arXiv:2105.00462 · Journal of Mathematical Physics

Improved quantum hypercontractivity inequality for the qubit depolarizing channel

Salman Beigi

Published Dec 1, 2021• 4 citations

Abstract

The hypercontractivity inequality for the qubit depolarizing channel Ψt states that ‖Ψt⊗n(X)‖p≤‖X‖q, provided that p ≥ q > 1 and t≥lnp−1q−1. In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber–Krahn inequality on the hypercube and a new quantum Schwartz–Zippel lemma.

DOI: 10.1063/5.0056388

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arXiv:2105.00462 · Journal of Mathematical Physics

Improved quantum hypercontractivity inequality for the qubit depolarizing channel

Salman Beigi

Published Dec 1, 2021• 4 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/5.0056388

Related papers

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(Φ_t)_{t\ge 0}

and every tensor power

n

, we show that the log-Sobolev constant of the product semigroup

Φ_t^{\otimes n}

is at least

2/(3\ln 2)\approx 0.96

times the log-Sobolev constant of the single-site semigroup

Φ_t

, independently of

n

and the local dimension

d

. The proof first establishes an exact tensorization of the

(q,2)

-hypercontractive inequality for integer

q

, in particular

q=3

, and then extends the estimate to all real

q>2

by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp

(q,2)

-hypercontractivity estimates for generalized qubit depolarizing channels.

\mathbb {L}_p

L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated

\mathbb {L}_p

L p norms are not uniformly convex for

1\le p \le 2

1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

σ

, we obtain the modified Log-Sobolev bound

α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}

where

λ

is the spectral gap. This removes the assumption of

L_p

-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.

L_p

spaces to obtain a general hypercontractive inequality from a particular

p \rightarrow q

-norm bound. Then, we derive a bound on the

2 \rightarrow 4

-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an

N

-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time

{\rm poly}(\log(N))

by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.