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arXiv:quant-ph/0410063

Multiplicativity of Maximal p-Norms in Werner-Holevo channels for $1 \le p \le 2$

Nilanjana Datta

Published Oct 8, 2004• 6 citations
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Abstract

Recently, King and Ruskai [1] conjectured that the maximal p-norm of the Werner--Holevo channel is multiplicative for all $1\le p \le 2$. In this paper we prove this conjecture. Our proof relies on certain convexity and monotonicity properties of the p--norm.

DOI
10.48550/arxiv.quant-ph/0410063

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Multiplicativity of Maximal p-Norms in Werner-Holevo channels for 1≤p≤21 \le p \le 21≤p≤2
Recently, King and Ruskai [1] conjectured that the maximal p-norm of the Werner--Holevo channel is multiplicative for all 1≤p≤21\le p \le 21≤p≤2. In this paper we prove this conjecture. Our proof relies on certain convexity and monotonicity properties of the p--norm.
We suggest a tight inequality for norms in ddd-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for d=3d=3d=3 and provide a detailed numerical check for d≤200d\leq 200d≤200 confirming the conjecture. We conclude with a brief survey of solutions for kin problems which anyhow concern minimization of the output entropy of certain quantum channel and rely upon the symmetry properties of the problem. Key words and phrases: $l_p $-norm, Rényi entropy, tight inequality, maximization of a convex function.
In this paper, we present a class of maps for which the multiplicativity of the maximal output p-norm holds for p=2 and p≥4. The class includes all positive trace-preserving maps from B(C3) to B(C2). In this sense, the result is a generalization of the corresponding result in the work of King and Koldan [“New multiplicativity results for qubit maps,” J. Math. Phys. 47, 042106 (2006)], where the multiplicativity was proved for all positive trace-preserving maps from B(C2) to B(C2) with p=2 and p≥4. Interestingly, by contrast, the multiplicativity of p-norm was investigated in the context of quantum information theory and shown not to hold, in general, for high dimensional quantum channels [Hayden, P. and Winter, A., “Counterexamples to the maximal p-norm multiplicativity conjecture for all p>1,” Commun. Math. Phys. 284, 263 (2008)]. Moreover, the Werner–Holevo channel, which is a map from B(C3) to B(C3), is a counterexample for p>4.79 [Werner and Holevo, J. Math. Phys. 43, 4353 (2002).].
We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal ℓp-norm with p a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when p is large relative to the dimension d, the condition suggests that multiplicativity holds when d⩾2p−1. This conjecture is proved explicitly for p=2,3,4. Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for p=2.
It is conjectured that the Holevo capacity of a product channel Ω⊗Φ is achieved when product states are used as input. Amosov, Holevo, and Werner have also conjectured that the maximal lp norm of a product channel is achieved with product input states. In this article we establish both of these conjectures in the case that Ω is arbitrary and Φ is a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ω is arbitrary and either Φ is a qubit channel and p=2, or Φ is a unital qubit channel and p is integer. Our proofs involve a new conjecture for the norm of an output state of the half-noisy channel I⊗Φ, when Φ is a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.
For all 1 < p < 2, we demonstrate the existence of quantum channels with non-multiplicative maximal p-norms. Equivalently, the minimum output Renyi entropy of order p of a quantum channel is not additive for all 1 < p < 2. The violations found are large. As p approaches 1, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of maximal p-norm multiplicativity.