Measurable entanglement

Measurable entanglement

Alexander Klyachko, Barış Öztop, and Alexander S. Shumovsky

Citation: Appl. Phys. Lett. 88, 124102 (2006); doi: 10.1063/1.2187398 View online: http://dx.doi.org/10.1063/1.2187398

View Table of Contents: http://aip.scitation.org/toc/apl/88/12

Published by the American Institute of Physics

Articles you may be interested in

Entanglement guides quantum computation

Measurable entanglement

Alexander Klyachko, Barış Öztop,a兲and Alexander S. Shumovsky Faculty of Science, Bilkent University, Bilkent, Ankara 06800, Turkey

共Received 22 December 2005; accepted 2 February 2006; published online 21 March 2006兲 The amount of entanglement carried by a quantum bipartite state is usually evaluated in terms of concurrence关S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 共1997兲; P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev. A 64, 042315 共2001兲兴. We give a physical interpretation of concurrence that reveals a way of its direct measurement and discuss possible generalizations. © 2006 American Institute of Physics.关DOI:10.1063/1.2187398兴

Entanglement, which has been considered for decades in the context of fundamentals of quantum mechanics, turns now more and more into a key tool of practical realization of quantum information technologies. The quantum key distribution1for completely secured communications should be mentioned here, first of all共e.g., see Ref. 2兲.

The design and manufacturing of generators of en-tangled states require a control of amount of entanglement carried by the states. For pure state ␺ of bipartite systems HA丢HB of format n⫻n 共n=dim HA,B; n = 2 corresponds to

qubits, n = 3 corresponds to qutrits, etc.兲, this quantity is given by the concurrence

C共␺兲 =

冑

that has been proposed in Ref. 3. Here, ␳r denotes the

re-duced 共single-party兲 density matrix, corresponding to the state ␺, and we use the normalization factor v = n / n − 1, to reduce the concurrence to the interval 关0,1兴. See Ref. 4 for further discussion.

The aim of this letter is to give a natural physical inter-pretation of the concurrence, and thus to show a way of direct measurement of the amount of bipartite entanglement in terms of mean values of certain physical quantities. Our approach also suggests a general definition of the concur-rence for multipartite systems discussed below.

It has been shown in Ref. 5 that entanglement, like co-herence and squeezing, can be associated with quantum fluc-tuations or quantum uncertainties, which are minimal for co-herent 共separable兲 states and maximal for completely entangled states. The fluctuations are measured by the total variance defined by the equation

V共␺兲 =

兺

2_{兩␺典 − 具␺兩X}

␣兩␺典2, 共2兲

where the sum is extended over orthonormal basis X_␣of Lie algebra of local observables. The basic observables X_␣act on one of the components: X_␣= X_iAor X_␣= X_jB, where X_iA and XB_j are are orthonormal bases in the space of traceless Hermitian operators inHAandHB, respectively. It is important to

real-ize that the total variance is independent of the choice of the basic observables X_␣.

The total uncertainty of all basic observables attains its maximal value in the case of completely entangled states 共like Bell states of two qubits兲.

The first sum in the right-hand side of Eq. 共2兲 is inde-pendent of the state␺. In fact, the sum

C =

兺

X_␣2

known as Casimir operator,6,7acts as a multiplication by a scalar CAB共equal to 6 for two qubits, for example兲. Thus,

V共␺兲 = CAB−

兺

具␺兩Xi兩␺典2, 共3兲

so that the measurement of the total uncertainty is reduced to the measurement of mean values of basic observables in the right-hand side of Eq.共3兲. In the case of complete entangle-ment,

具␺兩X␣兩␺典 = 0

for all ␣ 共see Ref. 5兲, so that the total uncertainty 共3兲 achieves its maximum.

We now show that concurrence 共1兲 can be equivalently expressed in terms of the total uncertainty共3兲 in the case of bipartite systems. Consider first the case of two qubits with the state 兩␺典 =

兺

⬘

兺

where兩ᐉ,ᐉ

⬘

⬘

easily seen that the concurrence共1兲 is then cast to the form C共␺兲 = 2兩␺00␺11−␺01␺10兩, =2关兩␺00兩2兩␺11兩2+兩␺01兩2兩␺10兩2,− 2 Re共␺00␺11␺01 * _␺ 10 *_兲兴1/2_. 共5兲 On the other hand, using Pauli operators,

␴x=兩0典具1兩 + 兩1典具0兩,

␴y= − i共兩0典具1兩 − 兩1典具0兩兲,

␴z=兩0典具0兩 − 兩1典具1兩 共6兲

as the basic local observables Xi A and Xj B , one gets V共␺兲 = 4 + 4关兩␺₀₀兩2兩␺₁₁兩2+兩␺₀₁兩2兩␺₁₀兩2 − 2 Re共␺00␺11␺01* ␺10* 兲兴. 共7兲

Comparing now Eqs.共5兲 and 共7兲 and taking into account that V_max= 6 and V_min= 4 in the case of completely entangled and

a兲_{Electronic mail: boztop@fen.bilkent.edu.tr}

APPLIED PHYSICS LETTERS 88, 124102共2006兲

unentangled states of two qubits, respectively, we get C共␺兲 =

冑

Vmax− Vmin

共8兲 in the case of the general two-qubit state 共4兲. Thus, the amount of entanglement carried by a pure two-qubit state can be determined by measurement of mean values of the basic observables given by Pauli operators共6兲. These observables can be directly measured in experiments, say by the Stern– Gerlach apparatus in the case of spins, or by means of polar-izers in the case of photons, etc.8

As a matter of fact, this expression 共8兲 is equivalent to 共1兲 for any bipartite system.9

For example, in the representa-tion of basic observables for qutrits共n=3兲 given in Ref. 10, the maximal and minimal values of total uncertainty in bi-partite system are Vmax= 32/ 3 and Vmin= 8, respectively. A

possible realization of qutrits is provided by biphotons.11 Equation共8兲 allows us to interpret concurrence 共1兲 as a square root of the normalized total uncertainty of basic ob-servables, specifying the system. In view of Eq.共3兲, the latter can be determined in terms of measurement of expectation values of the basic observables具␺兩X_␣兩␺典. In other words, Eq. 共8兲 provides an operational definition of measure of bipartite entanglement. Note also that Eq.共8兲 allows one to define the concurrence for any multipartite system.

Our considerations so far have applied to the pure bipar-tite states. In connection with mixed states, we now note that the uncertainty of an observable Xi can be interpreted as a

specific Wigner–Yanase “quantum information” about a state

␺extracted from the macroscopic measurement of Xiin this

state.12 The generalization of Wigner–Yanase “information” on the case of mixed states with the density matrix␳has the form

Ii共␳兲 = −

1

2Tr共关Xi,␳

1/2_兴2_{兲 艌 0. 共9兲}

It can be easily seen that in the case of pure states when ␳ =兩␺典具␺兩 the total amount of Wigner–Yanase skew informa-tion

I共␳兲 =

兺

i

Ii共␳兲 共10兲

coincides with the total uncertainty 共2兲. The supposition is that Eq. 共10兲 can represent a reasonable estimation from above for the amount of concurrence in the mixed bipartite state.9

One of the authors共B.O.兲 would like to acknowledge the Scientific and Technical Research Council of Turkey 共TÜBİ-TAK兲 for financial support.

1_{C. H. Bennett and G. Brassard, in Proceedings of the IEEE International}

Conference on Computers, Systems, and Signal Processing共IEEE, New

York, 1984兲, p. 175; A. Ekert, Phys. Rev. Lett. 67, 1661 共1991兲. 2_{J. Ouellette, Ind. Phys. 10, 22共2004兲.}

3_{S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022共1997兲; P. Rungta,} V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev. A 64, 042315共2001兲.

4_{F. Minnert, M. Kuś, and A. Buchleitner, Phys. Rev. Lett. 92, 167902} 共2004兲.

5_{M. A. Can, A. A. Klyachko, and A. S. Shumovsky, Phys. Rev. A 66,} 02111共2002兲; A. A. Klyachko and A. S. Shumovsky, J. Opt. B: Quantum Semiclassical Opt. 5, S322共2003兲; A. A. Klyachko and A. S. Shumovsky,

ibid. 6, S29共2004兲.

6_{D. Kaszlikowski, D. K. L. Oi, M. Christandl, K. Chang, A. Ekert, L. C.} Kwek, and C. H. Oh, Phys. Rev. A 67, 012310共2003兲; T. Durt T., N. Cerf, N. Gisin, and M. Zukowski, ibid. 67, 012311共2003兲; T. Durt, D. Kasz-likowski, J.-L. Chen, and L. C. Kwek, ibid. 69, 032313共2004兲. 7_{A. Bohm, Quantum Mechanics: Foundations and Applications共Springer,}

New York, 1993兲.

8_{M. Nielsen and I. Chuang, Quantum Computation and Quantum}

Informa-tion共Cambridge University Press, New York, 2000兲.

9_{A. A. Klyachko, B. Öztop, and A. S. Shumovsky共unpublished兲.} 10_{C. M. Caves and G. J. Milburn, Opt. Commun. 179, 439共2000兲.} 11_{A. V. Burlakov, M. V. Chechova, O. A. Karabutova, D. N. Klyshko, and S.}

P. Kulik, Phys. Rev. A 60, R4209 共1999兲; Y. I. Bogdanov, M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, A. A. Zhukov, C. H. Oh, and M. K. Tey, Phys. Rev. Lett. 93, 230503共2004兲.

12_{E. P. Wigner, Z. Phys. 131, 101共1952兲; E. P. Wigner and M. M. Yanase,} Proc. Natl. Acad. Sci. U.S.A. 19, 910共1963兲.