Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

Abstract

In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

Appendices

Appendix

Proof of Proposition 7

(i) As mentioned in [16] (and explicitly worked out in [5]) for $p\ge 1$, contractivity can be proven using the Riesz–Thorin interpolation theorem. So we focus on $p\in (-\infty , -1]\cup [1/2, 1)$. First let $p=-q\in (-\infty , -1]$, and $X> 0$. We note that

$$\begin{aligned} \Vert \Phi _t(X)\Vert _{p, \sigma } = \Vert \Phi _t(X)^{-1}\Vert _{q, \sigma }^{-1}. \end{aligned}$$

On the other hand, $\Phi _t$ is completely positive and unital, and $z\mapsto z^{-1}$ is operator convex. Therefore, by operator Jensen’s inequality $\Phi _t(X^{-1})\ge \Phi _t(X)^{-1}$ and by the monotonicity of the norm we have $\Vert \Phi _t(X)^{-1}\Vert _{q,\sigma } \le \Vert \Phi _t(X^{-1})\Vert _{q,\sigma }$. We conclude that

$$\begin{aligned} \Vert \Phi _t(X)\Vert _{p, \sigma } \ge \Vert \Phi _t(X^{-1})\Vert _{q,\sigma }^{-1} \ge \Vert X^{-1}\Vert _{q, \sigma }^{-1} = \Vert X\Vert _{p, \sigma }, \end{aligned}$$

where for the second inequality we use q-contractivity of $\Phi _t$ for $q\ge 1$.

Now suppose that $p\in [1/2, 1)$. We note that its Hölder conjugate ${\hat{p}}\in (-\infty , -1]$, and that $\Phi _t$ is reverse ${\hat{p}}$-contractive. Then using Hölder’s duality, for $X>0$ we have

$$\begin{aligned} \Vert \Phi _t(X)\Vert _p&= \inf _{Y>0: \Vert Y\Vert _{{\hat{p}}, \sigma }\ge 1} \langle Y, \Phi _t(X)\rangle _\sigma \\&= \inf _{Y>0: \Vert Y\Vert _{{\hat{p}}, \sigma }\ge 1} \langle \widehat{\Phi }_t(Y), X\rangle _\sigma \\&\ge \inf _{Z>0: \Vert Z\Vert _{{\hat{p}}, \sigma }\ge 1} \langle Z, X\rangle _\sigma \\&= \Vert X\Vert _{p, \sigma }, \end{aligned}$$

where $\widehat{\Phi }_t$ is the adjoint of $\Phi _t$ with respect to $\langle .,.\rangle _\sigma$, for each $t\ge 0$. Here the first equality follows from Lemma 6, and the inequality follows from the ${\hat{p}}$-contractivity of $\Phi _t$, i.e, $\Vert \Phi _t(Y)\Vert _{{\hat{p}}, \sigma } \ge \Vert Y\Vert _{{\hat{p}}, \sigma } \ge 1$.

(ii) As worked out in [14] this is an immediate consequence of the operator Jensen inequality.

Second Proof of Theorem 14

The proof is very similar to the one used in [3] to prove the strong $L_p$-regularity of the Dirichlet forms. Before stating the proof we need some definitions.

For a compact set I we let C(I) to be the Banach space of continuous, complex valued functions on I (equipped with the supremum norm). Then the Banach space $C(I\times I)$ becomes a $*$-algebra when endowed with the natural involution $f\mapsto f^*$ with $f^*(x,y)=\overline{f(x,y)}$. Thus $C(I\times I)$ is a $C^*$-algebra.

We endow $\mathcal {B}(\mathcal {H})$ with a Hilbert space structure by equipping it with the Hilbert–Schmidt inner product:

$$\begin{aligned} \langle X, Y\rangle _{\text {HS}}:= \text {tr}(X^\dagger Y). \end{aligned}$$

Fix $X,Y\in \mathcal {B}_{sa}(\mathcal {H})$, and let I be a compact interval containing the spectrum of both X and Y. We define a $*$-representation $\pi _{X,Y}: C(I\times I)\rightarrow \mathcal {B}\big (\mathcal {B}(\mathcal {H})\big )$ that is uniquely determined by its action on tensor products of functions as follows. For $f, g\in C(I)$ we define $\pi _{X, Y}(f\otimes g)\in \mathcal {B}\big ( \mathcal {B}(\mathcal {H}) \big )$ by

$$\begin{aligned} \pi _{X,Y}(f\otimes g) (Z)=f(X) Zg(Y),\qquad Z\in \mathcal {B}(\mathcal {H}). \end{aligned}$$

The following lemma can be found in [3] (see Lemma 4.2):

Lemma 34

$\pi _{XY}$ is a $*$-representation between $C^*$-algebras. That is,

• (i) $\pi _{XY}(1)=\mathcal {I}$, where 1 is the constant function on $I\times I$ equal to 1.

• (ii) $\pi _{XY}(f^*g)=\pi _{XY}(f)^*\pi _{XY}(g)$ for all $f,g\in C(I\times I)$.

• (iii) If $f\in C(I\times I)$, is a non-negative function, then $\pi _{XY}(f)$ is a positive semi-definite operator on $\mathcal {B}(\mathcal {H})$ for the Hilbert–Schmidt inner product, i.e., $\pi _{X, Y}(f)\in \mathcal {P}\big ( \mathcal {B}(\mathcal {H}) \big )$.

Now, for any function $f\in C(I)$, define ${\tilde{f}}$ to be the function in $C(I\times I)$ defined by

$$\begin{aligned} {\tilde{f}}(s,t)=\left\{ \begin{aligned}&\frac{f(s)-f(t)}{s-t}\qquad s\ne t\\&f'(s)\qquad ~~~~~~~~~ s=t. \end{aligned}\right. \end{aligned}$$

(55)

The following lemma, proved in [3] (see Lemma 4.2), gives a generalization of the chain rule formula to a derivation.

Lemma 35

Let $X, Y\in \mathcal {B}_{sa}(\mathcal {H})$ and let I be a compact interval containing the spectrums of X, Y. Let $f\in C(I)$ be a continuously differentiable function such that $f(0)=0$. Then for all $V\in \mathcal {B}(\mathcal {H})$ we have

$$\begin{aligned} Vf(Y)-f(X)V=\pi _{XY}({\tilde{f}})(VY-XV), \end{aligned}$$

where ${\tilde{f}}$ is defined by (55).

We can now prove the theorem. By the result of [11] (an extension of Lemma 13), there are superoperators $\partial _j:\mathcal {B}(\mathcal {H})\rightarrow \mathcal {B}(\mathcal {H})$ of the form

$$\begin{aligned} \partial _j(X)= [V_j, X] = V_j X- XV_j, \end{aligned}$$

where $V_j\in \mathcal {B}(\mathcal {H})$, such that

$$\begin{aligned} \langle X,\mathcal {L}(Y)\rangle _\sigma = \sum _{j}\langle \partial _jX,\partial _j Y\rangle _{\sigma }. \end{aligned}$$

(56)

Moreover, $V_j$’s are such that there are $\omega _j\ge 0$ with

$$\begin{aligned} \sigma V_j = \omega _j V_j \sigma . \end{aligned}$$

Using the above equation one can show [3] that

$$\begin{aligned}&\partial _j\big (I_{q,p}(X)\big )=\Gamma _\sigma ^{-\frac{1}{q}}\bigg (V_j \Big (\Gamma _{\sigma }^{\frac{1}{p}}\big (\omega _j^{-\frac{1}{2p}}X\big )\Big )^{\frac{p}{q}}-\Big (\Gamma ^{\frac{1}{p}}_\sigma \big (\omega _j^{\frac{1}{2p}}X\big )\Big )^{\frac{p}{q}}V_j\bigg ). \end{aligned}$$

(57)

For arbitrary $X> 0$ define $Y_j:= \omega _j^{-1/4}\, \Gamma _{\sigma }^{\frac{1}{2}}(X)$ and ${Z}_j:= \omega _j^{1/4}\,\Gamma _{\sigma }^{\frac{1}{2}}(X)$. Using (57) we compute

$$\begin{aligned} \mathcal {E}_{q,\mathcal {L}}\big (I_{q,2}(X)\big )&=\frac{q{\hat{q}}}{4}\big \langle I_{\hat{q},q}\big (I_{q,2}(X)\big ),\mathcal {L}\big (I_{q,2}(X)\big )\big \rangle _\sigma \nonumber \\&=\frac{q{\hat{q}}}{4}\big \langle I_{\hat{q},2}(X),\mathcal {L}\big (I_{q,2}(X)\big )\big \rangle _\sigma \nonumber \\&=\frac{q{\hat{q}}}{4} \sum _{j} \langle \partial _j I_{\hat{q},2}(X),\partial _j I_{q,2}(X)\rangle _\sigma \end{aligned}$$

(58)

$$\begin{aligned}&=\frac{q{\hat{q}}}{4} \sum _{j} \Big \langle \Gamma _\sigma ^{-\frac{1}{\hat{q}}} \Big (V_j {Y_j}^{2/\hat{q}}-{Z}_j^{2/\hat{q}}V_j\Big ), \Gamma ^{-\frac{1}{q}} \Big (V_jY_j^{2/q}-{Z}_j^{2/q}V_j\Big )\Big \rangle _\sigma \end{aligned}$$

(59)

$$\begin{aligned}&=\frac{q{\hat{q}}}{4} \sum _{j} \Big \langle V_j {Y_j}^{2/\hat{q}}-{Z}_j^{2/\hat{q}}V_j, V_j{Y}_j^{2/q}-{Z}_j^{2/q}V_j\Big \rangle _{\text {HS}}\nonumber \\&=\frac{q{\hat{q}}}{4}\sum _{j} \Big \langle \pi _{{Z}_j,{Y}_j}\big ({\tilde{f}}_{2/\hat{q}}\big )(V_j{Y}_j-{Z}_jV_j),\pi _{{Z}_j,{Y}_j}\big ({\tilde{f}}_{2/q}\big )(V_j{Y}_j-{Z}_jV_j)\Big \rangle _{\text {HS}}\end{aligned}$$

(60)

$$\begin{aligned}&=\frac{q{\hat{q}}}{4}\sum _{j} \Big \langle V_j{Y}_j-{Z}_jV_j, \pi _{{Z}_j,{Y}_j}\big ({\tilde{f}}_{2/\hat{q}}\big )^*\pi _{{Z}_j,{Y}_j}\big ({\tilde{f}}_{2/q}\big )(V_j{Y}_j-{Z}_jV_j)\Big \rangle _{\text {HS}}\nonumber \\&= \frac{q{\hat{q}}}{4} \sum _{j} \Big \langle V_j {Y}_j-Z_jV_j,\pi _{Z_j,Y_j}\big ({\tilde{f}}_{2/\hat{q}}^*{\tilde{f}}_{2/q}\big )(V_jY_j-Z_jV_j)\Big \rangle _{\text {HS}}, \end{aligned}$$

(61)

where in (58) we used (56), in (59) we used (57), and in (60) we used the chain rule formula of Lemma 35 for the functions $f_\alpha$ with $f_\alpha (x)=x^{\alpha }$. Finally, in (61) we used part (ii) of Lemma 34.

Now, using the proofs of Theorem 2.1 and Lemma 2.4 of [36], for any $x,y\ge 0$ and $0\le p\le q\le 2$ we have

$$\begin{aligned} q\hat{q}(x^{1/\hat{q}}-y^{1/\hat{q}})(x^{1/q}-y^{1/q})\le p\hat{p}(x^{1/\hat{p}}-y^{1/\hat{p}})(x^{1/p}-y^{1/p}). \end{aligned}$$

(62)

This means that for all x, y we have

$$\begin{aligned} q{\hat{q}} \big ({\tilde{f}}_{2/\hat{q}}^*{\tilde{f}}_{2/q}\big )(x, y) \le p{\hat{p}} \big ({\tilde{f}}_{2/\hat{p}}^*{\tilde{f}}_{2/p}\big )(x, y). \end{aligned}$$

Hence, by part (iii) of Lemma 34 we have

$$\begin{aligned} \mathcal {E}_{q,\mathcal {L}}(I_{q,2}(X))&\le \frac{p \hat{p}}{4} \sum _{j} \Big \langle V_j Y_j-Z_jV_j,\pi _{Z_j,Y_j}({\tilde{f}}_{2/\hat{p}}^*{\tilde{f}}_{2/p})(V_jY_j-Z_jV_j)\Big \rangle _{\text {HS}}\\&=\mathcal {E}_{p,\mathcal {L}}(I_{p,2}(X)). \end{aligned}$$

Remark 9

The difference with the proof of $L_p$-regularity of [3] lies in the choice of the inequality (62) used at the end of the proof.

Proof of Theorem 25

Since both $\text {Ent}_{2, \sigma }(X)$ and $\mathcal {E}_{2, \mathcal {L}}(X)$ are homogenous of degree two in X, to prove a log-Sobolev inequality, without loss of generality we can assume that X is of the form $X=\Gamma _\sigma ^{-1/2}(\sqrt{\rho })$ where $\rho$ is a density matrix. In this case

$$\begin{aligned} \text {Ent}_{2, \sigma }(X) = D(\rho \Vert \sigma ), \qquad \langle X, \mathcal {L}X\rangle _\sigma = 1- \big [\text {tr}\big (\sqrt{\sigma }\sqrt{\rho }\big )\big ]^2. \end{aligned}$$

Let $\sigma = \sum _{i=1}^d s_i|i\rangle \langle i|$ and $\rho =\sum _{k=1}^d r_k |{{\tilde{k}}}\rangle \langle {{\tilde{k}}}|$ be the eigen-decompositions of $\sigma$ and $\rho$. Then

$$\begin{aligned} \text {Ent}_{2, \sigma }(X) = \sum _{k=1}^d r_k \log r_k - \sum _{i, k=1}^d |\langle i| {{\tilde{k}}}\rangle |^2r_k \log s_i, \end{aligned}$$

and

$$\begin{aligned} \langle X, \mathcal {L}X\rangle _\sigma = 1- \Big ( \sum _{i, k=1}^d |\langle i| {{\tilde{k}}}\rangle |^2 \sqrt{s_i r_k} \Big )^2. \end{aligned}$$

Let $A=(a_{ik})_{d\times d}$ be a $d\times d$ matrix whose entries are given by

$$\begin{aligned} a_{ik} = |\langle i| {{\tilde{k}}}\rangle |^2. \end{aligned}$$

Observe that, fixing the eigenvalues $s_i$’s and $r_k$’s, the entropy $\text {Ent}_{2, \sigma }(X)$ is a linear function of A and $\mathcal {E}_{2, \mathcal {L}}(X)$ is concave function of A. On the other hand, since both $\{|1\rangle , \dots , |d\rangle \}$ and $\{|{{\tilde{1}}}\rangle , \dots , |{{\tilde{d}}}\rangle \}$ form orthonormal bases, A is a doubly stochastic matrix. Then by Birkhoff’s theorem, A can be written as a convex combination of permutations matrices. We conclude that if an inequality of the form

$$\begin{aligned} \beta \Big (\sum _{k=1}^d r_k \log r_k - \sum _{i, k=1}^d a_{ik}r_k \log s_i\Big )\le 1- \Big ( \sum _{i, k=1}^d a_{ik} \sqrt{s_i r_k} \Big )^2, \end{aligned}$$

holds for all permutation matrices A, then it holds for all doubly stochastic A, and then for all $\sigma , \rho$ with the given eigenvalues. We note that A is a permutation matrix when $\{|1\rangle , \dots , |d\rangle \}$ and $\{|{{\tilde{1}}}\rangle , \dots , |{{\tilde{d}}}\rangle \}$ are the same bases (up to some permutation) which means that $\sigma$ and $\rho$ commute. Therefore, a log-Sobolev inequality of the form

$$\begin{aligned} \beta \text {Ent}_{2, \sigma } \big ( \Gamma _{\sigma }^{-1/2}(\rho ) \big ) \le \mathcal {E}_{2, \mathcal {L}}\big (\Gamma _\sigma ^{-1/2}(\rho )\big ), \end{aligned}$$

holds for all $\rho$ if and only if it holds for all $\rho$ that commute with $\sigma$. That is, to find the log-Sobolev constant

$$\begin{aligned} \alpha _2(\mathcal {L}) = \inf _{\rho } \frac{\mathcal {E}_{2, \mathcal {L}}\big (\Gamma _\sigma ^{-1/2}(\rho )\big )}{\text {Ent}_{2, \sigma } \big ( \Gamma _{\sigma }^{-1/2}(\rho ) \big ) }, \end{aligned}$$

we may restrict to those $\rho$ that commute with $\sigma$. This optimization problem over such $\rho$ is equivalent to computing the 2-log-Sobolev constant of the classical simple Lindblad generator, and has been solved in Theorem A.1 of [18]. $\quad \square$