# Agents, Subsystems, and the Conservation of Information

^{1}

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## Abstract

**:**

## 1. Introduction

- quantum subsystems associated with the tensor product of two Hilbert spaces,
- subsystems associated with an subalgebra of self-adjoint operators on a given Hilbert space,
- classical systems of quantum systems,
- subsystems associated with the action of a group representation on a given Hilbert space.

## 2. Related Works

## 3. Constructing Subsystems

#### 3.1. A Pre-Operational Framework

**Example**

**1**(Closed quantum systems)

**.**

**Example**

**2**(Open quantum systems)

**.**

#### 3.2. Agents

**Definition**

**1**(Agents)

**.**

**Definition**

**2.**

**Remark**

**1**(Commutation of transformations vs. commutation of observables)

**.**

#### 3.3. Adversaries and Degradation

**Definition**

**3**(Adversary)

**.**

**Definition**

**4.**

- the identity map ${\mathcal{I}}_{S}$ commutes with all operations in $\mathsf{Act}(A;S)$, and
- if $\mathcal{B}$ and ${\mathcal{B}}^{\prime}$ commute with every operation in $\mathsf{Act}(A;S)$, then also their composition $\mathcal{B}\circ {\mathcal{B}}^{\prime}$ will commute with all the operations in $\mathsf{Act}(A;S)$.

#### 3.4. The States of the Subsystem

**Rule**

**1.**

**Proposition**

**1.**

**Proof.**

#### 3.5. The Transformations of a Subsystem

**Rule**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

**Proposition**

**3.**

**Proof.**

## 4. Examples of Agents, Adversaries, and Subsystems

#### 4.1. Tensor Product of Two Quantum Systems

**Proposition**

**4.**

**Proof.**

#### 4.2. Subsystems Associated with Finite Dimensional Von Neumann algebras

**Proposition**

**5.**

#### 4.3. Coherent Superpositions vs. Incoherent Mixtures in Closed-System Quantum Theory

**Proposition**

**6.**

**Proof.**

#### 4.4. Classical Subsystems in Open-System Quantum Theory

**Theorem**

**1.**

**Proposition**

**7.**

- 1.
- ρ and σ are equivalent for agent A,
- 2.
- $\mathcal{D}\left(\rho \right)=\mathcal{D}\left(\sigma \right)$, where $\mathcal{D}$ is the completely dephasing channel $\mathcal{D}(\xb7):={\sum}_{k}\phantom{\rule{0.166667em}{0ex}}|k\rangle \langle k|\xb7|k\rangle \langle k|$.

**Proof.**

#### 4.5. Classical Systems From Free Operations in the Resource Theory of Coherence

- Strictly incoherent operations [41], i.e., quantum channels $\mathcal{T}$ with the property that, for every Kraus operator ${T}_{i}$, the map ${\mathcal{T}}_{i}(\xb7)={T}_{i}\xb7{T}_{i}$ satisfies the condition $\mathcal{D}\circ {\mathcal{T}}_{i}={\mathcal{T}}_{i}\circ \mathcal{D}$, where $\mathcal{D}$ is the completely dephasing channel.
- Phase covariant channels [40], i.e., quantum channels $\mathcal{T}$ satisfying the condition $\mathcal{T}\circ {\mathcal{U}}_{\phi}={\mathcal{U}}_{\phi}\circ \mathcal{T}$, $\forall \phi \in [0,2\pi )$, where ${\mathcal{U}}_{\phi}$ is the unitary channel associated with the unitary matrix ${U}_{\phi}={\sum}_{k}\phantom{\rule{0.166667em}{0ex}}{e}^{ik\phi}\phantom{\rule{0.166667em}{0ex}}|k\rangle \langle k|$.
- Physically incoherent operations [38,39], i.e., quantum channels that are convex combinations of channels $\mathcal{T}$ admitting a Kraus representation where each Kraus operator ${T}_{i}$ is of the form$$\begin{array}{c}\hfill {T}_{i}={U}_{{\pi}_{i}}\phantom{\rule{0.166667em}{0ex}}{U}_{{\theta}_{i}}\phantom{\rule{0.166667em}{0ex}}{P}_{i}\phantom{\rule{0.166667em}{0ex}},\end{array}$$

**Theorem**

**2.**

- 1.
- contains the set of classical channels, and
- 2.
- commutes with the dephasing channel

## 5. Key Structures: Partial Trace and No Signalling

#### 5.1. The Partial Trace and the No Signalling Property

**Definition**

**5.**

#### 5.2. A Baby Category

## 6. Non-Overlapping Agents, Causality, and the Initialization Requirement

#### 6.1. Dual Pairs of Agents

**Definition**

**6.**

#### 6.2. Non-Overlapping Agents

**Definition**

**7.**

**Proposition**

**8.**

- 1.
- A and B are non-overlapping,
- 2.
- $\mathsf{Act}(A;S)$ has trivial center,
- 3.
- $\mathsf{Act}(B;S)$ has trivial center.

**Proof.**

**Proposition**

**9.**

- 1.
- system S admits a dual pair of non-overlapping agents,
- 2.
- the monoid $\mathsf{Transf}\left(S\right)$ has trivial center.

**Proof.**

**Definition**

**8**(Non-Overlapping Agents)

**.**

#### 6.3. Causality

**Proposition**

**10.**

- 1.
- ${S}_{{A}_{min}}$ is the trivial system,
- 2.
- one has ${Tr}_{{A}_{max}}\left[\rho \right]={Tr}_{{A}_{max}}\left[\sigma \right]$ for every pair of states $\rho ,\sigma \in \mathsf{St}\left(S\right)$.

**Proof.**

#### 6.4. The Initialization Requirement

**Definition**

**9.**

**Proposition**

**11.**

**Proof.**

## 7. The Conservation of Information

#### 7.1. Logically Invertible vs. Physically Invertible

**Definition**

**10.**

**Definition**

**11**(Logical Conservation of Information)

**.**

**Definition**

**12.**

**Definition**

**13**(Physical Conservation of Information)

**.**

**Example**

**3**(Conservation of Information in closed-system quantum theory)

**.**

#### 7.2. Systems Satisfying the Physical Conservation of Information

**Definition**

**14.**

**Proposition**

**12.**

**Proof.**

#### 7.3. Subsystems of Systems Satisfying the Physical Conservation of Information

## 8. Closed Systems

**Definition**

**15.**

- 1.
- every transformation is logically invertible,
- 2.
- there exists a state ${\psi}_{0}\in \mathsf{St}\left(S\right)$ such that, for every other state $\psi \in \mathsf{St}\left(S\right)$, one has $\psi =\mathcal{V}{\psi}_{0}$ for some suitable transformation $\mathcal{V}\in \mathsf{Transf}\left(S\right)$.

**Example**

**4.**

- system S is a qubit,
- the states are pure states, of the form $|\psi \rangle \langle \psi |$ for a generic unit vector $|\psi \rangle \in {\mathbb{C}}^{2},$
- the transformations are unitary channels $V\xb7{V}^{\u2020}$, where the unitary matrix V has real entries.

**Proposition**

**13**(Transitive action on the pure states)

**.**

**Proof.**

## 9. Purification

#### 9.1. Purification in Systems Satisfying the Physical Conservation of Information

**Proposition**

**14**(Purification)

**.**

**Proof.**

#### 9.2. Purification in Systems Satisfying the Logical Conservation of Information

- $|{\psi}^{\prime}\rangle =({I}_{A}\otimes {V}_{B})\phantom{\rule{0.166667em}{0ex}}|\psi \rangle $ for some isometry ${V}_{B}$ acting on system ${S}_{B},$
- $|\psi \rangle =({I}_{A}\otimes {V}_{B})\phantom{\rule{0.166667em}{0ex}}|{\psi}^{\prime}\rangle $ for some isometry ${V}_{B}$ acting on system ${S}_{B}$.

**Definition**

**16.**

- 1.
- for every pair of states $\psi ,{\psi}^{\prime}\in \mathsf{St}\left(S\right)$, the condition ${\mathsf{Deg}}_{\mathsf{M}}\left(\psi \right)\cap {\mathsf{Deg}}_{\mathsf{M}}\left({\psi}^{\prime}\right)\ne \varnothing $ implies that there exists a transformation $\mathcal{U}\in \mathsf{M}$ such that ${\psi}^{\prime}=\mathcal{U}\psi $ or $\psi =\mathcal{U}{\psi}^{\prime}$,
- 2.
- for every pair of transformations $\mathcal{V},{\mathcal{V}}^{\prime}\in \mathsf{M}$, there exists a transformation $\mathcal{W}\in \mathsf{M}$ such that $\mathcal{V}=\mathcal{W}\circ {\mathcal{V}}^{\prime}$ or ${\mathcal{V}}^{\prime}=\mathcal{W}\circ \mathcal{V}$.

**Example**

**5**(Isometric channels in quantum theory)

**.**

**Proposition**

**15.**

**Corollary**

**1**(Purification)

**.**

## 10. Example: Group Representations on Quantum State Spaces

**Theorem**

**3.**

#### Compact Connected Lie Groups

**Theorem**

**4.**

**Proposition**

**16.**

## 11. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof That Definitions (20) and (21) Are Well-Posed

**Proposition**

**A1.**

**Proof.**

## Appendix B. The Commutant of the Local Channels

## Appendix C. Subsystems Associated to Finite Dimensional Von Neumann Algebras

#### Appendix C.1. The Commutant of Chan(A)

**Theorem**

**A1.**

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Proof of Theorem A1.**Let $\mathcal{D}$ be a quantum channel in $\mathsf{Chan}{\left(\mathsf{A}\right)}^{\prime}$. Then, Lemma A4 guarantees that the adjoint ${\mathcal{D}}^{\u2020}$ preserves all operators in the algebra $\mathsf{A}$. Then, a result due to Lindblad [86] guarantees that all the Kraus operators of $\mathcal{D}$ belong to the algebra ${\mathsf{A}}^{\prime}$. This proves the inclusion $\mathsf{Chan}{\left(\mathsf{A}\right)}^{\prime}\subseteq \mathsf{Chan}\left({\mathsf{A}}^{\prime}\right)$.

#### Appendix C.2. States of Subsystems Associated to Finite Dimensional Von Neumann algebras

**Lemma**

**A5**(No signalling condition)

**.**

**Proof.**

**Proof of Proposition 5.**Suppose that $\rho $ and $\sigma $ are equivalent for A. By definition, this means that there exists a finite sequence $({\rho}_{1},{\rho}_{2},\cdots ,{\rho}_{n})$ such that

#### Appendix C.3. Transformations of Subsystems Associated to Finite Dimensional von Neumann algebras

**Lemma**

**A6.**

**Proof.**

**Lemma**

**A7.**

- 1.
- ${Tr}_{\mathsf{B}}\circ \phantom{\rule{0.166667em}{0ex}}\mathcal{C}={Tr}_{\mathsf{B}}\circ \phantom{\rule{0.166667em}{0ex}}{\mathcal{C}}^{\prime},$
- 2.
- ${\mathcal{A}}_{k}={\mathcal{A}}_{k}^{\prime}\phantom{\rule{3.33333pt}{0ex}}$ for every k.

**Proof.**

**Lemma**

**A8.**

**Proof.**

**Corollary**

**A1.**

- 1.
- $\mathcal{C}$ and ${\mathcal{C}}^{\prime}$ are equivalent for A,
- 2.
- ${\u2a01}_{k}{\mathcal{A}}_{k}={\u2a01}_{k}{\mathcal{A}}_{k}^{\prime}$.

**Proof.**

**Proposition**

**A2.**

**Proof.**

**Proposition**

**A3.**

**Proof.**

## Appendix D. Basis-Preserving and Multiphase-Covariant Channels

#### Appendix D.1. Proof of Theorem 1

**Lemma**

**A9.**

**Proof.**

**Lemma A10**(Characterization of $\mathsf{MultiPCov}\left(S\right)$)

**.**

**Proof.**

**Lemma**

**A11.**

**Proof.**

**Corollary**

**A2.**

**Lemma**

**A12.**

**Proof.**

**Corollary**

**A3.**

#### Appendix D.2. Proof of Equation (55)

**Lemma**

**A13.**

**Proof.**

**Lemma**

**A14.**

**Proof.**

## Appendix E. Classical Systems and the Resource Theory of Coherence

#### Appendix E.1. Operations That Lead to Classical Subsystems

- Strictly incoherent operations [41], i.e., quantum channels $\mathcal{T}$ with the property that, for every Kraus operator ${T}_{i}$, the map ${\mathcal{T}}_{i}(\xb7)={T}_{i}\xb7{T}_{i}$ satisfies the condition $\mathcal{D}\circ {\mathcal{T}}_{i}={\mathcal{T}}_{i}\circ \mathcal{D}$, where $\mathcal{D}$ is the completely dephasing channel.
- Phase covariant channels [40], i.e., quantum channels $\mathcal{T}$ satisfying the condition $\mathcal{T}\circ {\mathcal{U}}_{\phi}={\mathcal{U}}_{\phi}\circ \mathcal{T}$, $\forall \phi \in [0,2\pi )$, where ${\mathcal{U}}_{\phi}$ is the unitary channel associated with the unitary matrix ${U}_{\phi}={\sum}_{k}\phantom{\rule{0.166667em}{0ex}}{e}^{ik\phi}\phantom{\rule{0.166667em}{0ex}}|k\rangle \langle k|$.
- Physically incoherent operations [38,39], i.e., quantum channels that are convex combinations of channels $\mathcal{T}$ admitting a Kraus representation where each Kraus operator ${T}_{i}$ is of the form$$\begin{array}{c}\hfill {T}_{i}={U}_{{\pi}_{i}}\phantom{\rule{0.166667em}{0ex}}{U}_{{\theta}_{i}}\phantom{\rule{0.166667em}{0ex}}{P}_{i}\phantom{\rule{0.166667em}{0ex}},\end{array}$$
- Classical channels i.e., channels satisfying $\mathcal{T}=\mathcal{D}\circ \mathcal{T}\circ \mathcal{D}$.

**Lemma**

**A15.**

**Proof.**

**Lemma**

**A16.**

**Proof.**

**Lemma**

**A17.**

**Proof.**

**Proposition**

**A4.**

**Proof.**

**Proposition**

**A5.**

**Proof.**

#### Appendix E.2. Operations That Do Not Lead to Classical Subsystems

- The maximally incoherent operations are the quantum channels $\mathcal{T}$ that map diagonal density matrices to diagonal density matrices, namely $\mathcal{T}\circ \mathcal{D}=\mathcal{D}\circ \mathcal{T}\circ \mathcal{D}$, where $\mathcal{D}$ is the completely dephasing channel.
- The Incoherent operations are the quantum channels $\mathcal{T}$ with the property that, for every Kraus operator ${T}_{i}$, the map ${\mathcal{T}}_{i}(\xb7)={T}_{i}\xb7{T}_{i}$ sends diagonal matrices to diagonal matrices, namely ${\mathcal{T}}_{i}\circ \mathcal{D}=\mathcal{D}\circ {\mathcal{T}}_{i}\circ \mathcal{D}$.

**Lemma**

**A18.**

**Proof.**

## Appendix F. Enriching the Sets of Transformations

**Definition**

**A1.**

## Appendix G. The Total System as a Subsystem

**Proposition**

**A6.**

**Proof.**

## Appendix H. Proof of Proposition 15

- ${\psi}_{n}=\mathcal{V}\psi $ and ${\psi}_{n+1}={\mathcal{V}}^{\prime}{\psi}_{n}$. In this case, we have ${\psi}_{n+1}=({\mathcal{V}}^{\prime}\circ \mathcal{V})\psi $, which proves the desired statement.
- ${\psi}_{n}=\mathcal{V}\psi $ and ${\psi}_{n}={\mathcal{V}}^{\prime}{\psi}_{n+1}$. In this case, we have $\mathcal{V}\psi ={\mathcal{V}}^{\prime}{\psi}_{n+1}$, or equivalently ${\mathsf{Deg}}_{B}\left(\psi \right)\cap {\mathsf{Deg}}_{B}\left({\psi}_{n+1}\right)\ne \varnothing $. Applying the induction hypothesis to the sequence $(\psi ,{\psi}_{n+1})$, we obtain the desired statement.
- $\psi =\mathcal{V}{\psi}_{n}$ and ${\psi}_{n+1}={\mathcal{V}}^{\prime}{\psi}_{n}$. Using the second regularity condition, we obtain that there exists a transformation $\mathcal{W}\in \mathsf{Act}(B;S)$ such that at least one of the relations $\mathcal{V}=\mathcal{W}\circ {\mathcal{V}}^{\prime}$ and ${\mathcal{V}}^{\prime}=\mathcal{W}\circ \mathcal{V}$ holds. Suppose that $\mathcal{V}=\mathcal{W}\circ {\mathcal{V}}^{\prime}$. In this case, we have$$\begin{array}{c}\hfill \psi =\mathcal{V}{\psi}_{n}=(\mathcal{W}\circ {\mathcal{V}}^{\prime}){\psi}_{n}=\mathcal{W}{\psi}_{n+1}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Alternatively, suppose that ${\mathcal{V}}^{\prime}=\mathcal{W}\circ \mathcal{V}$. In this case, we have$$\begin{array}{c}\hfill {\psi}_{n+1}={\mathcal{V}}^{\prime}{\psi}_{n}=(\mathcal{W}\circ \mathcal{V}){\psi}_{n}=\mathcal{W}\psi \phantom{\rule{0.166667em}{0ex}}.\end{array}$$In both cases, we proved the desired statement.
- $\psi =\mathcal{V}{\psi}_{n}$ and ${\psi}_{n}={\mathcal{V}}^{\prime}{\psi}_{n+1}$. In this case, we have $\psi =(\mathcal{V}\circ {\mathcal{V}}^{\prime}){\psi}_{n+1}$, which proves the desired statement.

## Appendix I. Characterization of the Adversarial Group

**Lemma**

**A19**(Canonical form of the elements of the adversarial group)

**.**

- 1.
- The map ${U}^{\left(j\right)}\mapsto \omega \phantom{\rule{0.166667em}{0ex}}{U}^{\left(j\right)}$ is a permutation of the set $\mathsf{Irr}\left(U\right)$, denoted as $\pi :\mathsf{Irr}\left(U\right)\to \mathsf{Irr}\left(U\right)$. In other words, for every irrep ${U}^{\left(j\right)}$ with $j\in \mathsf{Irr}\left(U\right)$, the irrep $\omega \phantom{\rule{0.166667em}{0ex}}{U}^{\left(j\right)}$ is equivalent to an irrep $k\in \mathsf{Irr}\left(U\right)$, and the correspondence between j and k is bijective.
- 2.
- The multiplicity spaces ${\mathcal{M}}_{j}$ and ${\mathcal{M}}_{\pi \left(j\right)}$ have the same dimension.
- 3.
- The unitary operator V has the canonical form $V={U}_{\pi}{V}_{0}$, where ${V}_{0}$ is an unitary operator in the commutant ${U}^{\prime}$ and ${U}_{\pi}$ is a permutation operator satisfying$$\begin{array}{c}\hfill {U}_{\pi}\left({\mathcal{R}}_{j}\otimes {\mathcal{M}}_{j}\right)=\left({\mathcal{R}}_{\pi \left(j\right)}\otimes {\mathcal{M}}_{\pi \left(j\right)}\right)\phantom{\rule{2.em}{0ex}}\forall j\in \mathsf{Irr}\left(U\right)\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

**Proof.**

- the irreps $\omega \phantom{\rule{0.166667em}{0ex}}{U}^{\left(k\right)}$ and ${U}^{\pi \left(k\right)}$ are equivalent,
- the multiplicity spaces ${\mathcal{M}}_{k}$ and ${\mathcal{M}}_{\pi \left(k\right)}$ are unitarily isomorphic.

- $\omega \phantom{\rule{0.166667em}{0ex}}{U}^{\left(k\right)}$ and ${U}^{\left(\pi \right(k\left)\right)}$ are equivalent irreps,
- ${\mathcal{M}}_{k}$ and ${\mathcal{M}}_{\pi \left(k\right)}$ are unitarily equivalent,

**Lemma**

**A20.**

**Proof.**

**Proof of Theorem 3.**For different permutations in $\mathsf{A}$, we can choose the isomorphisms ${S}_{\pi \left(k\right),k}:{\mathcal{M}}_{k}\to {\mathcal{M}}_{\pi \left(k\right)}$ such that the following property holds:

## Appendix J. Example: The Phase Flip Group

## Appendix K. Proof of Theorem 4

## Appendix L. Proof of Proposition 16

**Proof.**

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Chiribella, G. Agents, Subsystems, and the Conservation of Information. *Entropy* **2018**, *20*, 358.
https://doi.org/10.3390/e20050358

**AMA Style**

Chiribella G. Agents, Subsystems, and the Conservation of Information. *Entropy*. 2018; 20(5):358.
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**Chicago/Turabian Style**

Chiribella, Giulio. 2018. "Agents, Subsystems, and the Conservation of Information" *Entropy* 20, no. 5: 358.
https://doi.org/10.3390/e20050358