on all copies of %′, whichgives directly the estimate of the lowest eigenvalue.

We start with the quantum interferometry, presented here as a quantum networkshown in Figure 1. A typical interferometric set-up for a single qubit — the Hadamardgate, phase shift ϕ, the Hadamard gate, followed by a measurement — is modified byinserting in between the Hadamard gates a controlled-U operation, with its controlon the qubit and with U acting on a quantum system described by some unknowndensity operator ρ. (N.B. we do not assume anything about the form of ρ; it can,for example, describe several entangled or separable sub-systems.) The action of thecontrolled-U on ρ modifies the interference pattern by the factor,

TrρU = veiα, (5.1)

where v is the new visibility and α is the shift of the interference fringes, also knownas the Pancharatnam phase [16]. Formula (10) has been derived, in the context ofgeometric phases, by Sjoqvist et al. [17].

The network can evaluate certain non-linear functionals of density operators.Let us choose ρ to be composed of two subsystems, ρ = %a ⊗ %b, and let U to be theexchange operator V , such that V |α〉 |β〉 = |β〉 |α〉 for any pure states of the twosubsystems. The interference pattern is now modified by the factor tr V (%a ⊗ %b) == tr %a%b. For ρ = % ⊗ %, we can estimate tr %2, which gives us an estimate of∑m

i=1 λ2i , where λi are the eigenvalues of %. N.B. tr %2 is real hence there is no need

to sweep the phase ϕ in the interferometer, it can be fixed at ϕ = 0.

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Fig. 1. Both the visibility and the shift of the interference patterns of a single qubit(top line) are affected by the controlled-U operation. This set-up allows to estimatetrUρ, the average value of U in state ρ.

In general, in order to calculate the spectrum of any m × m density matrix %we need to estimate m − 1 parameters Tr%2, Tr%3,... Tr%m. For this we need thecontrolled-shift operation. Given k systems of dimension m we define the shift V (k)

as

V (k) |φ1〉 |φ2〉 ... |φk〉 = |φk〉 |φ1〉 ... |φk−1〉 , (5.2)

for any states |φ〉. Such an operation can be easily constructed by cascading k − 1swaps V . This time, if we prepare ρ = %⊗k the interference will be modified by thefactor

Tr %⊗kV (k) = Tr %k =

m∑

i=1

λki . (5.3)

Thus, measuring the average values of V (k) for k = 2, 3...m gives us effectively thespectrum of %. In our case, in particular, we obtain the spectrum (and the lowest

eigenvalue) of %′ = [I ⊗ Λ](%) by estimating d2−1 parameters tr %′k, where k = 2...d2.Again, the phase in the interferometry can be fixed at ϕ = 0.

The interferometric scheme described above is conceptually simple and experi-mentally viable. However, if the simplicity of the implementation is not an issue, wecan choose to measure the estimate of the lowest eigenvalue directly. This requires ajoin measurement on all of the n pairs (see [10] and [18] for further details).

6. Conclusions

Let us summarize our findings. Given n copies of a bipartite d⊗d system, described bysome unknown density operator %, we can test for entanglement either by estimating %and applying criterion (2), or, more directly, by performing the measurements we havejust described. The state estimation involves estimating d4−1 parameters of %, mostof which are of no relevance for the entanglement detection. The optimal state esti-mations rely on joint measurements on all copies of %. However, one can construct less

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efficient but simpler state estimation methods which involve measurements only on in-dividual copies. Our more direct, interferometry based, method requires estimationsof only d2 − 1 parameters and joint operations on d copies of %′. The most demand-ing, from the experimental point of view, is our second method. It is a measurementwith an outcome which is an estimate of just one parameter, but, like the optimalstate estimation, the measurement involves joint operations on all copies of %′. Bothdirect and indirect entanglement detections have their own merits. Depending on thecontext, applications, and technologies involved, one can choose one or the other.

There are powerful mathematical tools to test the entanglement in bipartite andsome multipartite Gaussian states [25], and the present scheme can be modified todirectly detect it. However, in order to modify it efficiently, the present scheme wouldhave to be translated into the language of spectrum of covariance matrix, rather thenof the state itself. Those issues are related to more general and interesting questionsof inherently quantum decision problems on continuous variables domain [26]. Inparticular, it would be interesting to consider possible implications of such approach,on the so called quantum Gaussian channels.

Direct entanglement detections can be employed as sub-routines in quantumcomputation. For example, one may consider performing or not performing a quan-tum operation on a given quantum system, conditioned on some part of quantumdata being entangled or not. In fact, direct entanglement detections can be viewed asquantum computations solving an inherently quantum decision problem: given as aninput n copies of %, decide whether % is entangled. Here, the input data is quantumand such a decision problem cannot even be even formulated for classical computers.Nonetheless, the problem is perfectly well defined for quantum computers. Finally,let us add that the method presented here can be easily generalized to cover all linearmaps tests for arbitrary multiparticle entanglement [21] and the so called k-positivemap tests detecting Schmidt numbers of density matrices [22]. Modification of themethod to the case of two distant labs, under restrictions to local operations andclassical communication [8], is possible [24]).

Acknowledgements

A.K.E., L.C.K. and D.K. acknowledge financial support provided under the NSTBGrant No. 012-104-0040, P.H. and M.H. would like to acknowledge support from theEU (EQUIP, contract no. IST-1999-11053), C.M.A. is supported by the Fundacaopara a Ciencia e Tecnologia (Portugal) and D.K.L.O. would like to acknowledge thesupport of CESG (UK) and QAIP (contract no. IST-1999-11234).

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