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arXiv:1204.1813

Randomizing quantum states in Shatten ppp-norms

Kabgyun Jeong

Published Apr 9, 2012• 0 citations
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Abstract

In this paper, we formularize a method for randomizing quantum states with respect to the Schatten $p$-norm ($p\ge1$). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of $p>1$. 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 $p$) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.

DOI
10.48550/arxiv.1204.1813

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For large d, we study quantum channels on C d obtained by selecting ran- domly N independent Kraus operators according to a probability measure µ on the unitary group U(d). When µ is the Haar measure, we show that for N d/e 2 , such a channel is e-randomizing with high probability, which means that it maps every state within distance e/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discreti- zation argument. Moreover, for general µ, we obtain an e-randomizing channel provided N d (log d) 6 /e 2 .F ord = 2 k (k qubits), this includes Kraus operators obtained by ten- soring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.
Let Φ\PhiΦ be a super-operator, i.e., a linear mapping of the form Φ:L(F)→L(G)\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})Φ:L(F)→L(G) for finite dimensional Hilbert spaces F\mathcal{F}F and G\mathcal{G}G. This paper considers basic properties of the super-operator norms defined by ∥Φ∥q→p=sup⁡{∥Φ(X)∥p/∥X∥q : X≠0}\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}∥Φ∥q→p​=sup{∥Φ(X)∥p​/∥X∥q​:X=0}, induced by Schatten norms for 1≤p,q≤∞1\leq p,q\leq\infty1≤p,q≤∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if Φ\PhiΦ is completely positive, the value of the supremum in the definition of ∥Φ∥q→p\|\Phi\|_{q\rightarrow p}∥Φ∥q→p​ is achieved by a positive semidefinite operator XXX, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of p∈[1,∞]p\in [1,\infty]p∈[1,∞], there exists a super-operator Φ\PhiΦ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian XXX fail to achieve the supremum in the definition of ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p≥2p\geq 2p≥2, q≤2q\leq 2q≤2, and arbitrary Φ\PhiΦ, the norm ∥Φ∥q→p\|\Phi \|_{q\rightarrow p}∥Φ∥q→p​ is stable under tensoring Φ\PhiΦ with the identity super-operator, meaning that ∥Φ∥q→p=∥Φ⊗I∥q→p\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}∥Φ∥q→p​=∥Φ⊗I∥q→p​. For 1≤p<21\leq p < 21≤p<2, the norm ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​ may fail to be stable with respect to tensoring Φ\PhiΦ with the identity super-operator as just described, but ∥Φ⊗I∥1→p\|\Phi\otimes I\|_{1\rightarrow p}∥Φ⊗I∥1→p​ is stable in this sense for III the identity super-operator on L(H)\mathrm{L}(\mathcal{H})L(H) for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case p=1p=1p=1.
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 dlog⁡dd \log ddlogd 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 ddd.