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arXiv:0907.5446 · Journal of Mathematical Physics

Entanglement of random subspaces via the Hastings bound

Motohisa Fukuda, Christopher King

Published Apr 1, 2010• 25 citations
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Abstract

Recently, Hastings [“A counterexample to additivity of minimum output entropy,” Nat. Phys. 5, 255 (2009); e-print arXiv:0809.3972v3] proved the existence of random unitary channels, which violate the additivity conjecture. In this paper, we use Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of nonunital channels, which violate additivity of minimal output entropy.

DOI
10.1063/1.3309418

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We present a random construction of a pairs of channels which gives, with non-zero probability for sufficiently large dimensions, a counterexample to the minimum output entropy conjecture.As is known [4], this implies a violation of the additivity conjecture for the classical capacity of quantum channels.Thc violation of the minimum output cntropy conjecture is relatively small.

Physical Review Letters•Jan 1, 2008•21 citations
Recently, Hastings [“A counterexample to additivity of minimum output entropy,” Nat. Phys. 5, 255 (2009); e-print arXiv:0809.3972v3] proved the existence of random unitary channels, which violate the additivity conjecture. In this paper, we use Hastings’ method to derive new bounds for the entanglement of random subspaces of bipartite systems. As an application we use these bounds to prove the existence of nonunital channels, which violate additivity of minimal output entropy.
Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.
In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].
Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for all p≥0p\ge 0p≥0
The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order p>1p>1p>1, at the von Neumann point p=1p=1p=1, and for sufficiently small positive ppp, while much of the interval 0<p<10<p<10<p<1 has remained open. In this work, we prove that for every p>0p>0p>0 there exist finite-dimensional quantum channels whose minimum output ppp-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for p>3/4p>3/4p>3/4, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.
Non-Additivity of Minimum Output p-Reˊnyi\mathbf{R\acute{e}nyi}Reˊnyi Entropy
Hastings disproved additivity conjecture for minimum output entropy by using random unitary channels. In this note, we employ his approach to show that minimum output p−p-p−Rényi entropy is non-additive for p∈(0,p0)∪(1−p0,1)p\in(0,p_0)\cup(1-p_0,1)p∈(0,p0​)∪(1−p0​,1) where p0≈0.2855p_0\approx 0.2855p0​≈0.2855.