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Mar 2013

Volume 54, Issue 3, Articles (03xxxx)

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J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Andrew Neate and Aubrey Truman
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back to top Quantum Information and Computation

A new method to construct families of complex Hadamard matrices in even dimensions

D. Goyeneche

J. Math. Phys. 54, 032201 (2013); http://dx.doi.org/10.1063/1.4794068 (18 pages)

Online Publication Date: 13 March 2013

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We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.
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02.10.Yn Matrix theory

Tsirelson's problem and asymptotically commuting unitary matrices

Narutaka Ozawa

J. Math. Phys. 54, 032202 (2013); http://dx.doi.org/10.1063/1.4795391 (8 pages)

Online Publication Date: 15 March 2013

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In this paper, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.
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02.10.Yn Matrix theory
03.65.Fd Algebraic methods

Perturbation bounds for quantum Markov processes and their fixed points

Oleg Szehr and Michael M. Wolf

J. Math. Phys. 54, 032203 (2013); http://dx.doi.org/10.1063/1.4795112 (10 pages)

Online Publication Date: 19 March 2013

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We investigate the stability of quantum Markov processes with respect to perturbations of their transition maps. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. In the second part, we consider quantum Markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times. In this way we obtain a linear relation between the mixing time of a quantum Markov process and the sensitivity of its fixed point with respect to perturbations of the transition map.
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03.67.Hk Quantum communication
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.50.Ga Markov processes

Universal quantum state merging

I. Bjelaković, H. Boche, and G. Janßen

J. Math. Phys. 54, 032204 (2013); http://dx.doi.org/10.1063/1.4795243 (32 pages)

Online Publication Date: 19 March 2013

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We determine the optimal entanglement rate of quantum state merging when assuming that the state is unknown except for its membership in a certain set of states. We find that merging is possible at the lowest rate allowed by the individual states. Additionally, we establish a lower bound for the classical cost of state merging under state uncertainty. To this end we give an elementary proof for the cost in case of a perfectly known state which makes no use of the “resource framework.” As applications of our main result, we determine the capacity for one-way entanglement distillation if the source is not perfectly known. Moreover, we give another achievability proof for the entanglement generation capacity over compound quantum channels.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Hk Quantum communication
03.67.Mn Entanglement measures, witnesses, and other characterizations
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