Permutations and quantum entanglement

Transcription

1 Journal of Physics: Conference Series Permutations and quantum entanglement To cite this article: D Chruciski and A Kossakowski 2008 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Partial Transposition on Bipartite System Ren Xi-Jun, Han Yong-Jian, Wu Yu-Chun et al. - On separable decompositions of quantum states with strong positive partial transposes B Bylicka, D Chruciski and J Jurkowski - A class of symmetric Bell diagonal entanglement witnesses a geometric perspective Dariusz Chruciski This content was downloaded from IP address on 24/11/2017 at 20:25

2 Permutations and Quantum Entanglement Dariusz Chruściński and Andrzej Kossakowski Institute of Physics, Nicolaus Copernicus University, Grudzi adzka 5/7, Toruń, Poland Abstract. We construct a large class of quantum d d states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space which is governed by by cyclic permutation from the symmetric group S d 1. It turns out that partial transposition maps any such decomposition into another one corresponding to complementary permutation. This class contains many well known examples of PPT states from the literature and gives rise to a huge family of completely new states. 1. Introduction Quantum Entanglement is one of the key features which distinguish quantum mechanics from the classical one. Recent development of Quantum Information Theory [1] shows that quantum entanglement does have important practical applications and it serves as a basic resource for quantum cryptography, quantum teleportation, dense coding and quantum computing. A fundamental problem in Quantum Information Theory is to test whether a given state of a composite quantum system is entangled or separable. Let us recall that a state represented by a density operator ρ living in the Hilbert space H A H B is separable iff it can be represented as the following convex combination ρ = k p k ρ (A) k ρ (B) k, (1) where ρ (A) k and ρ (B) k are density operators living in H A and H B, respectively, and {p k } stands for a probability distribution, that is, p k 0 and k p k = 1. States which are not separable are called entangled. Surprisingly, this so called separability problem has no simple solution. Several operational criteria have been proposed to identify entangled states. Each of these criterion is only necessary and in general one needs to perform an infinite number of tests to be sure that a given state is separable (see e.g. [2] for the recent review). The most famous Peres-Horodecki criterion [3, 4] is based on the operation of partial transposition: for any operator X living in H A H B one defines its partial transposition X τ = (1l τ)x by X τ iα,jβ := X iβ,jα, (2) where the corresponding matrix elements are computed with respect to fixed bases {e i } in H A and {f α } in H B. It is clear that this operation depends upon chosen bases {e i } and {f α }. c 2008 Ltd 1

3 However, as was observed by Peres [3], the positivity of X τ is a universal property which is basis-independent. One calls a state ρ to be PPT (Positive Partial Transpose) if X τ 0. Now, if a state ρ is separable then its partial transposition (1l τ)ρ is evidently positive but the converse is not true. It was shown by Horodecki et al. [5] that PPT condition is both necessary and sufficient for separability for 2 2 and 2 3 systems (actually, this observation is based on the old mathematical result [6]). Since all separable states belong to a set of PPT states, the structure of this set is of primary importance in Quantum Information Theory. Unfortunately, this structure is still unknown, that is, one may check whether a given state is PPT but we do not know how to construct a general quantum state with PPT property. There are several examples of PPT entangled states [4], [7] [16] and its mathematical structure was studied in [17, 18, 19]. In the present paper we propose a very simple construction of PPT states which is based on certain decomposition of the total Hilbert space C d C d into direct sum of d-dimensional subspaces. This decomposition is controlled by some cyclic property, that is, knowing one subspace, say Σ 0, the remaining subspaces Σ 1,...,Σ d 1 are uniquely determined by applying a cyclic shift to elements from Σ 0. Now, we call a density matrix ρ a circulant state if ρ is a convex combination of density matrices supported on Σ α. The crucial observation is that a partial transposition of the circulant state has again a circular structure corresponding to another direct sum decomposition Σ 0... Σ d 1. It turns out that different circular decompositions are related by permutations and to each permutation from the symmetric group S d 1 there corresponds a circulant decomposition of C d C d. For more detailed discussion we refer to our recent paper [23]. 2. Two qubits Consider an operator living in C 2 C 2 of the following form where ρ 0 and ρ 1 are supported on two orthogonal subspaces ρ = ρ 0 + ρ 1, (3) Σ 0 = span {e 0 e 0, e 1 e 1 }, Σ 1 = span {e 0 e 1, e 1 e 0 }, (4) where {e 0, e 1 } is a computational base in C 2. It is clear that {Σ 0, Σ 1 } defines the direct sum decomposition of C 2 C 2, that is Σ 0 Σ 1 = C 2 C 2. We call it a circulant decomposition because its structure is determined by the cyclic shift S : C 2 C 2 defined by S e i = e i+1, (mod 2). (5) and hence One has therefore ρ 0 = ρ 1 = Σ 1 = (1l S)Σ 0. (6) 1 1 a ij e ij e ij, (7) b ij e ij S e ij S, (8) where e ij := e i e j, and one adds modulo 2. Now, since ρ 0 and ρ 1 are supported on two orthogonal subspaces Σ 0 and Σ 1 one has an obvious 2

4 Proposition 1 ρ defined in (3) is a density matrix iff a = [a ij ] and b = [b ij ] are 2 2 semi-positive matrices, and Tr(a + b) = 1. Now, the crucial observation is that both the original density matrix ρ and the partially transposed matrix ρ τ = (1l τ)ρ have exactly the same structure (in order to have more transparent pictures we replaced all vanishing matrix elements by dots and we use this convention through out this paper): a 00 a 01 ã 00 ã 01 ρ = b 00 b 01 b 10 b 11, ρτ = b00 b01 b10 b11, (9) a 10 a 11 ã 10 ã 11 where the matrices ã = [ã ij ] and b = [ b ij ] read as follows ( ) ( ) a00 b ã = 01 b00 a, b = 01 b 10 a 11 a 10 b 11 that is, both ρ and ρ τ are circulant bipartite operators. Therefore, one arrives at, (10) Theorem 1 A circulant state represented by (3) is PPT iff ã = [ã ij ] and b = [ b ij ] are 2 2 semi-positive definite matrices. Note, that matrices ã and b may be rewritten in the following transparent way and similarly ã = a I + b S, (11) b = b I + a S, (12) where x y denotes the Hadamard product of two matrices x and y. This simple construction reproduces many well known examples of 2-qubit states. (i) Bell states: These states give rise to the following circulant projectors 1 ±1 ψ ± ψ ± = 1 2 ±1 1 ψ ± = 1 2 ( 00 ± 11 ), (13) ϕ ± = 1 2 ( 01 ± 10 ). (14), ϕ± ϕ ± = 1 2 (ii) Bell diagonal state corresponds to ( ) p1 + p a = 2 p 1 p 2, b = p 1 p 2 p 1 + p 2 1 ±1 ±1 1 ( ) p3 + p 4 p 3 p 4 p 3 p 4 p 3 + p 4 with p k 0 and p 1 + p 2 + p 3 + p 4 = 1. The positivity of ã and b leads to. (15), (16) p 1 + p 2 p 3 p 4, p 3 + p 4 p 1 p 2, (17) which are equivalent to the well known condition p k 1/2. 3

5 (iii) Werner state [20] W = p 1 + p 2p 2p 1 + p 1 p with 1/3 p 1. PPT condition implies p 1/3. (iv) Isotropic state [21] I = p 2p 1 p 1 p 2p 1 + p, (18), (19) with 1/3 p 1. PPT condition implies again the well known result p 1/3. (v) O(2) invariant state [22] (mixture of Werner and isotropic states): a + 2b 2b a O = 1 a + 2c a 2c 4 a 2c a + 2c, (20) 2b a a + 2b with a, b, c 0 and a + b + c = 1. It is clear that O 0 and O is PPT iff b = c which implies and hence reproduces the well known result [22]. b 1 2, c 1 2, (21) 3. Two qutrits The construction of circulant states for two qubits may be easily generalized for two qutrits. It is based on the following direct sum decomposition where and with S : C 3 C 3 being a shift operator defined by One easily finds C 3 C 3 = Σ 0 Σ 1 Σ 2, (1) Σ 0 = span {e 0 e 0, e 1 e 1, e 2 e 2 }, (2) Σ 1 = (1l S)Σ 0, Σ 2 = (1l S 2 )Σ 0, (3) S e i = e i+1, (mod 3). (4) Σ 1 = span {e 0 e 1, e 1 e 2, e 2 e 0 }, Σ 2 = span {e 0 e 2, e 1 e 0, e 2 e 1 }. (5) Consider now an operator living in C 3 C 3 of the following form ρ = ρ 0 + ρ 1 + ρ 2, (6) 4

6 where ρ 0 = ρ 1 = ρ 2 = 2 a ij e ij e ij, 2 b ij e ij S e ij S, (7) 2 c ij e ij S 2 e ij S 2. Proposition 2 ρ defined in (6) is a density matrix iff a = [a ij ], b = [b ij ] and c = [c ij ] are 3 3 semi-positive matrices, and Tr(a + b + c) = 1. The circulant 3 3 state ρ has therefore the following form a 00 a 01 a 02 b 00 b 01 b 02 c 00 c 01 c 02 c 10 c 11 c 12 ρ = a 10 a 11 a 12 b 10 b 11 b 12 b 20 b 21 b 22 c 20 c 21 c 22 a 20 a 21 a 22 and hence its partially transposed counterpart ρ τ is given by ρ τ = ã 00 ã 01 ã 02 b00 b01 b02 c 00 c 01 c 02 b10 b11 b12 c 10 c 11 c 12 ã 10 ã 11 ã 12 c 20 c 21 c 22 ã 20 ã 21 ã 22 b20 b21 b22, (8), (9) where the matrices ã = [ã ij ], b = [ b ij ] and c = [ c ij ] read as follows ã = a 00 c 01 b 02 c 10 b 11 a 12 b 20 a 21 c 22, b = b 00 a 01 c 02 a 10 c 11 b 12 c 20 b 21 a 22, c = c 00 b 01 a 02 b 10 a 11 c 12 a 20 c 21 b 22. (10) Note, that ρ τ = ρ 0 + ρ 1 + ρ 2, (11) 5

7 where ρ k are supported on three orthogonal subspaces of C 3 C 3 : Σ 0 = span {e 0 e 0, e 1 e 2, e 2 e 1 }, Σ 1 = span {e 0 e 1, e 1 e 0, e 2 e 2 }, (12) Σ 2 = span {e 0 e 2, e 1 e 1, e 2 e 0 }. It is clear that Σ α define again circulant direct sum decomposition C 3 C 3 = Σ 0 Σ 1 Σ 2, (13) and Σ α = (1l S α ) Σ 0. (14) It shows that partial transposition (1l τ) maps any circulant density matrix ρ = ρ 0 + ρ 1 + ρ 2 into another circulant matrix ρ τ = ρ 0 + ρ 1 + ρ 2. Now, contrary to the d = 2 case, both circulant structures are different. One has therefore Theorem 2 A circulant 3 3 state ρ is PPT iff the matrices ã, b and c are semi-positive definite. 4. Two qudits Now, we are ready to construct circular states in d d. The basic idea is to decompose the total Hilbert space C d C d into a direct sum of d orthogonal d-dimensional subspaces related by a certain cyclic property. It turns out that there are (d 1)! different cyclic decompositions and it is therefore clear that they may be labeled by permutations from the symmetric group S d 1. Take any permutation π from the symmetric group S d and fix its action on 0 by π(0) = 0. This way it is effectively a permutation from S d 1. Let us define Σ π 0 which is spanned by Moreover, let us define d 1 orthogonal subspaces by e 0 e π(0), e 1 e π(1),..., e d 1 e π(d 1). (15) where S is a circulant matrix corresponding to shift in C d : and α = 0, 1,...,d 1. It is clear that Σ π α = (1l S α )Σ π 0, (16) S e i = e i+1, (mod d), (17) Σ π 0 Σ π 1... Σ π d 1 = Cd C d. (18) To construct a circulant state corresponding to this decomposition let us introduce d positive d d matrices a (α) = [a (α) ij ] ; α = 0,1,...,d 1. Now, define d positive operators ρπ α supported on Σ π α by ρ π α = = a (α) ij e ij S α e π(i),π(j) S α a (α) ij e ij e π(i)+α,π(j)+α. (19) 6

8 Finally, ρ π = ρ π 0 + ρ π ρ π d 1, (20) defines the circulant state (corresponding to π). Normalization of ρ π is equivalent to the following condition for matrices a (α) ( Tr a (0) + a (1) a (d 1)) = 1. Theorem 3 If ρ π is a circulant state corresponding to permutation π with π(0) = 0, then its partial transposition ρ τ π is also circulant with respect to another decomposition corresponding to permutation π such that π(i) + π(i) = d, (21) for i = 1, 2,...,d 1, and π(0) = 0, that is (1l τ)ρ π = α=0 ã (α) ij e ij S α e π(i), π(j) S α. (22) Moreover where we add modulo d. ã (α) ij = a (α+[ π(j) π(i)]) ij, (23) 5. Conclusions We have constructed a large class of PPT states in d d which correspond to circular decompositions of C d C d into direct sums of d-dimensional subspaces. This class significantly enlarges the previous class defined in [15] constructed by a completely different method. It contains several known examples from the literature. The simplest one is a set of generalized Bell states in d d: for any µ, ν {0,1,...,d 1} let us define ψ µν = (Ω µ S ν )ψ + d, (24) where ψ + d is a canonical maximally entangled state in Cd C d, that is and Ω is defined by ψ + d = 1 e k e k, (25) d with ω = e 2πi/d. It is easy to see that each 1-dimensional projector k=0 defines a circulant state. It is therefore clear that any mixture Ωe k = ω k e k, (26) P µν = ψ µν ψ µν, (27) ρ = µ,ν=0 with c µν 0 and d 1 µ,ν=0 c µν = 1 is again circulant. c µν P µν, (28) 7

9 Other examples are provided by Werner and isotropic states, entangled PPT states in 4 4 constructed by Ha [17] and Fei et. al. [13] (for detailed discussion we refer to our recent paper [23]). It is clear that the presented class of circulant states produces a highly nontrivial family of new states (we stress however that the seminal Horodecki 3 3 entangled PPT state [4] does not belong to our class). There are still many open problems connected with this class: the basic question is how to detect entanglement within this class of PPT states. One may expect that there is special class of entanglement witnesses which are sensitive to entanglement encoded into circular decompositions, that is, one may look for an entanglement witness which has again a circulant structure but now not all matrices a (α) are positive. The condition that entanglement witness should be positive on separable states has to replace positivity of all a (α) by some weaker conditions (for recent discussion of entanglement witnesses see [24]). It is interesting to explore the possibility of other characteristic decompositions leading to new classes of PPT states. Finally, it would be very interesting to look for multipartite generalization, that is, to construct circulant states in C d... C d for an arbitrary number of copies. Acknowledgments This work was partially supported by the Polish Ministry of Science and Higher Education Grant No 3004/B/H03/2007/33. References [1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, [2] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, arxiv: quantph/ [3] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). [4] P. Horodecki, Phys. Lett. A 232, 333 (1997). [5] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223, 1 (1996); [6] S. L. Woronowicz, Rep. Math. Phys. 10, 165 (1976). [7] P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. Lett. 82, 1056 (1999). [8] A. C. Doherty, P.A. Parrilo, and F. M. Spedalieri, Phys. Rev. Lett. 88, (2002). [9] D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal and A. V. Thapliyal, Phys. Rev. A 61, (2000). [10] P. Horodecki and M. Lewenstein, Phys. Rev. Lett. 85, 2657 (2000). [11] D. Bruss and A. Peres, Phys. Rev. A 61, (R) (2000). [12] S. Yu and N. Liu, Phys. Rev. Lett, 95, (2005). [13] S.-M. Fei, X. Li-Jost, B.-Z. Sun, Phys. Lett. A 352, 321 (2006). [14] M. Piani and C. Mora, Phys. Rev. A 75, (2007). [15] D. Chruściński and A. Kossakowski, Phys. Rev. A 74, (2006). [16] L. Clarisse, Phys. Lett. A 359 (2006) 603. [17] K.-C. Ha, Publ. RIMS, Kyoto Univ. 34, 591 (1998) [18] K. Ha, S.-H Kye, and Y. Park, Phys. Lett. A 313, 163 (2003). [19] K. Ha, S.-H. Kye, Phys. Lett. A (2004); J. Phys. A: Math. Gen. 38, 9039 (2005). [20] R.F. Werner, Phys. Rev. A 40, 4277 (1989). [21] M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999). [22] K.G.H. Vollbrecht and R.F. Werner, Phys. Rev. A 64, (2001). [23] D. Chruściński and A. Kossakowski, Phys. Rev. A 76, (2007). [24] D. Chruściński and A. Kossakowski, Open Systems and Inf. Dynamics, 14, 275 (2007). 8