The quantum relative entropy as a rate function and information criteria

Abstract

We prove that the quantum relative entropy is a rate function in large deviation principle. Next, we define information criteria for quantum states and estimate the accuracy of the use of them. Most of the results in this paper are essentially based on Hiai-Ohya-Tsukada theorem.

Proofs of Theorems

Proof of Theorem 7

It suffices to prove for $|\alpha |<1$. It is proved by discussion in [17, p. 129] that

$$\begin{aligned} QF_t\left(\varphi ^R,\psi ^L\right)(1,1)=\left\langle \Psi ,\Delta _{\Phi ,\Psi }^{1-t} \Psi \right\rangle . \end{aligned}$$

Therefore, we have

$$\begin{aligned} S^{(\alpha )}(\varphi \Vert \psi )_\mathrm{Uhlmann }&= \frac{4}{1-\alpha ^2}\left(1-QF_{\frac{1+\alpha }{2}}\left(\varphi ^R,\psi ^L\right)(1,1)\right) \\&= \frac{4}{1-\alpha ^2}\left(1-\langle \Psi ,\Delta _{\Phi ,\Psi }^{1-\frac{1+\alpha }{2}} \Psi \rangle \right) \\&= S^{(\alpha )}(\varphi \Vert \psi )_\mathrm{Araki }. \end{aligned}$$

$\square$

Proof of Theorem 8

It suffices to prove for $|\alpha |<1$. By [17, Lemma 3.1], it is proved that

$$\begin{aligned} QF_t(\varphi ^R,\psi ^L)(A,A) =QF_t(\tilde{\varphi }^R,\tilde{\psi }^L)(\pi (A),\pi (A)), \end{aligned}$$

where $\pi =\pi _{\varphi +\psi }$. Thus we have

$$\begin{aligned} S^{(\alpha )}(\varphi \Vert \psi )&= \frac{4}{1-\alpha ^2}\Bigg (1-QF_{\frac{1+\alpha }{2}}(\varphi ^R,\psi ^L)(1,1)\Bigg ) \\&= \frac{4}{1-\alpha ^2}\Bigg (1-QF_{\frac{1+\alpha }{2}} \Bigg (\tilde{\varphi }^R,\tilde{\psi }^{L}\Bigg )(\pi (1),\pi (1)\Bigg )\Bigg ) \\&= S^{(\alpha )}\left(\tilde{\varphi }\Vert \tilde{\psi }\right). \end{aligned}$$

$\square$

Lemma 4

Let $\varphi |_\mathcal{B }, \psi |_\mathcal{B }$ be the restrictions of states $\varphi ,\psi$ on a von Neumann algebra $\mathcal M$ to a $*$-subalgebra $\mathcal B$ to $\mathcal M$. Then,

$$\begin{aligned} S^{(\alpha )}_\mathcal{B }(\varphi \Vert \psi ):=S^{(\alpha )}(\varphi |_\mathcal{B }\Vert \psi |_\mathcal{B }) \le S^{(\alpha )}(\varphi \Vert \psi ). \end{aligned}$$

Proof

It suffices to prove it for $|\alpha |<1$. $|_\mathcal{B }:E_\mathcal M \rightarrow E_\mathcal B$ is the dual of the embedding $\mathcal B \ni B \mapsto B\in \mathcal M$. Using [50, Proposition 9] and Theorem 8,

$$\begin{aligned} QF_t\Bigg (\varphi ^R\Vert \psi ^L\Bigg )(1,1)&= QI_t\Bigg (\varphi ^R\Vert \psi ^L\Bigg )(1)^2 \\&\le QI_t\Bigg ((\varphi |_\mathcal{B })^R\Vert (\psi |_\mathcal{B })^L\Bigg )(1)^2 \\&= QF_t\Bigg ((\varphi |_\mathcal{B })^R\Vert (\psi |_\mathcal{B })^L\Bigg )(1,1). \end{aligned}$$

Thus we have

$$\begin{aligned} S^{(\alpha )}(\varphi \Vert \psi )&= \frac{4}{1-\alpha ^2}\left(1-QF_{\frac{1+\alpha }{2}}\left(\varphi ^R,\psi ^L\right)(1,1)\right) \\&\ge \frac{4}{1-\alpha ^2}\left(1-QF_{\frac{1+\alpha }{2}} \left((\varphi |_\mathcal{B })^R\Vert (\psi |_\mathcal{B })^L\right)(1,1)\right) \\&= S^{(\alpha )}(\varphi |_\mathcal{B }\Vert \psi |_\mathcal{B })=S^{(\alpha )}_\mathcal{B }(\varphi \Vert \psi ). \end{aligned}$$

$\square$

Proof of Theorem 9

It suffices to prove for $|\alpha |<1$.

We define $\Gamma : \mathfrak A \rightarrow C(E_\mathfrak A )$ by $\Gamma (A)(\rho )=\rho (A)$, and its dual map $\Gamma ^*:M_1(E_\mathfrak A )\rightarrow E_\mathfrak A$ by

$$\begin{aligned} (\Gamma ^*\lambda )(A)=\int \rho (A)\;d\lambda (\rho )=\lambda (\Gamma (A)), \end{aligned}$$

for $\lambda \in M_1(E_\mathfrak A )$. $\Gamma$ satisfies

$$\begin{aligned} \Gamma (1)=1,\;\Gamma (A^*) =\Gamma (A)^*,\; \Gamma (A^*)\Gamma (A) \le \Gamma (A^*A), \end{aligned}$$

for any $A\in \mathfrak A$. Thus,

$$\begin{aligned} S^{(\alpha )}(\varphi \Vert \psi ) =S^{(\alpha )}(\Gamma ^*\mu \Vert \Gamma ^*\nu ) \le D^{(\alpha )}(\mu \Vert \nu ). \end{aligned}$$

By assumption, $m$ is a subcentral measure $\mu _\mathcal{C }$ associated with a von Neumann subalgebra $\mathcal C$ of the center $\mathfrak Z _\chi (\mathfrak A )=\pi _{\chi }(\mathfrak A )^{\prime \prime }\cap \pi _{\chi }(\mathfrak A )^{\prime }$ of a state $\chi \in E_\mathfrak{A }$. We define a $*$-isomorphism $\kappa _m:L^\infty (E_\mathfrak A ,m)\rightarrow \mathcal C$ by

$$\begin{aligned} \langle \Omega _\chi ,\pi _\chi (A)\kappa _m(f) \Omega _\chi \rangle =\int f(\rho )\;\rho (A)\;dm(\rho ). \end{aligned}$$

It is then proved in [17, Example 2.6 and Theorem 3.2] that

$$\begin{aligned} \tilde{\varphi }(\kappa _m(f))&= \int f(\rho )\;d\mu (\rho ),\\ \tilde{\psi }(\kappa _m(f))&= \int f(\rho )\;d\nu (\rho ), \end{aligned}$$

where $\tilde{\varphi }, \tilde{\psi }$ are the normal extensions of $\varphi , \psi$ to $\pi _{\chi }(\mathfrak A )^{\prime \prime }$. By Lemma 4,

$$\begin{aligned} S^{(\alpha )}(\varphi \Vert \psi )&= S^{(\alpha )}(\tilde{\varphi }\Vert \tilde{\psi }) \\&\ge S^{(\alpha )}_\mathcal{C }(\tilde{\varphi }\Vert \tilde{\psi })=D^{(\alpha )}(\mu \Vert \nu ). \end{aligned}$$

$\square$