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arXiv:0802.4193

A remark on the paper ``Randomizing quantum states: Constructions and applications''

Guillaume Aubrun

Published Feb 28, 2008• 1 citation
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Abstract

The concept of $\e$-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly $d \log d$ random unitary operators provide examples of such channels on $\C^d$. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly $d$.

DOI
10.48550/arxiv.0802.4193

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The concept of \e\e\e-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly dlog⁡dd \log ddlogd random unitary operators provide examples of such channels on \Cd\C^d\Cd. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly ddd.
The randomization of quantum states in Quantum Information Theory is an important issue and has many applications on information processing. Such a method for randomizing quantum states is given by a completely positive and trace-preserving map known as an epsilon-randomizing map or a random unitary channel.\nIn this thesis, we represent a general formula for the\nepsilon-randomizing maps with respect to the p-norm of\nd dimensional quantum states and introduce some applications. Recently, the epsilon-randomizing channel together with its complementary channel gives a key technique for the proof of the additivity conjecture, that is, the additivity violation for the classical\ncapacity of quantum channel, which is the motivation of our work.\n\nThere are plenty of applications of the epsilon-randomizing channels in the view of reducing communication resources as well as the approximate encryption and decryption of quantum states. We explicitly investigate several applications such as\napproximate quantum state sharing, approximate state merging, and quantum sequential transmission protocol.
This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for εεε-randomizing maps, n+2log⁡(1/ε)+cn+2\log(1/ε)+cn+2log(1/ε)+c bits required to εεε-randomize an arbitrary nnn-qubit state by improving a scheme of Ambainis and Smith \cite{AS04} based on small bias spaces \cite{NN90, AGHP92}. We show by a probabilistic argument that in fact the great majority of random schemes using slightly more than this many bits of key are also εεε-randomizing. We provide the first known non-trivial lower bound for εεε-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future.
In this paper, we formularize a method for randomizing quantum states with respect to the Schatten ppp-norm (p≥1p\ge1p≥1). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of p>1p>1p>1. We exploit the methods of a net construction on the unit sphere and McDiard's inequality in probability theory, and then we prove certain general cases (all ppp) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.
Randomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/ε2) Pauli operators to map any d‐dimensional state to a state that is within trace distance ε of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and Smith.Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small ε.
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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].