# Agents, Subsystems, and the Conservation of Information

^{1}

^{2}

^{3}

^{4}

## 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$$\begin{array}{c}\hfill {U}_{\pi}\left(\right)open="("\; close=")">{\mathcal{R}}_{j}\otimes {\mathcal{M}}_{j}=\left(\right)open="("\; close=")">{\mathcal{R}}_{\pi \left(j\right)}\otimes {\mathcal{M}}_{\pi \left(j\right)}& \phantom{\rule{2.em}{0ex}}\forall j\in \mathsf{Irr}\left(U\right)\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

**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.**

## References

- Nielsen, M.; Chuang, I. Quantum information and computation. Nature
**2000**, 404, 247. [Google Scholar] - Kitaev, A.Y.; Shen, A.; Vyalyi, M.N. Classical and Quantum Computation; Number 47; American Mathematical Society: Providence, RI, USA, 2002. [Google Scholar]
- Einstein, A.; Podolsky, B.; Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.
**1935**, 47, 777. [Google Scholar] [CrossRef] - Schrödinger, E. Discussion of probability relations between separated systems. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambrdige, UK, 1935; Volume 31, pp. 555–563. [Google Scholar]
- Hardy, L. Quantum theory from five reasonable axioms. arXiv, 2001; arXiv:quant-ph/0101012. [Google Scholar]
- Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A. Generalized no-broadcasting theorem. Phys. Rev. Lett.
**2007**, 99, 240501. [Google Scholar] [CrossRef] [PubMed] - Barrett, J. Information processing in generalized probabilistic theories. Phys. Rev. A
**2007**, 75, 032304. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.; Perinotti, P. Probabilistic theories with purification. Phys. Rev. A
**2010**, 81, 062348. [Google Scholar] [CrossRef] - Barnum, H.; Wilce, A. Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci.
**2011**, 270, 3–15. [Google Scholar] [CrossRef] - Hardy, L. Foliable operational structures for general probabilistic theories. In Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed.; Cambridge University Press: Cambrdige, UK, 2011; p. 409. [Google Scholar]
- Hardy, L. A formalism-local framework for general probabilistic theories, including quantum theory. Math. Struct. Comput. Sci.
**2013**, 23, 399–440. [Google Scholar] [CrossRef] - Chiribella, G. Dilation of states and processes in operational-probabilistic theories. In Proceedings of the 11th workshop on Quantum Physics and Logic, Kyoto, Japan, 4–6 June 2014; Coecke, B., Hasuo, I., Panangaden, P., Eds.; Electronic Proceedings in Theoretical Computer Science. Volume 172, pp. 1–14. [Google Scholar]
- Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Quantum from principles. In Quantum Theory: Informational Foundations and Foils; Springer: Dordrecht, The Netherlands, 2016; pp. 171–221. [Google Scholar]
- Hardy, L. Reconstructing quantum theory. In Quantum Theory: Informational Foundations and Foils; Springer: Dordrecht, The Netherlands, 2016; pp. 223–248. [Google Scholar]
- Mauro D’Ariano, G.; Chiribella, G.; Perinotti, P. Quantum Theory from First Principles. In Quantum Theory from First Principles; D’Ariano, G.M., Chiribella, G., Perinotti, P., Eds.; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Abramsky, S.; Coecke, B. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, Turku, Finland, 17 July 2004; pp. 415–425. [Google Scholar]
- Coecke, B. Kindergarten quantum mechanics: Lecture notes. In Proceedings of the AIP Conference Quantum Theory: Reconsideration of Foundations-3, Växjö, Sweden, 6–11 June 2005; American Institute of Physics: Melville, NY, USA, 2006; Volume 810, pp. 81–98. [Google Scholar]
- Coecke, B. Quantum picturalism. Contemp. Phys.
**2010**, 51, 59–83. [Google Scholar] [CrossRef] - Abramsky, S.; Coecke, B. Categorical quantum mechanics. In Handbook of Quantum Logic and Quantum Structures: Quantum Logic; Elsevier Science: New York, NY, USA, 2008; pp. 261–324. [Google Scholar]
- Coecke, B.; Kissinger, A. Picturing Quantum Processes; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Selinger, P. A survey of graphical languages for monoidal categories. In New Structures for Physics; Springer: Berlin/Heidelberg, Germany, 2010; pp. 289–355. [Google Scholar]
- Haag, R. Local Quantum Physics: Fields, Particles, Algebras; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Viola, L.; Knill, E.; Laflamme, R. Constructing qubits in physical systems. J. Phys. A Math. Gen.
**2001**, 34, 7067. [Google Scholar] [CrossRef] - Zanardi, P.; Lidar, D.A.; Lloyd, S. Quantum tensor product structures are observable induced. Phys. Rev. Lett.
**2004**, 92, 060402. [Google Scholar] [CrossRef] [PubMed] - Palma, G.M.; Suominen, K.A.; Ekert, A.K. Quantum computers and dissipation. Proc. R. Soc. Lond. A
**1996**, 452, 567–584. [Google Scholar] [CrossRef] - Zanardi, P.; Rasetti, M. Noiseless quantum codes. Phys. Rev. Lett.
**1997**, 79, 3306. [Google Scholar] [CrossRef] - Lidar, D.A.; Chuang, I.L.; Whaley, K.B. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett.
**1998**, 81, 2594. [Google Scholar] [CrossRef] - Knill, E.; Laflamme, R.; Viola, L. Theory of quantum error correction for general noise. Phys. Rev. Lett.
**2000**, 84, 2525. [Google Scholar] [CrossRef] [PubMed] - Zanardi, P. Stabilizing quantum information. Phys. Rev. A
**2000**, 63, 012301. [Google Scholar] [CrossRef] - Kempe, J.; Bacon, D.; Lidar, D.A.; Whaley, K.B. Theory of decoherence-free fault-tolerant universal quantum computation. Phys. Rev. A
**2001**, 63, 042307. [Google Scholar] [CrossRef] - Zanardi, P. Virtual quantum subsystems. Phys. Rev. Lett.
**2001**, 87, 077901. [Google Scholar] [CrossRef] [PubMed] - Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics 1; Springer: Berlin/Heidelberg, Germany, 1987. [Google Scholar]
- Kraemer, L.; Del Rio, L. Operational locality in global theories. arXiv, 2017; arXiv:1701.03280. [Google Scholar]
- Åberg, J. Quantifying superposition. arXiv, 2006; arXiv:quant-ph/0612146. [Google Scholar]
- Baumgratz, T.; Cramer, M.; Plenio, M. Quantifying coherence. Phys. Rev. Lett.
**2014**, 113, 140401. [Google Scholar] [CrossRef] [PubMed] - Levi, F.; Mintert, F. A quantitative theory of coherent delocalization. New J. Phys.
**2014**, 16, 033007. [Google Scholar] [CrossRef] - Winter, A.; Yang, D. Operational resource theory of coherence. Phys. Rev. Lett.
**2016**, 116, 120404. [Google Scholar] [CrossRef] [PubMed] - Chitambar, E.; Gour, G. Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett.
**2016**, 117, 030401. [Google Scholar] [CrossRef] [PubMed] - Chitambar, E.; Gour, G. Comparison of incoherent operations and measures of coherence. Phys. Rev. A
**2016**, 94, 052336. [Google Scholar] [CrossRef] - Marvian, I.; Spekkens, R.W. How to quantify coherence: Distinguishing speakable and unspeakable notions. Phys. Rev. A
**2016**, 94, 052324. [Google Scholar] [CrossRef] - Yadin, B.; Ma, J.; Girolami, D.; Gu, M.; Vedral, V. Quantum processes which do not use coherence. Phys. Rev. X
**2016**, 6, 041028. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.; Perinotti, P. Informational derivation of quantum theory. Phys. Rev. A
**2011**, 84, 012311. [Google Scholar] [CrossRef] - Hardy, L. Reformulating and reconstructing quantum theory. arXiv, 2011; arXiv:1104.2066. [Google Scholar]
- Masanes, L.; Müller, M.P. A derivation of quantum theory from physical requirements. New J. Phys.
**2011**, 13, 063001. [Google Scholar] [CrossRef] - Dakic, B.; Brukner, C. Quantum Theory and Beyond: Is Entanglement Special? In Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed.; Cambridge University Press: Cambridge, UK, 2011; pp. 365–392. [Google Scholar]
- Masanes, L.; Müller, M.P.; Augusiak, R.; Perez-Garcia, D. Existence of an information unit as a postulate of quantum theory. Proc. Natl. Acad. Sci. USA
**2013**, 110, 16373–16377. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wilce, A. Conjugates, Filters and Quantum Mechanics. arXiv, 2012; arXiv:1206.2897. [Google Scholar]
- Barnum, H.; Müller, M.P.; Ududec, C. Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys.
**2014**, 16, 123029. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.; Perinotti, P. Quantum Theory, namely the pure and reversible theory of information. Entropy
**2012**, 14, 1877–1893. [Google Scholar] [CrossRef] - Chiribella, G.; Yuan, X. Quantum theory from quantum information: The purification route. Can. J. Phys.
**2013**, 91, 475–478. [Google Scholar] [CrossRef] - Chiribella, G.; Scandolo, C.M. Conservation of information and the foundations of quantum mechanics. In EPJ Web of Conferences; EDP Sciences: Les Ulis, France, 2015; Volume 95, p. 03003. [Google Scholar]
- Chiribella, G.; Scandolo, C.M. Entanglement and thermodynamics in general probabilistic theories. New J. Phys.
**2015**, 17, 103027. [Google Scholar] [CrossRef] - Chiribella, G.; Scandolo, C.M. Microcanonical thermodynamics in general physical theories. New J. Phys.
**2017**, 19, 123043. [Google Scholar] [CrossRef] - Chiribella, G.; Scandolo, C.M. Entanglement as an axiomatic foundation for statistical mechanics. arXiv, 2016; arXiv:1608.04459. [Google Scholar]
- Lee, C.M.; Selby, J.H. Generalised phase kick-back: The structure of computational algorithms from physical principles. New J. Phys.
**2016**, 18, 033023. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H. Deriving Grover’s lower bound from simple physical principles. New J. Phys.
**2016**, 18, 093047. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H.; Barnum, H. Oracles and query lower bounds in generalised probabilistic theories. arXiv, 2017; arXiv:1704.05043. [Google Scholar]
- Susskind, L. The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics; Hachette UK: London, UK, 2008. [Google Scholar]
- Takesaki, M. Theory of Operator Algebras I; Springer: New York, NY, USA, 1979. [Google Scholar]
- Barnum, H.; Knill, E.; Ortiz, G.; Somma, R.; Viola, L. A subsystem-independent generalization of entanglement. Phys. Rev. Lett.
**2004**, 92, 107902. [Google Scholar] [CrossRef] [PubMed] - Barnum, H.; Knill, E.; Ortiz, G.; Viola, L. Generalizations of entanglement based on coherent states and convex sets. Phys. Rev. A
**2003**, 68, 032308. [Google Scholar] [CrossRef] - Barnum, H.; Ortiz, G.; Somma, R.; Viola, L. A generalization of entanglement to convex operational theories: entanglement relative to a subspace of observables. Int. J. Theor. Phys.
**2005**, 44, 2127–2145. [Google Scholar] [CrossRef] - Del Rio, L.; Kraemer, L.; Renner, R. Resource theories of knowledge. arXiv, 2015; arXiv:1511.08818. [Google Scholar]
- Del Rio, L. Resource Theories of Knowledge. Ph.D. Thesis, ETH Zürich, Zürich, Switzerland, 2015. [Google Scholar] [CrossRef]
- Kraemer Gabriel, L. Restricted Agents in Thermodynamics and Quantum Information Theory. Ph.D. Thesis, ETH Zürich, Zürich, Switzerland, 2016. [Google Scholar] [CrossRef]
- Brassard, G.; Raymond-Robichaud, P. The equivalence of local-realistic and no-signalling theories. arXiv, 2017; arXiv:1710.01380. [Google Scholar]
- Holevo, A.S. Statistical Structure of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 2003; Volume 67. [Google Scholar]
- Kraus, K. States, Effects and Operations: Fundamental Notions of Quantum Theory; Springer: Berlin/Heidelberg, Germany, 1983. [Google Scholar]
- Haag, R.; Schroer, B. Postulates of quantum field theory. J. Math. Phys.
**1962**, 3, 248–256. [Google Scholar] [CrossRef] - Haag, R.; Kastler, D. An algebraic approach to quantum field theory. J. Math. Phys.
**1964**, 5, 848–861. [Google Scholar] [CrossRef] - Buscemi, F.; Chiribella, G.; D’Ariano, G.M. Inverting quantum decoherence by classical feedback from the environment. Phys. Rev. Lett.
**2005**, 95, 090501. [Google Scholar] [CrossRef] [PubMed] - Buscemi, F.; Chiribella, G.; D’Ariano, G.M. Quantum erasure of decoherence. Open Syst. Inf. Dyn.
**2007**, 14, 53–61. [Google Scholar] [CrossRef] - Selinger, P. Idempotents in dagger categories. Electron. Notes Theor. Comput. Sci.
**2008**, 210, 107–122. [Google Scholar] [CrossRef] - Coecke, B.; Selby, J.; Tull, S. Two Roads to Classicality. Electron. Proc. Theor. Comput. Sci.
**2018**, 266, 104–118. [Google Scholar] [CrossRef] - Coecke, B.; Lal, R. Causal categories: relativistically interacting processes. Found. Phys.
**2013**, 43, 458–501. [Google Scholar] [CrossRef] - Coecke, B. Terminality implies no-signalling... and much more than that. New Gener. Comput.
**2016**, 34, 69–85. [Google Scholar] [CrossRef] - Chiribella, G. Distinguishability and copiability of programs in general process theories. Int. J. Softw. Inform.
**2014**, 8, 209–223. [Google Scholar] - Fulton, W.; Harris, J. Representation Theory: A First Course; Springer: Berlin/Heidelberg, Germany, 2013; Volume 129. [Google Scholar]
- Marvian, I.; Spekkens, R.W. A generalization of Schur–Weyl duality with applications in quantum estimation. Commun. Math. Phys.
**2014**, 331, 431–475. [Google Scholar] [CrossRef] - Galley, T.D.; Masanes, L. Impossibility of mixed-state purification in any alternative to the Born Rule. arXiv, 2018; arXiv:1801.06414. [Google Scholar]
- Yngvason, J. Localization and entanglement in relativistic quantum physics. In The Message of Quantum Science; Springer: Berlin/Heidelberg, Germany, 2015; pp. 325–348. [Google Scholar]
- Murray, F.J.; Neumann, J.V. On rings of operators. Ann. Math.
**1936**, 37, 116–229. [Google Scholar] [CrossRef] - Murray, F.J.; von Neumann, J. On rings of operators. II. Trans. Am. Math. Soc.
**1937**, 41, 208–248. [Google Scholar] [CrossRef] - Uhlmann, A. The transition probability in the state space of a *-algebra. Rep. Math. Phys.
**1976**, 9, 273–279. [Google Scholar] [CrossRef] - Jozsa, R. Fidelity for mixed quantum states. J. Mod. Opt.
**1994**, 41, 2315–2323. [Google Scholar] [CrossRef] - Lindblad, G. A general no-cloning theorem. Lett. Math. Phys.
**1999**, 47, 189–196. [Google Scholar] [CrossRef] - D’Ariano, G.M.; Presti, P.L. Optimal nonuniversally covariant cloning. Phys. Rev. A
**2001**, 64, 042308. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.; Perinotti, P.; Cerf, N. Extremal quantum cloning machines. Phys. Rev. A
**2005**, 72, 042336. [Google Scholar] [CrossRef] - Coecke, B.; Selby, J.; Tull, S. Categorical Probabilistic Theories. Electron. Proc. Theor. Comput. Sci.
**2018**, 266, 367–385. [Google Scholar]

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/e20050358

**Chicago/Turabian Style**

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