Detecting the concurrence of an unknown state with a single observable
aa r X i v : . [ qu a n t - ph ] A p r Detecting the concurrence of an unknown state with a single observable
Zhi-Hao Ma , Zhi-Hua Chen , Jing-Ling Chen Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, P. R. ChinaDepartment of Science, Zhijiang college, Zhejiang University of technology, Hangzhou, 310024, P.R.China andTheoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, P.R.China
While the detection of entanglement has been proved already to be quite a difficult task, ex-perimental quantification of entanglement is even more challenging. In this work, we derive ananalytical lower bound for the concurrence of a bipartite mix quantum state in arbitrary dimension.The lower bound is experimentally implementable in a feasible way, which enabling quantificationof entanglement in a broad variety of cases.
PACS numbers: 03.67.Mn, 03.65.Ud
Entanglement is a distinctive feature of quantum me-chanics [1, 2], and an indispensable ingredient in vari-ous kinds of quantum information processing applicationsvary from quantum cryptography [3] and quantum tele-portation [4] to measurement-based quantum computing[5].The use of entanglement as a resource not only bearsthe question of how it can be detected, but also how itcan be quantified. For this purpose, several entangle-ment measures have been introduced, one of the mostprominent of which is the concurrence [6].However, calculation of the concurrence is a formidabletask as the Hilbert space dimension is increasing. Goodalgorithms and progresses have been obtained concern-ing lower bounds [9–12]. Considerable progress is madein order to give a purely algebraic lower bound and ex-perimental verifying it [13–23].We start with a generalized definition of concurrencefor a pure state | ψ i in the tensor product H A ⊗H B of twoHilbert spaces H A , H B for systems A, B .The concurrenceis defined by C ( | ψ i ) := q − Tr ρ A ) (1)where the reduced density matrix ρ A is obtained by trac-ing over the subsystem B .The concurrence is then ex-tended to mixed states ρ by the convex roof, C ( ρ ) ≡ min { p i , | ψ i i} X i p i C ( | ψ i i ) , (2)for all possible ensemble realizations ρ = P i p i | ψ i ih ψ i | ,where p i ≥ P i p i = 1. From definition, a state ρ is separable if and only if C ( ρ ) = 0.In this letter, we will show that strong lower bounds ofconcurrence can be derived by exploiting close relationsbetween concurrence and a recently introduced detectioncriteria for bipartite entanglement. The lower bound wefound is not only analytical, but also is physical accessi-ble, i.e.,we can easily use the existing experiments tech-nology to obtain this bound(e.g., [17–19, 25]).To disclose the connection between concurrence andentanglement detection criteria, let us first review the known criterions. Until now, many possible ways todetect entanglement have been proposed. These rangefrom Bell inequalities, entanglement witnesses and spinsqueezing inequalities to entropic inequalities, the mea-surement of nonlinear properties of the quantum stateand the approximation of positive maps, see[1, 2].Among so many methods, the PPT criterion and therealignment criterion are the two distinguished ones, theyare most powerful and widely used by quantum informa-tion community. The Peres-Horodecki criterion of posi-tivity under partial transpose (PPT criterion)[26, 27] saythat ρ T A ≥ ρ T A stands for a partial transpose with respect tothe subsystem A . This criterion is even to be sufficientfor 2 × × realignment criterion is very strongin detecting many of BES [29]. This criterion states thata realigned version R ( ρ ) of ρ should satisfy ||R ( ρ ) || ≤ ρ . Note that realignment crite-rion is in some sense “ dual ” to the PPT criterion, inthe following way: A density operator can be written as ρ = X ijkl h ij | ρ | kl i| ij ih kl | = X ijkl ρ ij,kl | i ih k | ⊗ | j ih l | (3)A partial transpose with respect to the first system A is (cid:0) ρ T A (cid:1) ij,kl := ρ kj,il . (4)While the realignment of ρ is defined as: (cid:0) ρ R (cid:1) ij,kl := ρ ik,jl . (5)Recently, a criterion was introduced, which was strictlystronger than the realignment criterion. It says that, if astate is separable, then the following must hold[30]: kR ( ρ − ρ A ⊗ ρ B ) k ≤ q (1 − Tr ρ A )(1 − Tr ρ B )] (6)We now derive the main result of this Letter. Theorem:
For any m ⊗ n ( m ≤ n ) mixed quantumstate ρ , the concurrence C ( ρ ) satisfies the following: C ( ρ ) ≥ s n ( n − n + 1) × f ( ρ ) . (7)Where f ( ρ ) := [ kR ( ρ − ρ A ⊗ ρ B ) k − p (1 − Tr ρ A )(1 − Tr ρ B )]. Proof.—
Without loss of generality, we suppose that apure m ⊗ n ( m ≤ n ) quantum state ρ := | ψ ih ψ | has thestandard Schmidt form | ψ i = X i √ µ i | a i b i i , (8)where √ µ i ( i = 1 , . . . m ) are the Schmidt coefficients, | a i i and | b i i are orthonormal basis in H A and H B , re-spectively. The two reduced density matrices ρ A and ρ B have the same eigenvalues of µ i . It follows C ( ρ ) = 2 (cid:16) − X i µ i (cid:17) = 4 X i 1) + n − n = q ( n − n +1) n × C ( ρ )Now, we will show that the inequality (7) also holdsfor mix state.Assume we have found the optimal de-composition P i p i ρ i for ρ to achieve the infimum of C ( ρ ), where ρ i are pure state density matrices. Then C ( ρ ) = P i p i C ( ρ i ) by definition. Now, we need to prove that after mixture, the bound will not become bigger,i.e., f ( P i p i ρ i ) ≤ P i p i f ( ρ i ).From the convex propertyof the trace norm, we know that kR ( ρ − ρ A ⊗ ρ B ) k isdecreasing after convex combination, so the only thingleft to prove is that p (1 − Tr ρ A )(1 − Tr ρ B ) is increasingafter convex combination. It is sufficient to consider thecase of ρ = ρ + ρ . Now, we will prove that q [1 − Tr( ρ A + ρ A ) ][1 − Tr( ρ B + ρ B ) ] ≥ [1 − Tr( ρ A )] + [1 − Tr( ρ A )] (10)For clear, denote Tr( ρ A ) = Tr( ρ B ) := x , Tr( ρ A ) =Tr( ρ B ) := x , Tr( ρ A ρ A ) := x ,Tr( ρ B ρ B ) := x , thenthe inequality (10) reduced to prove that the function F := − ( x + x ) + ( x + x )( x + x )+ x x − ( x + x ) + ( x + x ) ≥ F , wewill use the Lagrange multipliers method, and find that F attained its minimal value at the point ( x , x , x , x ) =(1 , , , F = 0.So we get F ≥ 0. Theorem is proved.The most prominent feature of this theorem is that itnot only allows to obtain a strong lower bound for theconcurrence without any numerical optimization proce-dure, but also this bound is measurable, i.e., it is directlyaccessible in currently existing laboratory experiments.We will explain this in detail. We will see that, ourbound is experimentally implementally by means of localobservables, use the method as that of [13, 18, 24, 25].First, note that the concurrence of a bipartite purestate has another representation as [13, 18] C ( | ψ i ) ≡ q − Tr ρ A ) = p h ψ | ⊗ h ψ | A | ψ i ⊗ | ψ i , (11)where A = 4 P (1) − ⊗ P (2) − . P ( i ) − is the projector on theantisymmetric subspace H i ∧ H i of the two copies of the i th subsystem H i ⊗ H i .Define K = 4 P (1) − ⊗ (2) and K = 4( (1) ⊗ P (2) − ), then 1 − Tr ρ A = Tr( ρ ⊗ ρK ),and 1 − Tr ρ B = Tr( ρ ⊗ ρK ), so we can obtain theterm p (1 − Tr ρ A )(1 − Tr ρ B )], provide that we have twocopies of the state, see [24, 25].On the other hand, the term kR ( ρ − ρ A ⊗ ρ B ) k can be obtained by generalized entanglement witness[27, 31].Entanglement witnesses (EW) are Hermitian op-erators that have positive averages on all separable states,but a negative one on at least one entangled state.It wasshown that, a state is entangled if and only if it is de-tected by some EW[27]. EW can be measured locally,and one can optimize such measurements in various as-pects [32]. Nowadays, entanglement witnesses are rou-tinely used in experiments to detect entanglement (seee.g., [33]). Using the similar method of [34], we can di-rectly measure the value of kR ( ρ − ρ A ⊗ ρ B ) k by a gen-eralized witness. The method is as follows: from theresult of matrix analysis[35], every operator has a singu-lar value decomposition (SVD), so we can get the SVDof R ( ρ − ρ A ⊗ ρ B ) as R ( ρ − ρ A ⊗ ρ B ) = U DV + , with U, V unitary matrices, D is a diagonal matrix. Then since thetrace norm has a variational representation as || A || =max {| Tr[ X + AY ] |} , where X, Y are unitary matrices, wecan get that kR ( ρ − ρ A ⊗ ρ B ) k = Tr[ V U + R ( ρ − ρ A ⊗ ρ B )].Now the only problem leaved is how to deal with therealignment operation. Note that every map φ on in-ner product space can induce its adjoint map φ ⋆ ,i.e.,Tr[ φ ( X ) Y + ] = Tr[ X ( φ ⋆ ( Y )) + ], and for our question, itreads that Tr[ V U + R ( ρ − ρ A ⊗ ρ B )] = Tr[ R ⋆ ( V U + )( ρ − ρ A ⊗ ρ B )], and the adjoint map of the realignment oper-ation is defined by R ⋆ ( V U + ) := [ R − ( V U + ) T ] T , where T is the transpose, and R − is the inverse map of the re-alignment operation. Now define W := [ R − ( V U + ) T ] T ,then we get that kR ( ρ − ρ A ⊗ ρ B ) k = Tr[ W ( ρ − ρ A ⊗ ρ B )],which is clearly physical accessible(e.g., see [33]).Next we consider some examples to illustrate furtherthe tightness and significance of our bound. To show thatour bound is close to the real concurrence, note that in[38], the authors find a fast optimal algorithm to calculatethe entanglement of formation of a mixed state, so we canuse the optimal algorithm of [38] to give the estimation ofthe concurrence, and comparing it with our bound. Theconcurrence using optimal method of [38] is representedby blue colors, our low bound is represented by greencolors. Example 1 Isotropic states are a class of U ⊗ U ∗ in-variant mixed states in d × d systems ρ F = 1 − Fd − (cid:0) I − | Ψ + ih Ψ + | (cid:1) + F | Ψ + ih Ψ + | , (12)where | Ψ + i ≡ p /d P di =1 | ii i and F = h Ψ + | ρ F | Ψ + i ,satisfying 0 ≤ F ≤ 1, is the fidelity of ρ F and | Ψ + i . FIG. 1: Figure of example 1 Example 2. Pawe l Horodecki introduced a 3 × ρ is real and symmetric, ρ = 18 a + 1 a a a a a a a a a a a √ − a a a a √ − a a , (13)where 0 < a < 1. Let us consider a mixture of this statewith white noise, ρ ( p ) = pρ + (1 − p ) , (14)and show the curves 1 − kR ( ρ ) k = 0, 1 − k τ k − (Tr ρ A +Tr ρ B ) / p (1 − Tr ρ A )(1 − Tr ρ B ) − kR ( ρ − ρ A ⊗ ρ B ) k = 0 with respect to the CCNR criterion, its op-timal nonlinear witness, and Theorem 1 in Fig. 2.It isfound that the state ρ ( p ) still has entanglement when p = 0 . a = 0 . p = 0 . a = 0 . 232 for ρ ( p ) which is still entangled. FIG. 2: Figure of example 2 When p = 0 . a = 0 . p = 0 . a = 0 . Example 3. Consider the following states introducedin Ref. [39]: ρ ( α ) = 27 | Ψ + ih Ψ + | + 27 σ + + 5 − α σ − , (15)where 2 ≤ α ≤ σ + := ( | i| ih |h | + | i| ih |h | + | i| ih |h | ), σ − := ( | i| ih |h | + | i| ih |h | + | i| ih |h | ). α FIG. 3: Figure of example 3 In summary, we have provided an analytical formulafor a lower bound of concurrence, by finding a connectionwith the currently most powerful detection criterion. Thebound is very close to the actual values of concurrence forsome special class of quantum states. Also, this bound isexperimentally implementable and computationally veryefficient, allowing to not only detect, but also to quantifyentanglement in an experimental scenario. Acknowledgment. This work is supported by NSF ofChina(10901103), partially supported by a grant of sci-ence and technology commission of Shanghai Municipal-ity (STCSM, No. 09XD1402500). [1] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki,Rev. Mod. Phys. , 865(2009).[2] O. G¨ u hne, G. Toth, Phys. Rep. , 1(2009).[3] A. K. Ekert, Phys. Rev. Lett. , 661(1991).[4] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. ,1895(1993).[5] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. ,5188(2001).[6] W.K. Wootters, Phys. Rev. Lett. , 2245 (1998).[7] B.M.Terhal and K. G. H. Vollbrecht, Phys. Rev. Lett. , 2625(2000).[8] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A ,062307(2001).[9] P.X. Chen, L.-M. Liang, C.-Z. Li, and M.-Q. Huang Phys.Lett. A Z yczkowski, T. Wellens,Europhys. Lett. , 052304(2001).[12] F. Mintert, M. Ku´s, A. Buchleitner,Phys. Rev. Lett. ,167902(2004).[13] F. Mintert, A.R.R.Carvalhoa, M. Ku´s, A. Buchleitner, Phys. Rep. , 207(2005).[14] K. Chen, S. Albeverio, S.-M. Fei, Phys. Rev. Lett. ,040504(2005).[15] H.-P. Breuer, J. Phys. A: Math. Gen. , 11847(2006).[16] J. I. de Vicente, Phys. Rev. A , 052320(2007); ibid ,039903(E)(2008).[17] S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F.Mintert,A. Buchleitner, Nature 440, 1022(2006).[18] F. Mintert and A. Buchleitner, Phys. Rev. Lett. ,140505(2007).[19] C. Schmid, N. Kiesel, W. Wieczorek, H. Weinfurter,F. Mintert, A. Buchleitner, Phys. Rev. Lett. ,260505(2008).[20] Z. Ma, F. Zhang, D.-L. Deng, J-L Chen, Phys. Lett. A , 1616(2009).[21] R. Augusiak, M. Lewenstein, Quant. Inf. Process 8,493(2009).[22] Z.-H.Ma,M.-L.Bao, Phys. Rev. A , 034305(2010).[23] I.Sargolzahi, S.Y.Mirafzali, M.Sarbishaei,Quantum Inf.Comput. , 0079(2011).[24] C.-J. Zhang, Y.-X. Gong, Y.-S. Zhang, G.-C. Guo, Phys.Rev. A 78, 042308(2008).[25] Y.-F. Huang, X.-L. Niu, Y.-X. Gong, J. Li, L. Peng, C.-J.Zhang, Y.-S. Zhang, G.-C. Guo,Phys. Rev. A 79, 052338(2009).[26] A. Peres, Phys. Rev. Lett. , 1413(1996).[27] M. Horodecki,P. Horodecki, R. Horodecki, Phys. Lett. A ,1(1996).[28] P. Horodecki, Phys. Lett. A 232, 333(1997).[29] K. Chen and L.-A. Wu, Quantum Inf. Comput. ,193(2003);O. Rudolph, Physical Review A , 032312(2003); S. Albeverio, K. Chen, and S.M. Fei, Phys. Rev.A u hne, P. Hyllus, J. Eisert, Phys. Rev. A 78,052319(2008).[31] D. Bruß, J. Math. Phys. 43, 4237(2002);G. T´ o th,O. G¨ u hne, Phys. Rev. Lett. 94, 060501(2005); D.Chru´ s ci´ n ski and A. Kossakowski, Open Systems and Inf.Dynamics 14, 275(2007); D. Chru´ s ci´ n ski and A. Kos-sakowski, J. Phys. A: Math. Theor. 41, 145301(2008);M.A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D55, 197(2009).[32] O. G¨ u hne, et al., Phys. Rev. A 66, 062305(2002);[33] M. Barbieri et al., Phys. Rev. Lett. 91, 227901(2003);M.Bourennane et al., Phys. Rev. Lett. , 087902(2004); H.H¨affner et al., Nature , 643 (2005).[34] K.Chen and L.A. Wu, Phys. Rev. A , 022312 (2004).[35] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, New York, 1991).[36] P. Rungta, C. M. Caves,Phys. Rev. A 67, 012307(2003).[37] C.H. Bennett et al., Phys. Rev. Lett. ,5385 (1999).[38] S. Ryu, W. Cai, and A. Caro, Phys. Rev. A77