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Towards a state minimizing the output entropy of a tensor product of random quantum channels

### Abstract

We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.

© 2012 American Institute of Physics

Received 04 December 2011
Accepted 25 February 2012
Published online 21 March 2012

Acknowledgments:
The three authors would like to thank in the first place the “Quantum Information Theory” program at the Mittag-Leffler Institute, where this collaboration was initiated. Our research was supported by NSERC Discovery grants and an ERA at the University of Ottawa (B.C.). The research of I.N. was supported by a PEPS grant from the Institute of Physics of the CNRS. The research of B.C. was also supported by the ANR Granma and this manuscript was finalized while he was visiting the RIMS at Kyoto University. The research of M.F. was supported by QuantumWorks and NSERC Discovery grant.

Article outline:

I. INTRODUCTION
A. Background
II. REVIEW ON RANDOM QUANTUM CHANNELS AND UNITARY INTEGRATION
A. Random quantum channels
B. The Hayden-Winter trick
C. Unitary integration
D. Graphical calculus
III. GENERALIZED BELL STATES FOR .
A. Well-behaved input
B. Consequence of Theorem 3.1
C. Ill-behaved input
IV. GENERALIZED BELL STATES FOR *U* × *U*
V. WHAT DISTINGUISHES in the graphical calculus
A. Conjugate versus identical channels
B. Two different models: *U* ⊗ *U** and *U* ⊗ *U* ^{ T }
VI. NON-PURE INPUT STATES FOR
A. Mixing a pure Bell state
B. Adapted mixed inputs for
VII. DISCUSSION

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2012-03-21

2016-02-08

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