On variational expressions for quantum relative entropies

Abstract

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for $\alpha \in (\frac{1}{2}, \infty )$ and strictly smaller for $\alpha \in [0,\frac{1}{2})$. The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for $\alpha < \frac{1}{2}$. Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.

Appendix A: Proofs for results in Sect. 5

Proof of Lemma 10

We start by writing out the relative entropy of recovery as a convex optimization program,

(A1)

where the notation is as in the proof of Lemma 8. Clearly, the first and last constraint can be relaxed to $\gamma _{AD} \le {{\mathrm{tr}}}_F\left[ \sqrt{\sigma _{AF}}\tau _{ADF}\sqrt{\sigma _{AF}}\right]$ and $\tau _{AF} \le \Pi ^\sigma _{AF}$ without changing the solution. The Lagrangian can then be written as

$$\begin{aligned} \mathcal {L}(\gamma , \tau , R, S)&= - {{\mathrm{tr}}}[\rho _{AD} \log \gamma _{AD}]+{{\mathrm{tr}}}\left[ R_{AD}\left( \gamma _{AD}-{{\mathrm{tr}}}_F\left[ \sqrt{\sigma _{AF}}\tau _{ADF}\sqrt{\sigma _{AF}}\right] \right) \right] \nonumber \\&\quad +{{\mathrm{tr}}}\left[ S_{AF}\left( \tau _{AF}-\Pi ^\sigma _{AF}\right) \right] \end{aligned}$$

(A2)

$$\begin{aligned}&= - {{\mathrm{tr}}}[\rho _{AD} \log \gamma _{AD}]+{{\mathrm{tr}}}[R_{AD}\gamma _{AD}]\nonumber \\&\quad -{{\mathrm{tr}}}\left[ \tau _{ADF}\left( \sqrt{\sigma _{AF}}R_{AD}\sqrt{\sigma _{AF}}-S_{AF}\right) \right] \nonumber \\&\quad -{{\mathrm{tr}}}\left[ S_{AF}\Pi ^\sigma _{AF}\right] \, , \end{aligned}$$

(A3)

where we introduced the variables $R_{AD} \ge 0$ and $S_{AF} \ge 0$. In order to compute the dual objective function, we should take the infimum of this quantity over $\gamma _{AD}>0$ and $\tau _{ADF} \ge 0$. Using the variational expression for the measured relative entropy (Lemma 1), we find

$$\begin{aligned} \inf _{\gamma _{AD}>0,\,\tau _{ADF} \ge 0} \mathcal {L}(\gamma , \theta , R, S) =-D^{\mathbb {M}}(\rho _{AD}\Vert R_{AD})+1-{{\mathrm{tr}}}[S_{AF}\Pi ^\sigma _{AF}], \end{aligned}$$

(A4)

when $S_{AF}\ge \sqrt{\sigma _{AF}}R_{AD}\sqrt{\sigma _{AF}}$. With Slater’s strong duality and a change of variable $\bar{S}_{AF}:=\sigma _{AF}^{-1/2}S_{AF}\sigma _{AF}^{-1/2}$ (but not including the bar in the following), we get

(A5)

Adding the constraint ${{\mathrm{tr}}}[S_{AF}\sigma _{AF}]=1$ as in the proof of Lemma 8 concludes the proof. $\square$

Proof of Proposition 11

We start by writing out the argument flipped relative entropy of recovery as a convex optimization program,

(A6)

where the notation is as in the proof of Lemma 8. The Lagrangian can be written as

$$\begin{aligned} \mathcal {L}(\gamma , \tau , T, S)&= {{\mathrm{tr}}}[\gamma _{AD} \log \gamma _{AD}]-{{\mathrm{tr}}}[\gamma _{AD} \log \rho _{AD}]\nonumber \\&\quad +{{\mathrm{tr}}}\left[ T_{AD}\left( \gamma _{AD}-{{\mathrm{tr}}}_F\left[ \sqrt{\sigma _{AF}}\tau _{ADF}\sqrt{\sigma _{AF}}\right] \right) \right] \nonumber \\&\quad +{{\mathrm{tr}}}\left[ S_{AF}\left( \tau _{AF}-\Pi ^\sigma _{AF}\right) \right] \end{aligned}$$

(A7)

$$\begin{aligned}&= {{\mathrm{tr}}}[\gamma _{AD} \log \gamma _{AD}]+{{\mathrm{tr}}}[\gamma _{AD}(T_{AD}-\log \rho _{AD})]\nonumber \\&\quad -{{\mathrm{tr}}}\left[ S_{AF}\Pi ^\sigma _{AF}\right] -{{\mathrm{tr}}}\left[ \tau _{ADF}\left( \sqrt{\sigma _{AF}}T_{AD}\sqrt{\sigma _{AF}}-S_{AF}\right) \right] , \end{aligned}$$

(A8)

where $T_{AD}$ and $S_{AF}$ are Hermitian operators. In order to compute the dual objective function, we should take the infimum of this quantity over $\gamma _{AD}>0$ and $\tau _{ADF} \ge 0$. From the last expression we get $S_{AF}\ge \sqrt{\sigma _{AF}}T_{AD}\sqrt{\sigma _{AF}}$ and we also know how to optimize the first expression as it is an entropy maximization question:

$$\begin{aligned} \inf _{\gamma _{AD}>0} {{\mathrm{tr}}}[\gamma _{AD} \log \gamma _{AD}]+{{\mathrm{tr}}}[\gamma _{AD}(T_{AD}-\log \rho _{AD})]. \end{aligned}$$

(A9)

It is optimized when $\gamma _{AD} = \exp (\log \rho _{AD} - T_{AD} + \alpha 1_{AD})$ for some $\alpha$ [9]. This means that this infimum is given by

$$\begin{aligned}&\inf _{\alpha } {{\mathrm{tr}}}[\exp (\log \rho _{AD} - T_{AD} + \alpha 1_{AD}) \left( \log \rho _{AD} - T_{AD} + \alpha 1_{AD} \right) ]\nonumber \\&\quad + {{\mathrm{tr}}}[\exp (\log \rho _{AD} - T_{AD} + \alpha 1_{AD}) \left( T_{AD} - \log \rho _{AD} \right) ]\nonumber \\&= \inf _{\alpha } \alpha {{\mathrm{tr}}}[\exp (\log \rho _{AD} - T_{AD} + \alpha 1_{AD})] \end{aligned}$$

(A10)

$$\begin{aligned}&= \inf _{\alpha } \alpha e^{\alpha } {{\mathrm{tr}}}[\exp (\log \rho _{AD} - T_{AD})] \end{aligned}$$

(A11)

$$\begin{aligned}&= - {{\mathrm{tr}}}[\exp (\log \rho _{AD} -T_{AD} - 1_{AD})]. \end{aligned}$$

(A12)

With Slater’s strong duality and a change of variable $\bar{S}_{AF}:=\sigma _{AF}^{-1/2}S_{AF}\sigma _{AF}^{-1/2}$ (but not including the bar in the following), we get

(A13)

We now do another change of variable (but not including the bar in the following)

$$\begin{aligned} R_{AD}:= \exp (\log \rho _{AD} - T_{AD} - 1_{AD})\quad \text {as well as}\quad 1_{D}\otimes \bar{S}_{AF}:=1_{D}\otimes S_{AF}+1_{ADF}, \end{aligned}$$

(A14)

and the program becomes

(A15)

Observe now that we can add the constraint ${{\mathrm{tr}}}[R_{AD}] = 1$. In fact, let

$$\begin{aligned} \bar{R}_{AD}:= \frac{R_{AD}}{{{\mathrm{tr}}}[R_{AD}]}\quad \mathrm {and}\quad \bar{S}_{AF}:= S_{AF} + \log {{\mathrm{tr}}}[R_{AD}]. \end{aligned}$$

(A16)

This solution satisfies the constraint and the objective value becomes

$$\begin{aligned} -{{\mathrm{tr}}}[\bar{S}_{AF}\sigma _{AF}]+1- {{\mathrm{tr}}}[\bar{R}_{AD}]&= - {{\mathrm{tr}}}[S_{AF}\sigma _{AF}] - \log {{\mathrm{tr}}}[R_{AD}] \nonumber \\&\ge - {{\mathrm{tr}}}[S_{AF}\sigma _{AF}] + 1 - {{\mathrm{tr}}}[R_{AD}]. \end{aligned}$$

(A17)

This concludes the proof of (96).

To prove (97), first note that it is immediate from the definition of the argument flipped relative entropy of recovery that

$$\begin{aligned} \bar{D}^{\mathrm {rec}}(\sigma _{AE}\otimes \omega _{A'E'}\Vert \rho _{AD}\otimes \tau _{A'D'})\le \bar{D}^{\mathrm {rec}}(\sigma _{AE}\Vert \rho _{AD})+\bar{D}^{\mathrm {rec}}(\omega _{A'E'}\Vert \tau _{A'D'}), \end{aligned}$$

(A18)

and in the following we prove inequality in the other direction using the dual representation (96). Given feasible operators $R_{AD},S_{AF}$ for the quantity $\bar{D}^{\mathrm {rec}}(\sigma _{AE}\Vert \rho _{AD})$ and feasible operators $R_{A'D'},S_{A'F'}$ for the quantity $\bar{D}^{\mathrm {rec}}(\omega _{A'E'}\Vert \tau _{A'D'})$, we have

$$\begin{aligned}&1_D\otimes S_{AF}\ge (\log \rho _{AD}-\log R_{AD})\otimes 1_F \wedge \ 1_{D'}\otimes S_{A'F'}\nonumber \\&\quad \ge (\log \tau _{A'D'}-\log R_{A'D'})\otimes 1_{F'} \nonumber \\&\implies 1_{DD'}\otimes (S_{AF}\otimes 1_{A'F'}+1_{AF}\otimes S_{A'F'})\nonumber \\&\quad \quad \;\;\ge \big ((\log \rho _{AD}-\log R_{AD})\otimes 1_{A'D'}+1_{AD}\otimes (\log \tau _{A'D'}-\log R_{A'D'})\big )\otimes 1_{FF'}\nonumber \\&\quad \quad \;\;=\big (\log (\rho _{AD}\otimes \tau _{A'D'})-\log (R_{AD}\otimes R_{A'D'})\big )\otimes 1_{FF'}, \end{aligned}$$

(A19)

just by multiplying with identities and adding the resulting operator inequalities. Moreover, we have ${{\mathrm{tr}}}[R_{AD}\otimes R_{A'D'}]=1$. Hence, $(R_{AD}\otimes R_{A'D'},S_{AF}\otimes 1_{A'F'}+1_{AF}\otimes S_{A'F'})$ is a feasible pair in the expression (96) for $\bar{D}^{\mathrm {rec}}(\sigma _{AE}\otimes \omega _{A'E'}\Vert \rho _{AD}\otimes \tau _{A'D'})$ and we get

$$\begin{aligned}&\bar{D}^{\mathrm {rec}}(\sigma _{AE}\otimes \omega _{A'E'}\Vert \rho _{AD}\otimes \tau _{A'D'})\nonumber \\&\quad \ge -{{\mathrm{tr}}}[(S_{AF}\otimes 1_{A'F'}+1_{AF}\otimes S_{A'F'})(\sigma _{AF}\otimes \omega _{A'F'})] \end{aligned}$$

(A20)

$$\begin{aligned}&\quad =-{{\mathrm{tr}}}[S_{AF}\sigma _{AF}]-{{\mathrm{tr}}}[S_{A'F'}\omega _{A'F'}]. \end{aligned}$$

(A21)

Taking the supremum over feasible $(R_{AD},S_{AF})$ and $(R_{A'D'},S_{A'F'})$, we find the claimed additivity. $\square$