Hypercontractivity for Semigroups of Unital Qubit Channels

Abstract

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form e −t 1 H 1 ⊗ · · · ⊗ e −tnHn to be a contraction from L p to L q , where L p is the algebra of 2 n -dimensional matrices equipped with the normalized Schatten norm, and each generator H j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.

Hypercontractivity in finite-dimensional matrix algebras

“…Actually, similar results have been already studied for semigroups consisting of tensor products of q-bits channels, i.e. of the form e −t 1 Hn ⊗ · · · ⊗ e −tnHn acting on M 2 n , where the tensor product structure was crucially exploited by the authors, see for instance [21,24].…”

Potential output purity of completely positive maps

“…Theorems 1 and 2 will be proved in the next section. Our next two results present some special cases where there is no gap between potential purity and purity, that is where equality holds in (12). In the paper [13] it was shown that equality in (12) holds at q = 2 for a class of entanglement-breaking maps which includes the CQ maps.…”

“…Since this holds for all Ω we immediately get Φ (pot) q→p ≤ Φ q→p . The reverse inequality (12) follows immediately from the definition, so we deduce that equality holds.…”

“…Our next Theorem extends Watrous' result to include all Hadamard maps (recall that the identity map is one example of a Hadamard map). Remarks: Equality in (12) has been shown for unital qubit channels [12] in the range 1 ≤ q ≤ 2 ≤ p. Equality is also known to hold for maps with entrywise non-negative Choi matrices when q = 2 and p is an even integer [14]. Except for Theorem 3 not much is known about conditions for equality outside the range of values 1 ≤ q ≤ 2 ≤ p. Hypercontractivity bounds can provide some related but weaker results [12]: for example, for the qubit depolarizing channel ∆ λ we have ∆ ⊗n…”

Tensorization of the strong data processing inequality for quantum chi-square divergences

“…The bijection maps that preserve the quantum χ 2 κα divergence (see Example 3 about the family κ α ) have been characterized in [4], which complements the study of the contraction of quantum states. There are other tools based on the functional perspective, including quantum (reverse) hypercontractivity and related quantum functional inequalities [3,7,10,11,19].…”