Interpolation Between Modified Logarithmic Sobolev and Poincaré Inequalities for Quantum Markovian Dynamics

Abstract

We define the quantum p-divergence and introduce Beckner’s inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative $L_p$-norm. We obtain a number of implications between Beckner’s inequalities and other quantum functional inequalities, as well as the hypercontractivity. In particular, we show that quantum Beckner’s inequalities interpolate between Sobolev-type inequalities and Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constant $\alpha _p$ in terms of the spectral gap and establish the stability of $\alpha _p$ with respect to the invariant state. As applications, we compute the Beckner constant for the depolarizing semigroup and discuss the mixing time. For symmetric quantum Markov semigroups, we derive the moment estimate, which further implies a concentration inequality. We introduce a new class of quantum transport distances $W_{2,p}$ interpolating the quantum 2-Wasserstein distance by Carlen and Maas (J Funct Anal 273(5):1810–1869, 2017) and a noncommutative ${\dot{H}}^{-1}$ Sobolev distance. We show that the quantum Markov semigroup with $\sigma$-GNS detailed balance is the gradient flow of a quantum p-divergence with respect to the metric $W_{2,p}$. We prove that the set of quantum states equipped with $W_{2,p}$ is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of p-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner’s inequality, from which a transport cost and Poincaré inequalities can follow.

Appendices

Appendix A: Some Additional Preliminaries

1.1 A.1: Quantum $\chi ^2$-Divergence

In this appendix, we briefly recall the quantum $\chi ^2$-divergences introduced in [104]. For any $\sigma \in \mathcal {D}_+(\mathcal {H})$ and $\kappa : (0,\infty ) \rightarrow (0,\infty )$, we define the operator:

$$\begin{aligned} \Omega _\sigma ^\kappa = R_\sigma ^{-1} \kappa (\Delta _\sigma ):\ \mathcal {B}(\mathcal {H})\rightarrow \mathcal {B}(\mathcal {H})\,. \end{aligned}$$

(A.1)

Recalling $J_\sigma ^f$ in (2.4), clearly there holds

$$\begin{aligned} (\Omega _\sigma ^\kappa )^{-1} = J_\sigma ^{1/\kappa }\,. \end{aligned}$$

(A.2)

The quantum $\chi _\kappa ^2$-divergence for $\rho \in \mathcal {D}(\mathcal {H})$ and $\sigma \in \mathcal {D}_+(\mathcal {H})$ is defined by

$$\begin{aligned} \chi _{\kappa }^2(\rho ,\sigma ) = \langle \rho - \sigma , \Omega _\sigma ^\kappa (\rho -\sigma )\rangle \,. \end{aligned}$$

(A.3)

To make the divergence $\chi _{\kappa }^2(\cdot ,\cdot )$ have nice properties, we usually consider $\kappa$ in the following functional class:

$$\begin{aligned} \mathcal {K} = \{\kappa : (0,\infty ) \rightarrow (0,\infty )\,;\ \kappa \ \text {is operator convex},\ x\kappa (x) = \kappa (x^{-1}),\ \kappa (1) = 1\}. \end{aligned}$$

For the purposes of this work, the following family of power difference means in $\mathcal {K}$ is of particular interest [67]:

$$\begin{aligned} \kappa _\alpha = \frac{\alpha }{\alpha -1} \frac{x^{\alpha - 1} - 1}{x^{\alpha } - 1}\,, \quad \alpha \in [-1,2]\,. \end{aligned}$$

(A.4)

In fact, the kernel function of the operator $J_\sigma ^{1/k}$ in (A.2) is given by

$$\begin{aligned} M_\alpha = y \kappa _\alpha ^{-1}(x/y) = \frac{\alpha - 1}{\alpha }\frac{x^\alpha - y^\alpha }{x^{\alpha - 1} - y^{\alpha - 1}}\,, \end{aligned}$$

which is the so-called A-L-G interpolation mean since $M_\alpha$, for $\alpha = -1$, $\alpha = 1/2$, $\alpha = 1$, and $\alpha = 2$, gives the harmonic mean, the geometric mean, the logarithmic mean, and the arithmetic mean, respectively.

1.2 A.2: Noncommutative Calculus

In this appendix, following [12, 36], we review some fundamentals about noncommutative calculus associated with the derivation $\partial _j$. Let $A,B \in \mathcal {B}_{sa}(\mathcal {H})$ admit the spectral decompositions:

$$\begin{aligned} A = \sum _{i = 1}^{d}\lambda _i A_i\,,\quad B = \sum _{k=1}^{d} \mu _k B_k\,, \end{aligned}$$

where $\lambda _i$ and $\mu _k$ are eigenvalues of A and B, respectively; $A_i$ and $B_i$ are the associated rank-one spectral projections. For a function $f \in C(I \times I)$ with I being a compact interval containing the spectra of A and B, we define the Schur multiplier (double sum operator) by [22, 45, 98]

$$\begin{aligned} f(A,B) = \sum _{i,k = 1}^{d} f(\lambda _i,\mu _k) L_{A_i}R_{B_k}\,, \end{aligned}$$

(A.5)

where $C(I \times I)$ is the Banach space of complex-valued continuous functions on $I \times I$. It was observed in [12] that given $A,B \in \mathcal {B}_{sa}(\mathcal {H})$, f(A, B) is $*$-representation between $C(I \times I)$ and $\mathcal {B}(\mathcal {B}(\mathcal {H}))$. Indeed, we have the following lemma from [12, Lemma 4.1] and [36, Lemma 6.6].

Lemma A.1

Let $A,B \in \mathcal {B}_{sa}(\mathcal {H})$ and the compact interval I contain the spectra of A and B. It holds that

• 1. $f(A,B)g(A,B) = (fg)(A,B)$ for $f,g \in C(I \times I)$.

• 2. If $f \in C(I \times I)$ is non-negative, then f(A, B) is a positive semidefinite operator on $B(\mathcal {H})$ with respect to the inner product $\langle \cdot , \cdot \rangle$. It follows that if f is strictly positive, the sequilinear form $\langle \cdot , f(A,B)(\cdot )\rangle$ defines an inner product on $\mathcal {B}(\mathcal {H})$.

In this work, we mainly consider the case where f is the divided difference of some differentiable function $\varphi$ on I:

$$\begin{aligned} \varphi ^{[1]}(\lambda ,\mu ) = {\left\{ \begin{array}{ll} \frac{\varphi (\lambda )-\varphi (\mu )}{\lambda -\mu }\,, \quad &{} \lambda \ne \mu \,, \\ \varphi '(\lambda )\,,\quad &{} \lambda = \mu \,, \end{array}\right. } \end{aligned}$$

(A.6)

which is closely related to the chain rule for $\partial _j$ (cf. [12, Lemma 4.2] and [36, Proposition 6.2]).

Lemma A.2

Under the same assumption as in Lemma A.1, for any $f: I \rightarrow \mathbb {C}$, we have, for $V \in \mathcal {B}(\mathcal {H})$,

$$\begin{aligned} V f(B) - f(A) V = f^{[1]}(A,B)(V B - A V)\,. \end{aligned}$$

(A.7)

Then, by Lemma A.2, for a differentiable curve $A(t): (a,b) \rightarrow \mathcal {B}(\mathcal {H})$ and function f, we have

$$\begin{aligned}&\frac{d}{d t} f(A(t)) = f^{[1]}(A(t),A(t))(A'(t))\,. \end{aligned}$$

(A.8)

We also need a multiple operator version of (A.8). We recall that for a differentiable function $\varphi : \mathbb {R}^n \rightarrow \mathbb {C}$, the partial divided difference $\delta _j \varphi : \mathbb {R}^{n+1} \rightarrow \mathbb {C}$ with respect to the variable $x_j$ is defined by

$$\begin{aligned} (\delta _{j} \varphi )(x_1,\ldots ,x_{j-1},(\lambda ,\mu ),x_{j+1},\ldots ,x_n)= \left( \varphi (x_1,\ldots ,x_{j-1},\cdot ,x_{j+1},\ldots ,x_n)\right) ^{[1]}(\lambda ,\mu )\,. \end{aligned}$$

(A.9)

Let $A^{(k)}$, $k = 1,\ldots ,n$, be self-adjoint operators with the spectral decompositions: $A_i = \sum _{i} \lambda _{i}^{(k)}A_{i}^{(k)}$, where $\lambda _i^{(k)}$ are eigenvalues and $A_{i}^{(k)}$ are the associated rank-one spectral projections. For a function $\varphi : I \times \cdots \times I \rightarrow \mathbb {C}$ with the interval I containing the spectra of $A^{(i)}$, the multiple operator sum is defined as:

$$\begin{aligned} \varphi (A_1,\ldots ,A_n) = \sum _{i_1,\cdots ,i_n = 1}^{d} \varphi (\lambda ^{(1)}_{i_1},\cdots ,\lambda ^{(n)}_{i_n})A_{i_1}^{(1)}\otimes \cdots \otimes A_{i_n}^{(n)}. \end{aligned}$$

(A.10)

The following chain rule from [36, Proposition 6.8] shall be useful in the expression of the geodesic equations for the generalized quantum transport distance.

Lemma A.3

Let the curves $A_t, B_t: (a,b) \rightarrow \mathcal {B}_{sa}(\mathcal {H})$ be differentiable, and let $\varphi : I \times I \rightarrow \mathbb {C}$ be differentiable with I containing the spectra of $A_t$ and $B_t$ for all $t \in (a,b)$. Then there holds

$$\begin{aligned} \partial _t \varphi (A_t,B_t)(\cdot ) = (\delta _1 \varphi ) ((A_t,A_t),B_t)[\partial _t A_t, \cdot ] + (\delta _2 \varphi )(A_t,(B_t,B_t))[\cdot , \partial _t B_t]. \end{aligned}$$

Appendix B: Note on the Detailed Balance Condition

Noting (4.13), it follows from Lemma 2.2 that the QMS satisfying $\sigma$-GNS DBC is also self-adjoint with respect to the inner product $\langle \cdot , \cdot \rangle _{[\sigma ]_{p,0}}$. In this appendix, we modify the discussions in [36, Appendix B] and [20, Appendix B] to show that the $[\sigma ]_{p,0}$-DBC, for $p \in (1,2)$, and KMS DBC are not comparable, and that there exists a primitive QMS satisfying $[\sigma ]_{p,0}$-DBC but not $\sigma$-GNS DBC.

Let $\{\left| 0\right\rangle ,\left| 1\right\rangle \}$ be the standard basis of $\mathbb {C}^2$, and $\left| v_1\right\rangle = \frac{1}{\sqrt{2}}(\left| 0\right\rangle + \left| 1\right\rangle )$ and $\left| v_2\right\rangle = \frac{1}{\sqrt{5}}(\left| 0\right\rangle + 2\left| 1\right\rangle )$ be an another basis of $\mathbb {C}^2$. We define the quantum channel $\Phi (X) = K_1^* X K_1 + K_2^* X K_2$ with $K_1 = \left| v_1\right\rangle \left\langle 0\right|$ and $K_2 = \left| v_2\right\rangle \left\langle 1\right|$. It is easy to see that the associated unique invariant state of $\Phi ^\dag$ is

$$\begin{aligned} \sigma = \frac{1}{7}\left[ \begin{matrix} 2 &{} 3 \\ 3 &{} 5 \end{matrix} \right] , \end{aligned}$$

and the spectrum of $\Phi ^\dag$ is given by $\{1, \frac{3}{10}, 0\}$ with 0 of multiplicity two. We denote by $\Phi ^{\dagger }_{\textrm{KMS}}$ the adjoint of $\Phi$ with respect to the KMS inner product and define $\Psi = \Phi ^\dag _{\textrm{KMS}} \Phi$, which is also a quantum channel with $\Psi ^\dag (\sigma ) = \sigma$ but satisfying the KMS DBC. Note that for a linear map $\Phi$ on $\mathcal {B}(\mathcal {H})$ satisfying both $[\sigma ]_{p,0}$-DBC and KMS DBC, there holds $g(\Delta _\sigma ) \circ \Phi = \Phi \circ g(\Delta _\sigma )$, where $g(x) = \kappa _{1/p}(x) x^{1/2}$ and $g(\Delta _\sigma ) = [\sigma ]_{p,0}^{-1} \Gamma _\sigma$. By definition (A.4), we have $g(1/x) = g(x)$. Then, by exactly the same argument as in [20, Appendix B], we can show that the operator $\Psi$ defined above does not commute with $g(\Delta _\sigma )$ and hence the Lindbladian $\Psi - \textrm{id}$ satisfies the KMS DBC but not $[\sigma ]_{p,0}$-DBC.

We next consider $\widetilde{\Psi }:= [\sigma ]_{p,0}^{-1} \circ \Psi ^\dag \circ \Gamma _\sigma$, which is a quantum channel, since the operator $[\sigma ]_{p,0}^{-1}$ is completely positive for $p \in (1,2)$ by [68, Example 4.7]. Then we can check from definition that $\widetilde{\Psi }$ satisfies the $[\sigma ]_{p,0}$-DBC. Again, since $\Psi$ defined above does not commute with $g(\Delta _\sigma )$, neither does $\widetilde{\Psi }$. Hence, the Lindbladian $\widetilde{\Psi } - \textrm{id}$ satisfies $[\sigma ]_{p,0}$-DBC but not KMS DBC.

We conclude with an example of a primitive QMS with $[\sigma ]_{p,0}$-DBC but not $\sigma$-GNS DBC. The construction is modified from [20] as well. We define, for some $\eta \in (0,1/2)$,

$$\begin{aligned} K_1 = \left[ \begin{array}{ll} \sqrt{\eta } &{} 0 \\ 0 &{} \sqrt{1 - \eta } \end{array} \right] \,,\quad K_2 = \left[ \begin{array}{ll} 0 &{} \sqrt{\eta } \\ \sqrt{1 - \eta } &{} 0 \end{array} \right] \,, \end{aligned}$$

and the associated quantum channel $\Phi (X) = K_1^* X K_1 + K_2^* X K_2$, which has the unique invariant state:

$$\begin{aligned} \sigma = \left[ \begin{matrix} \eta &{} 0 \\ 0 &{} 1 - \eta \end{matrix} \right] . \end{aligned}$$

It is direct to verify $[\sigma ]_{p,0}^{-1} \circ \Phi ^\dag = \Phi \circ [\sigma ]_{p,0}^{-1}$, i.e., $\Phi$ satisfies the $[\sigma ]_{p,0}$-DBC. However, it was shown in [20] that $\sigma$-GNS DBC does not hold for $\Phi$. It follows that the generator $\Phi - \textrm{id}$ is the desired example.

Appendix C: Beckner’s Inequality for Non-primitive QMS

In this appendix, we introduce p-Beckner’s inequality with $p \in (1,2]$ for the non-primitive QMS and show that it holds for any QMS satisfying GNS DBC, which extends [63, Theorem 3.3] for MLSI.

Let $\mathcal {P}_t = e^{t \mathcal {L}}$ be a non-primitive QMS with a full-rank invariant state $\sigma \in \mathcal {D}_+(\mathcal {H})$, which may be non-unique. We introduce the fixed-point algebra $\mathcal {F}:= \left\{ X \in \mathcal {B}(\mathcal {H});\ \forall t \ge 0,\, \mathcal {P}_t(X) = X \right\}$, and denote the associated conditional expectation by $E_\mathcal {F}$. It is known from [58, 63] that $E_\mathcal {F} \mathcal {P}_t = \mathcal {P}_t E_\mathcal {F} = E_\mathcal {F}$ and for $X \in \mathcal {B}(\mathcal {H})$,

$$\begin{aligned} \lim _{t \rightarrow \infty } \mathcal {P}_t (X - E_{\mathcal {F}}(X)) = \lim _{t \rightarrow \infty } \mathcal {P}_t (X) - E_{\mathcal {F}}(X) = 0\,. \end{aligned}$$

(C.1)

Similarly, Beckner’s inequality quantifies the convergence rate of (C.1) in terms of the p-divergence (3.1). We consider QMS that satisfies $\sigma$-GNS DBC as in Definition 2.1, which is well-defined since the self-adjointness of $\mathcal {L}$ with respect to $\langle \cdot ,\cdot \rangle _{\sigma ,1}$ is independent of the choice of invariant state $\sigma$; see [63, Lemma 2.6]. We compute the entropy production of $\mathcal {F}_{p,\sigma }(\rho )$ as in (3.16): for $\rho _t = \mathcal {P}_t^\dag (\rho )$ with $\rho \in \mathcal {D}_+(\mathcal {H})$,

$$\begin{aligned} \frac{d}{dt}\Big |_{t = 0} \mathcal {F}_{p,E_\mathcal {F}^\dag (\rho )}(\rho _t) = -\frac{4}{p^2} \mathcal {E}_{p,\mathcal {L}}(X)\,,\quad X = \Gamma _{E_\mathcal {F}^\dag (\rho )}^{-1}(\rho ) \,, \end{aligned}$$

where $\mathcal {E}_{p,\mathcal {L}}$ is given in (2.17) with $\sigma = E_\mathcal {F}^\dag (\rho )$. Hence, we can define the non-primitive Beckner’s inequality as (3.18): for some $\alpha _p > 0$ and any $\rho \in \mathcal {D}_+(\mathcal {H})$,

$$\begin{aligned} \alpha _p \mathcal {F}_{p,E_{\mathcal {F}}^\dag (\rho )}(\rho ) \le p^{-2} \mathcal {E}_{p,\mathcal {L}}\big (\Gamma _{E_\mathcal {F}^\dag (\rho )}^{-1}(\rho )\big )\,, \end{aligned}$$

(C.2)

which is equivalent to the exponential convergence: $\mathcal {F}_{p,E_{\mathcal {F}}^\dag (\rho )} (\mathcal {P}_t^\dag (\rho )) \le e^{- 4 \alpha _p t} \mathcal {F}_{p,E_{\mathcal {F}}^\dag (\rho )}(\rho )$.

We introduce the subalgebra index for the fixed-point algebra $\mathcal {F}$ by

$$\begin{aligned} C(E_\mathcal {F}) = \inf \big \{c> 0 \,;\ \rho \le c E_{\mathcal {F}}^\dag (\rho )\,,\ \forall \rho \in \mathcal {D}(\mathcal {H})\big \}\,, \end{aligned}$$

(C.3)

which is finite in the finite-dimensional setting [97, Theorem 6.1]. For a primitive QMS with the unique invariant state $\sigma \in \mathcal {D}_+(\mathcal {H})$, the index (C.3) reduces to

$$\begin{aligned} C(\sigma )&:= \inf \{c > 0\,;\ \rho \le c \sigma \ \, \text {for all}\ \rho \in \mathcal {D}(\mathcal {H}) \}\,, \end{aligned}$$

(C.4)

which is closely related to the max-relative entropy $D_{\infty }$ in (2.32) and can be explicitly represented by (2.33),

$$\begin{aligned} C(\sigma ) = \sup _{\rho \in \mathcal {D}(\mathcal {H})} \exp \left( D_{\infty }(\rho \Vert \sigma ) \right) = \sigma _{\min }^{-1}\,. \end{aligned}$$

(C.5)

We also recall the spectral gap (Poincaré constant) $\lambda (\mathcal {L})$ for a non-primitive Lindbladian $\mathcal {L}$:

$$\begin{aligned} \lambda (\mathcal {L}) = \inf _{X \in \mathcal {B}(\mathcal {H})} \frac{-\langle X, \mathcal {L}(X) \rangle _{\sigma ,f}}{\left\Vert X - E_\mathcal {F}(X) \right\Vert _{\sigma ,f}^2}\,, \end{aligned}$$

for an invariant state $\sigma \in \mathcal {D}_+(\mathcal {H})$ and function $f:(0,\infty ) \rightarrow (0,\infty )$, where the inner product $\langle \cdot ,\cdot \rangle _{\sigma ,f}$ is given in (2.5). It was proved in [63, Lemma 3.2] that $\lambda (\mathcal {L})$ is independent of the choice of $\sigma$ and f. We are now prepared to give the following result.

Theorem C.1

Let $\mathcal {P}_t = e^{t \mathcal {L}}$ be a QMS satisfying $\sigma$-GNS DBC for some $\sigma \in \mathcal {D}_+(\mathcal {H})$. Then the Beckner’s inequality (C.2) holds for all $p \in (1,2]$ with constant $\alpha _p(\mathcal {L})$ satisfying the estimate:

$$\begin{aligned} \alpha _p(\mathcal {L}) \ge \frac{p}{4} C(E_\mathcal {F})^{p-2} \lambda (\mathcal {L})\,. \end{aligned}$$

(C.6)

Proof

We consider the relative density $X = \Gamma _{E_\mathcal {F}^\dag (\rho )}^{-1}(\rho )$ for $\rho \in \mathcal {D}_+(\mathcal {H})$ and then have

$$\begin{aligned}&p^2 \mathcal {F}_{p,E_{\mathcal {F}}^\dag (\rho )}(\rho ) \nonumber \\&= \frac{p}{p-1}\big (\left\Vert X \right\Vert _{p,E_{\mathcal {F}}^\dag (\rho )}^p - 1\big ) \le p \left\Vert X - \textbf{1} \right\Vert _{E_{\mathcal {F}}^\dag (\rho ),\varphi _p}^2 \le - \lambda (\mathcal {L})^{-1} p \langle X, \mathcal {L} X \rangle _{E_{\mathcal {F}}^\dag (\rho ),\varphi _p}, \end{aligned}$$

(C.7)

by using $E_{\mathcal {F}}^\dag \Gamma _\sigma = \Gamma _\sigma E_\mathcal {F}$ for any invariant state $\sigma$ and the upper estimate in (3.12). By definition (C.3) and formulas (2.19) and (3.10), it follows from Lemma 3.2 that

$$\begin{aligned}&- \left\langle X, \mathcal {L} X \right\rangle _{\sigma ,\varphi _p}\nonumber \\&= \left\langle \Gamma _\sigma ^{1/p} \big (\partial _j X \big ), f_p^{[1]}\left( e^{\omega _j/2p}\sigma ^{1/p}, e^{-\omega _j/2p}\sigma ^{1/p}\right) \Gamma _\sigma ^{1/p} \big (\partial _j X\big ) \right\rangle \nonumber \\&\le C(E_{\mathcal {F}})^{2-p} \left\langle \Gamma _\sigma ^{1/p} \big (\partial _j X\big ), f_p^{[1]}\left( e^{\omega _j/2p} \Gamma _\sigma ^{1/p}(X), e^{-\omega _j/2p} \Gamma _\sigma ^{1/p}(X) \right) \Gamma _\sigma ^{1/p}\big (\partial _j X\big ) \right\rangle \nonumber \\&\le 4 p^{-2} C(E_{\mathcal {F}})^{2-p} \mathcal {E}_{p,\mathcal {L}}(X)\,, \end{aligned}$$

(C.8)

where $\sigma = E_{\mathcal {F}}^\dag (\rho )$ and $X = \Gamma _{E_\mathcal {F}^\dag (\rho )}^{-1}(\rho )$. Therefore, by (C.7) and (C.8), we obtain

$$\begin{aligned} p^2 \mathcal {F}_{p,E_{\mathcal {F}}^\dag (\rho )}(\rho ) \le \frac{4}{p \lambda (\mathcal {L})} C(E_{\mathcal {F}})^{2-p} \mathcal {E}_{p,\mathcal {L}}\big (\Gamma _{E_\mathcal {F}^\dag (\rho )}^{-1}(\rho )\big ). \end{aligned}$$

The proof is complete by (C.2). $\square$

Remark C.2

If $\mathcal {P}_t$ in Theorem C.1 is primitive, then we have $\alpha _p \ge p\,\sigma _{\min }^{2-p} \lambda /4$, which is asymptotically worse than the one in (3.43) for fixed $\sigma _{\min } \le 1/2$ and p close to 1 or fixed $p \in (1,2]$ and small enough $\sigma _{\min }$.

Remark C.3

When $p \rightarrow 1^+$, the lower bound (C.6) reduces to the one in [63, Theorem 3.3] for MLSI constant, which has been improved very recently in [60, Theorem 4.18] by Gao et al. with a logarithmic dependence on the complete version of subalgebra index. One may hence expect a similar improvement for $\alpha _p(\mathcal {L})$ as well, which we leave for future investigation.