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Entanglement thresholds for random induced states. (English) Zbl 1296.81014
In this work the authors consider the question whether a state obtained by partial tracing a random pure state is typically separable or typically entangled. A rigorous proof is presented in terms of asymptotic geometric analysis. The main result states the existence of effectively computable constants \(C, c>0\) and a function \(s_0(d)\) satisfying certain conditions such that, from the text, “if \(\rho\) is a random state on \(\mathbb{C}^d\otimes\mathbb{C}^d\) distributed according to the measure \(\mu_{d^2,s}\) then for any \(\epsilon>0\), i) If \(s\leq (1-\epsilon)s_0(d)\), we have \[ P(\rho\text{ is separable})\leq 2\exp(-c(\epsilon)d^3), \] ii) If \(s\geq(1+\epsilon)s_0(d)\), we have \[ P(\rho\text{ is entangled})\leq 2\exp(-c(\epsilon)s)." \]

MSC:
81P40 Quantum coherence, entanglement, quantum correlations
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
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References:
[1] Anderson, An introduction to random matrices 118 (2010) · Zbl 1184.15023
[2] Arveson, The probability of entanglement, Comm. Math. Phys. 286 (1) pp 283– (2009) · Zbl 1176.81011
[3] Aubrun, Partial transposition of random states and non-centered semicircular distributions, Random Matrices Theory Appl. 1 (2) pp 29– (2012) · Zbl 1239.60002
[4] Aubrun, Tensor products of convex sets and the volume of separable states on N qudits, Phys. Rev. A 73 (2) (2006)
[5] Aubrun, Hastings’s additivity counterexample via Dvoretzky’s theorem, Comm. Math. Phys. 305 (1) pp 85– (2011) · Zbl 1222.81131
[6] Aubrun, Phase transitions for random states and a semicircle law for the partial transpose, Phy. Rev. A 85 (3) (2012)
[7] Bai, Convergence to the semicircle law, Ann. Probab. 16 (2) pp 863– (1988) · Zbl 0648.60030
[8] Banaszczyk, The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces, Math. Oper. Res. 24 (3) pp 728– (1999) · Zbl 0965.52009
[9] Bengtsson, Geometry of quantum states (2006)
[10] Bhatia, Matrix analysis 169 (1997) · Zbl 0863.15001
[11] Bourgain, On martingales transforms in finite-dimensional lattices with an appendix on the K-convexity constant, Math. Nachr. 119 pp 41– (1984) · Zbl 0568.42010
[12] Buchleitner, organizers. Mini-workshop: geometry of quantum entanglement, Oberwolfach Rep. 6 (4) pp 2993– (2009) · Zbl 1192.00052
[13] Collins, The absolute positive partial transpose property for random induced states, Random Matrices: Theory and Applications 1 (3) (2012) · Zbl 1260.60014
[14] Davidson, Local operator theory, random matrices and Banach spaces. Handbook of the geometry of Banach spaces 1 pp 317– (2001)
[15] Davidson, Addenda and corrigenda to: ”Local operator theory, random matrices and Banach spaces.” 2 pp 1819– (2003)
[16] Einstein, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47 (10) pp 777– (1935) · Zbl 0012.04201
[17] Fawzi , O. Hayden , P. Sen , P. From low-distortion norm embeddings to explicit uncertainty relations and efficient information locking. STOC’11 - Proceedings of the 43rd ACM Symposium on Theory of Computing (San Jose, 2011) 773 782 2011 · Zbl 1288.81026
[18] Figiel, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math. 33 (2) pp 155– (1979) · Zbl 0427.46010
[19] Galambos, Advanced probability theory 3 (1988)
[20] Gurvits , L. Classical deterministic complexity of Edmond’s problem and quantum entanglement. Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (San Diego, 2003) 10 19 2003 · Zbl 1192.68252
[21] Gurvits, Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A 66 (6) (2002)
[22] Gurvits, Better bound on the exponent of the radius of the multipartite separable ball, Phys. Rev. A 72 (3) (2005)
[23] Haagerup, Random matrices with complex Gaussian entries, Expo. Math. 21 (4) pp 293– (2003) · Zbl 1041.15018
[24] Hastings, Superadditivity of communication capacity using entangled inputs, Nature Physics 5 (4) pp 255– (2009)
[25] Hayden, Aspects of generic entanglement, Comm. Math. Phys. 265 (1) pp 95– (2006) · Zbl 1107.81011
[26] Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232 (5) pp 333– (1997) · Zbl 1053.81504
[27] Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1-2) pp 1– (1996) · Zbl 1037.81501
[28] Horodecki, Mixed-state entanglement and distillation: Is there a ”bound” entanglement in nature?, Phys. Rev. Lett. 80 (24) pp 5239– (1998) · Zbl 0947.81005
[29] Kendon, Bounds on entanglement in qudit subsystems, Phys. Rev. A 66 (6) pp 062310– (2002)
[30] Latała, Gaussian measures of dilatations of convex symmetric sets, Ann. Probab. 27 (4) pp 1922– (1999) · Zbl 0966.60037
[31] Ledoux, The concentration of measure phenomenon 89 (2001) · Zbl 0995.60002
[32] Lévy, Problèmes concrets d’analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino (1951)
[33] Li, Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods 19 pp 533– (2001)
[34] Marčhenko, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N. S.) 72 (114) pp 507– (1967)
[35] Mehta, Random matrices (1991)
[36] Milman, Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. Math. 152 (2) pp 314– (2000) · Zbl 0974.52004
[37] Milman, Asymptotic theory of finite dimensional normed spaces 1200 (1986)
[38] Nielsen, Quantum computation and quantum information (2000) · Zbl 1049.81015
[39] Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (8) pp 1413– (1996) · Zbl 0947.81003
[40] Pisier, Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert, Ann. Sci. École Norm. Sup. (4) 13 pp 23– (1980)
[41] Pisier, The volume of convex bodies and Banach space geometry 94 (1989) · Zbl 0698.46008
[42] Rogers, Convex bodies associated with a given convex body, J. London Math. Soc. 33 pp 270– (1958) · Zbl 0083.38402
[43] Rudelson, Distances between non-symmetric convex bodies and the M M*-estimate, Positivity 4 (2) pp 161– (2000) · Zbl 0959.52008
[44] Ruskai, Bipartite states of low rank are almost surely entangled, J. Phys. A 42 (9) pp 15– (2009) · Zbl 1159.81018
[45] Serre, Linear representations of finite groups 42 (1977) · Zbl 0355.20006
[46] Shor, Algorithms for quantum computation: discrete logarithms and factoring pp 124– (1994)
[47] Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab. 13 (4) pp 1364– (1985) · Zbl 0591.60025
[48] Stormer, Positive linear maps of operator algebras, Acta Math. 110 pp 233– (1963) · Zbl 0173.42105
[49] Szarek, The volume of separable states is super-doubly-exponentially small in the number of qubits, Phys. Rev. A (3) 72 (3) pp 10– (2005)
[50] Szarek, On the structure of the body of states with positive partial transpose, J. Phys. A 39 (5) pp L119– (2006) · Zbl 1085.81035
[51] Szarek, Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive, J. Math. Phys. 49 (3) pp 21– (2008) · Zbl 1153.81441
[52] Szarek, How often is a random quantum state k-entangled?, J. Phys. A 44 (4) pp 15– (2011) · Zbl 1206.81023
[53] Walgate, Generic local distinguishability and completely entangled subspaces, J. Phys. A 41 (37) pp 15– (2008) · Zbl 1147.81011
[54] Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a -variable model, Phys. Rev. A 40 (8) pp 4277– (1989) · Zbl 1371.81145
[55] Woronowicz, Positive maps of low dimensional matrix algebra, Rep. Math. Phys. 10 (2) pp 165– (1976) · Zbl 0347.46063
[56] Ye, On the Bures volume of separable quantum states, J. Math. Phy. 50 (8) pp 14– (2009) · Zbl 1298.81030
[57] Ye, On the comparison of volumes of quantum states, J. Phys. A 43 (31) pp 17– (2010) · Zbl 1194.81027
[58] \.Zyczkowski, Induced measures in the space of mixed quantum states, J. Phys. A 34 (35) pp 7111– (2001) · Zbl 1031.81011
[59] \.Zyczkowski, Hilbert-Schmidt volume of the set of mixed quantum states, J. Phys. A 36 (39) pp 10115– (2003) · Zbl 1052.81012
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