# Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Setting of Quantum Systems

real r.v. | Hermitian operator | self-adjoint element | |

observable | in | A on $\mathcal{H}$ | A in |

$M(\Omega )$ | (self adjoint operator | C*-algebra $\mathcal{A}$ | |

in $B(\mathcal{H})$) | |||

state | probability measure | density operator | p.l.fnal $\phi \in \mathfrak{S}$ |

$\mu \in P(\Omega )$ | ρ on $\mathcal{H}$ | with $\phi (I)=1$ | |

expectation | ${\int}_{\Omega}fd\mu $ | $\mathrm{tr}\rho A$ | $\phi (A)$ |

## 3. Communication Processes

**a channel**, which reflects the property of a physical device. With a decoding $\stackrel{\sim}{\Xi},$ the whole information transmission process is written as

**channel**in the sequel. This channel (the dual of $\Gamma )$ is expressed by a completely positive mapping ${\Gamma}^{*},$ in the sense of Chapter 5 , from the state space of X to that of $\stackrel{\sim}{X}$ , hence the output coded quantum state $\stackrel{\sim}{\sigma}$ is ${\Gamma}^{*}\sigma .$ Since the information transmission process can be understood as a process of state (probability) change, when Ω and $\stackrel{\sim}{\Omega}$ are classical and X and $\stackrel{\sim}{X}$ are quantum, the process is written as

## 4. Quantum Entropy for Density Operators

**Theorem 1**

- (1)
- Positivity : $S(\rho )\ge 0$.
- (2)
- Symmetry : Let ${\rho}^{\prime}=U\rho {U}^{*}$ for an unitary operator U. Then$$S({\rho}^{\prime})=S(\rho )$$
- (3)
- Concavity : $S(\lambda {\rho}_{1}+(1-\lambda ){\rho}_{2})\ge \lambda S({\rho}_{1})+(1-\lambda )S({\rho}_{2})$ for any ${\rho}_{1},{\rho}_{2}\in \mathfrak{S}(\mathcal{H})$ and $\lambda \in [0,1]$.
- (4)
- Additivity : $S({\rho}_{1}\otimes {\rho}_{2})=S({\rho}_{1})+S({\rho}_{2})$ for any ${\rho}_{k}\in \mathfrak{S}({\mathcal{H}}_{k})$.
- (5)
- Subadditivity : For the reduced states ${\rho}_{1},{\rho}_{2}$ of $\rho \in \mathfrak{S}({\mathcal{H}}_{1}\otimes {\mathcal{H}}_{2})$,$$S(\rho )\le S({\rho}_{1})+S({\rho}_{2})$$
- (6)
- Lower Semicontinuity : If $\parallel {\rho}_{n}{-\rho \parallel}_{1}\to 0(\equiv \mathrm{tr}|{\rho}_{n}-\rho |\to 0)$ as $n\to \infty $, then$$S(\rho )\le \underset{n\to \infty}{lim}infS({\rho}_{n})$$
- (7)
- Continuity : Let ${\rho}_{n},\rho $ be elements in $\mathfrak{S}(\mathcal{H})$ satisfying the following conditions : (i) ${\rho}_{n}\to \rho $ weakly as $n\to \infty $, (ii) ${\rho}_{n}\le A\phantom{\rule{4pt}{0ex}}(\forall n)$ for some compact operator A, and (iii) $-{\sum}_{k}{a}_{k}log{a}_{k}<+\infty $ for the eigenvalues $\{{a}_{k}\}$ of A. Then $S({\rho}_{n})\to S(\rho )$.
- (8)
- Strong Subadditivity : Let $\mathcal{H}={\mathcal{H}}_{1}\otimes {\mathcal{H}}_{2}\otimes {\mathcal{H}}_{3}$ and denote the reduced states ${\mathrm{tr}}_{{\mathcal{H}}_{i}\otimes {\mathcal{H}}_{j}}\rho $ by ${\rho}_{k}$ and ${\mathrm{tr}}_{{\mathcal{H}}_{k}}\rho $ by ${\rho}_{ij}$. Then $S(\rho )+S({\rho}_{2})\le S({\rho}_{12})+S({\rho}_{23})$ and $S({\rho}_{1})+S({\rho}_{2})\le S({\rho}_{13})+S({\rho}_{23}).$
- (9)
- Entropy increasing: (i) Let $\mathcal{H}$ be finite dimensional space. If the channel ${\Lambda}^{*}$ is unital, that is, for the dual map Λ of ${\Lambda}^{*}$ satisfies $\Lambda I=I,$ then $S({\Lambda}^{*}\rho )\ge S(\rho ).$ (ii) For arbitrary Hilbert space $\mathcal{H}$, if the dual map Λ of the channel ${\Lambda}^{*}$ satisfies $\Lambda \left(\rho \right)\in \mathfrak{S}(\mathcal{H})$, then $S({\Lambda}^{*}\rho )\ge S(\rho ).$

**Lemma 2**

- (1)
- Klein’s inequality: tr$\{f(\rho )-f(\sigma )-(\rho -\sigma ){f}^{\prime}(\sigma )\}\phantom{\rule{0.166667em}{0ex}}\ge \phantom{\rule{0.166667em}{0ex}}0.$
- (2)
- Peierls inequality: ${\sum}_{k}f(<{x}_{k},\rho {x}_{k}>)\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}$ tr$f(\rho )$ for any CONS $\{{x}_{k}\}$ in $\mathcal{H}$. (Remark: $\rho \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\sum}_{n}{\mu}_{n}{E}_{n}\phantom{\rule{0.166667em}{0ex}}\u27f9\phantom{\rule{0.166667em}{0ex}}f(\rho )\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\sum}_{n}f({\mu}_{n}){E}_{n}.$)

## 5. Relative Entropy for Density Operators

**Theorem 3**

- (1)
- Positivity: $S(\rho ,\sigma )\ge 0,$$=0$ iff $\rho =\sigma .$
- (2)
- Joint Convexity : $S(\lambda {\rho}_{1}+(1-\lambda ){\rho}_{2},\lambda {\sigma}_{1}+(1-\lambda ){\sigma}_{2})\le \lambda S({\rho}_{1},{\sigma}_{1})+(1-\lambda )S({\rho}_{2},{\sigma}_{2})$ for any $\lambda \in [0,1]$.
- (3)
- Additivity : $S({\rho}_{1}\otimes {\rho}_{2},{\sigma}_{1}\otimes {\sigma}_{2})=S({\rho}_{1},{\sigma}_{1})+S({\rho}_{2},{\sigma}_{2})$.
- (4)
- Lower Semicontinuity : If ${lim}_{n\to \infty}{\parallel {\rho}_{n}-\rho \parallel}_{1}=0$ and ${lim}_{n\to \infty}{\parallel {\sigma}_{n}-\sigma \parallel}_{1}=0$, then $S(\rho ,\sigma )\le lim{inf}_{n\to \infty}S({\rho}_{n},{\sigma}_{n})$. Moreover, if there exists a positive number λ satisfying ${\rho}_{n}\le \lambda {\sigma}_{n}$, then ${lim}_{n\to \infty}S({\rho}_{n},{\sigma}_{n})=S(\rho ,\sigma )$.
- (5)
- Monotonicity : For a channel ${\Lambda}^{*}$ from $\mathfrak{S}$ to $\overline{\mathfrak{S}}$, $S({\Lambda}^{*}\rho ,{\Lambda}^{*}\sigma )\le S(\rho ,\sigma ).$
- (6)
- Lower Bound : ${\parallel \rho -\sigma \parallel}^{2}/2\le S(\rho ,\sigma ).$
- (7)
- Invaiance under the unitary mapping: ${S(U\rho U}^{*}{;U\sigma U}^{*})=S(\rho ;\sigma )$ where U is a unitary operator.

**Theorem 4**

## 6. Channel and Lifting

#### 6.1. Quantum Channels

**Definition 5**

- (1)
- ${\Lambda}^{*}$ is linear if ${\Lambda}^{*}(\lambda \phi +(1-\lambda )\psi )=\lambda {\Lambda}^{*}\phi +(1-\lambda ){\Lambda}^{*}\psi $ holds for any $\lambda \in [0,1]$.
- (2)
- ${\Lambda}^{*}$ is completely positive (CP) if ${\Lambda}^{*}$ is linear and its dual $\Lambda :\overline{\mathcal{A}}\to \mathcal{A}$ satisfies$$\sum _{i,j=1}^{n}{A}_{i}^{*}\Lambda ({\overline{A}}_{i}^{*}{\overline{A}}_{j}){A}_{j}\ge 0$$
- (3)
- ${\Lambda}^{*}$ is Schwarz type if $\Lambda ({\overline{A}}^{*})=\Lambda {(\overline{A})}^{*}$ and ${\Lambda}^{*}{(\overline{A})}^{*}{\Lambda}^{*}(\overline{A})\le {\Lambda}^{*}({\overline{A}}^{*}\overline{A})$.
- (4)
- ${\Lambda}^{*}$ is stationary if $\Lambda \circ {\alpha}_{t}={\overline{\alpha}}_{t}\circ \Lambda $ for any $t\in \mathbf{R}$.(Here ${\alpha}_{t}$ and ${\overline{\alpha}}_{t}$ are groups of automorphisms of the algebra $\mathcal{A}$ and $\overline{\mathcal{A}}$ respectively.)
- (5)
- ${\Lambda}^{*}$ is ergodic if ${\Lambda}^{*}$ is stationary and ${\Lambda}^{*}(exI(\alpha ))\subset exI(\overline{\alpha})$.(Here $exI(\alpha )$ is the set of extreme points of the set of all stationary states $I(\alpha )$.)
- (6)
- ${\Lambda}^{*}$ is orthogonal if any two orthogonal states ${\phi}_{1},{\phi}_{2}\in \mathfrak{S}(\mathcal{A})$ (denoted by ${\phi}_{1}\perp {\phi}_{2}$) implies ${\Lambda}^{*}{\phi}_{1}\perp {\Lambda}^{*}{\phi}_{2}$.
- (7)
- ${\Lambda}^{*}$ is deterministic if ${\Lambda}^{*}$ is orthogonal and bijective.
- (8)
- For a subset ${\mathfrak{S}}_{0}$ of $\mathfrak{S}(\mathcal{A})$, ${\Lambda}^{*}$ is chaotic for ${\mathfrak{S}}_{0}$ if ${\Lambda}^{*}{\phi}_{1}={\Lambda}^{*}{\phi}_{2}$ for any ${\phi}_{1},{\phi}_{2}\in {\mathfrak{S}}_{0}$.
- (9)
- ${\Lambda}^{*}$ is chaotic if ${\Lambda}^{*}$ is chaotic for $\mathfrak{S}(\mathcal{A})$.
- (10)
- Stinespring-Sudarshan-Kraus representation: a completely positive channel ${\Lambda}^{*}$ can be represented as$${\Lambda}^{*}\rho =\sum _{i}{A}_{i}\rho {A}_{i}^{*},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\sum _{i}{A}_{i}^{*}{A}_{i}\u2a7d1$$

**Unitary evolution:**Let H be the Hamiltonian of a system.

**Semigroup evolution:**Let ${V}_{t}(t\in {\mathbf{R}}^{+})$ be an one parameter semigroup on $\mathcal{H}$.

**Quantum measurement:**If a measuring apparatus is prepared by an positive operator valued measure $\left\{{Q}_{n}\right\}$ then the state ρ changes to a state ${\Lambda}^{*}\rho $ after this measurement,

**Reduction:**If a system ${\Sigma}_{1}$ interacts with an external system ${\Sigma}_{2}$ described by another Hilbert space $\mathcal{K}$ and the initial states of ${\Sigma}_{1}$ and ${\Sigma}_{2}$ are ρ and σ, respectively, then the combined state ${\theta}_{t}$ of ${\Sigma}_{1}$ and ${\Sigma}_{2}$ at time t after the interaction between two systems is given by

**Optical communication processes:**Quantum communication process is described by the following scheme [13].

**Attenuation process:**Based on the construction of the optical communication processes of (5), the attenuation channel is defined as follows [13]: Take ${\nu}_{0}=\left|0\right.\u232a\left.\u23290\right|=$ vacuum state and ${\pi}_{0}^{*}\left(\xb7\right)\equiv {V}_{0}\left(\xb7\right){V}_{0}^{*}$ given by

**Noisy optical channel:**Based on (5), the noisy optical channel is defined as follows [28]: Take a noise state $\nu =\left|{m}_{1}\right.\u232a\left.\u2329{m}_{1}\right|$, ${m}_{1}$ photon number state of ${\mathcal{K}}_{1}$ and a linear mapping $V:{\mathcal{H}}_{1}\otimes {\mathcal{K}}_{1}\to {\mathcal{H}}_{2}\otimes {\mathcal{K}}_{2}$ as

#### 6.2. Liftings

**Definition 6**

**Definition 7**

**Definition 8**

**Example 9 (1)**

**: Isometric lifting.**

**Example 10 (2)**

**: The compound lifting.**

**Example 11 (3) :**

**The attenuation (or beam splitting) lifting.**

**Example 12 (4)**

**Amplifier channel:**## 7. Quantum Mutual Entropy

**Theorem 13**

**Theorem 14**

**Theorem 15 (Shannon’s inequality)**

## 8. Some Applications to Statistical Physics

#### 8.1. Ergodic theorem

**Theorem 16**

**(1)**- If a channel ${\Lambda}^{*}$ is deterministic, then $I(\rho ;{\Lambda}^{*})=S(\rho )$.
**(2)**- If a channel ${\Lambda}^{*}$ is chaotic, then $I(\rho ;{\Lambda}^{*})=0$.
**(3)**- If ρ is a faithful state and the every eigenvalue of ρ is nondegenerate, then $I(\rho ;{\Lambda}^{*})=S({\Lambda}^{*}\rho )$.

#### 8.2. CCR and channel

**)**as $CCR(\mathcal{H})\otimes CCR(\mathcal{K}$

**),**and given a state ψ on CCR($\mathcal{H}$), a channeling transformation arises as

**Theorem 17**

**Theorem 18**

**Corollary 19**

#### 8.3. Irreversible processes

#### 8.4. Entropy change in linear response dynamics

**Theorem 20**

**Theorem 21**

#### 8.5. Time development of mutual entropy

**Theorem 22**

**Theorem 23**

## 9. Entropies for General Quantum States

**Definition 24**

**Theorem 25**

- (1)
- $S(\phi )=-\mathrm{tr}\rho log\rho $.
- (2)
- If φ is an α-invariant faithful state and every eigenvalue of ρ is non-degenerate, then ${S}^{I(\alpha )}(\phi )=S(\phi ),$ where $I\left(\alpha \right)$ is the set of all α-invariant faithful states.
- (3)
- If $\phi \in K(\alpha )$, then ${S}^{K\left(\alpha \right)}(\phi )=0$, where K$\left(\alpha \right)$ is the set of all KMS states.

**Theorem 26**

- (1)
- ${S}^{K\left(\alpha \right)}(\phi )\le {S}^{I(\alpha )}(\phi )$.
- (2)
- ${S}^{K\left(\alpha \right)}(\phi )\le S(\phi )$.

**Definition 27**

**Uhlmann’s relative entropy**>

**[10]**

- (1)
- For any $x\in \mathcal{L}$, ${p}_{t}(x)$ is continuous in t,
- (2)
- ${p}_{1/2}=QM(p,q)$,
- (3)
- ${p}_{t/2}=QM(p,{p}_{t})$,
- (4)
- ${p}_{(t+1)/2}=QM({p}_{t},q)$.

**Definition 28**

**Ohya’s mutual entropy**>

**[16]**

**Definition 29**

**Theorem 30**

- (1)
- Positivity : $S(\phi ,\psi )\ge 0\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}S(\phi ,\psi )=0\phantom{\rule{1.em}{0ex}},\mathrm{iff}\phantom{\rule{4pt}{0ex}}\phi =\psi $.
- (2)
- Joint Convexity : $S(\lambda {\psi}_{1}+(1-\lambda ){\psi}_{2},\lambda {\phi}_{1}+(1-\lambda ){\phi}_{2})\le \lambda S({\psi}_{1},{\phi}_{1})+(1-\lambda )S({\psi}_{2},{\phi}_{2})$ for any $\lambda \in [0,1]$.
- (3)
- Additivity : $S({\psi}_{1}\otimes {\psi}_{2},{\phi}_{1}\otimes {\phi}_{2})=S({\psi}_{1},{\phi}_{1})+S({\psi}_{2},{\phi}_{2})$.
- (4)
- Lower Semicontinuity : If ${lim}_{n\to \infty}\parallel {\psi}_{n}-\psi \parallel =0$ and ${lim}_{n\to \infty}\parallel {\phi}_{n}\to \phi \parallel =0$, then $S(\psi ,\phi )\le {lim}_{n\to \infty}infS({\psi}_{n},{\phi}_{n})$. Moreover, if there exists a positive number λ satisfying ${\psi}_{n}\le \lambda {\phi}_{n}$, then ${lim}_{n\to \infty}S({\psi}_{n},{\phi}_{n})=S(\psi ,\phi )$.
- (5)
- Monotonicity : For a channel ${\Lambda}^{*}$ from $\mathfrak{S}$ to $\overline{\mathfrak{S}}$,$$S({\Lambda}^{*}\psi ,{\Lambda}^{*}\phi )\le S(\psi ,\phi )$$
- (6)
- Lower Bound : ${\parallel \psi -\phi \parallel}^{2}/4\le S(\psi ,\phi ).$

**Remark 31**

**Connes-Narnhofer-Thirring Entropy**>

**Theorem 32.**

- (1)
- For any state φ on a unital C*-algebra $\mathcal{A}$,$$S(\phi )={H}_{\phi}(\mathcal{A})$$
- (2)
- Let $(\mathcal{A},G,\alpha )$ with a certain group G be a W*-dynamical system andf φ be a G-invariant normal state of $\mathcal{A}$, then$${S}^{I(\alpha )}(\phi )={H}_{\phi}({\mathcal{A}}^{\alpha})$$
- (3)
- Let $\mathcal{A}$ be the C*-algebra $C(\mathcal{H})$ of all compact operators on a Hilbert space $\mathcal{H}$, and G be a group, α be a *-automorphic action of G-invariant density operator. Then$${S}^{I(\alpha )}(\rho )={H}_{\rho}({\mathcal{A}}^{\alpha})$$
- (4)
- There exists a model such that$${S}^{I(\alpha )}(\phi )>{H}_{\phi}({\mathcal{A}}^{\alpha})=0$$

## 10. Entropy Exchange and Coherent Information

**Definition 33**

## 11. Comparison of various quantum mutual type entropies

**Theorem 34**

**Proof.**

**Theorem 35**

**Remark 36**

**Theorem 37**

- 1.
- $0\le I\left(\rho ;{\Lambda}_{0}^{*}\right)\le min\left\{S\left(\rho \right),S\left({\Lambda}_{0}^{*}\rho \right)\right\}$ (Ohya mutual entropy),
- 2.
- ${I}_{C}\left(\rho ;{\Lambda}_{0}^{*}\right)=0$ (coherent entropy),
- 3.
- ${I}_{LN}\left(\rho ;{\Lambda}_{0}^{*}\right)=S\left(\rho \right)$ (Lindblad-Nielsen entropy).

**Theorem 38**

- 1.
- $0\le I\left(\rho ;{\Lambda}_{0}^{*}\right)\le min\left\{S\left(\rho \right),S\left({\Lambda}_{0}^{*}\rho \right)\right\}$ (Ohya mutual entropy),
- 2.
- $-S\left(\rho \right)\le {I}_{C}\left(\rho ;{\Lambda}_{0}^{*}\right)\le S\left(\rho \right)$ (coherent entropy),
- 3.
- $0\le {I}_{LN}\left(\rho ;{\Lambda}_{0}^{*}\right)\le 2S\left(\rho \right)$ (Lindblad-Nielsen entropy).

## 12. Quantum Capacity and Coding

#### 12.1. Capacity of quantum channel

**.**

**Theorem 39**

**Remark 40**

#### 12.2. Capacity of classical-quantum-classical channel

#### 12.3. Bound of mutual entropy and capacity

**Theorem 41**

**the theorem 9.**

**Theorem 42**

**,**and each λ corresponds to a quantum coded state $\sigma (\lambda ),$ then

**.**

**Theorem 43**

## 13. Computation of Capacity

**Proposition 44**

**Example 45**

#### 13.1. Divergence center

**Definition 46**

**Lemma 47**

**Definition 48**

**Lemma 49**

**B**$(\mathcal{K})$ such that the Hilbert space $\mathcal{K}$ is finite dimensional and set ${\psi}_{\lambda}=(1-\lambda ){\psi}_{0}+\lambda {\psi}_{1}$ $(0\le \lambda \le 1)$. If $S({\psi}_{0},\omega )$, $S({\psi}_{1},\omega )$ are finite and

**Lemma 50**

**Theorem 51**

#### 13.2. Comparison of capacities

**Example 52**

**Lemma 53**

**Theorem 54**

**Theorem 55**

#### 13.3. Numerical computation of quantum capacity

**Theorem 56**