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### Citations

249 | Coecke A categorical semantics of quantum protocols
- Abramsky, B
(Show Context)
Citation Context ...he specific variant of Refs. [13, 14, 12], known as the framework of operational-probabilistic theories (OPTs). OPTs arise from the marriage of the graphical language of symmetric monoidal categories =-=[1, 2, 19, 20, 43]-=- with the toolbox of probability theory. Physical processes can be combined in sequence or in parallel, giving rise to circuits like the following ρ A A A ′ A ′ A ′′ a B B B ′ b . Here, A, A′, A′′, B,... |

202 |
Completely bounded maps and operator algebras
- Paulsen
- 2003
(Show Context)
Citation Context ...′AB, then Ψ′AB = (IA⊗UB)ΨAB, where UB is a reversible channel on B. In quantum theory, the validity of Purification lies at the foundation of all dilation theorems, such Stinespring’s [46], Naimark’s =-=[39]-=-, and Ozawa’s [38]. In the finite dimensional setting, these theorems have been reconstructed axiomatically in [13]. Finally, we introduce a new axiom, which we name Pure Sharpness. This axiom ensures... |

196 |
Foundations of Quantum Physics
- Piron
- 1976
(Show Context)
Citation Context ...tem has at least one pure effect occurring with unit probability on some state. Such a pure effect can be seen as part of a yes-no test designed to check an elementary property, in the sense of Piron =-=[40]-=-. In these terms, Pure Sharpness requires that for every system there exists at least one property, and at least one state that has such a property. Note that none of our axioms assumes that there exi... |

158 |
F.: Positive Functions on C∗-algebras
- Stinespring
- 1955
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Citation Context ...at TrBΨAB = TrBΨ′AB, then Ψ′AB = (IA⊗UB)ΨAB, where UB is a reversible channel on B. In quantum theory, the validity of Purification lies at the foundation of all dilation theorems, such Stinespring’s =-=[46]-=-, Naimark’s [39], and Ozawa’s [38]. In the finite dimensional setting, these theorems have been reconstructed axiomatically in [13]. Finally, we introduce a new axiom, which we name Pure Sharpness. Th... |

132 |
Über den Variabilitätsbereich der Koefficienten von Potenzreihen, die gegebene Werte nicht annehmen
- Carathéodory
- 1907
(Show Context)
Citation Context ...ation of ρ is a convex decomposition of ρ into perfectly distinguishable pure states. .The probabilities in such a convex decomposition will be called the eigenvalues of ρ . By Carathéodory’s theorem =-=[11, 45]-=-, we know that every diagonalization of ρ ∈ St1 (A) can have at most DA + 1 terms, where DA is the dimension of the vector space spanned by the state space St(A) (see Ref. [13]). Here we are not postu... |

110 |
Quantum theory from five reasonable axioms
- Hardy
- 2003
(Show Context)
Citation Context ...er, it would be desirable to have an axiomatic foundation of quantum thermodynamics based on purely operational axioms. The problem can be addressed in the framework of general probabilistic theories =-=[29, 21, 8, 3, 13, 14, 7, 31, 30, 12]-=-. The first step in this direction is to prove an operational version of the spectral theorem from basic principles, showing that every state can be decomposed as a mixture of perfectly distinguishabl... |

104 | A survey of graphical languages for monoidal categories
- Selinger
- 2010
(Show Context)
Citation Context ...he specific variant of Refs. [13, 14, 12], known as the framework of operational-probabilistic theories (OPTs). OPTs arise from the marriage of the graphical language of symmetric monoidal categories =-=[1, 2, 19, 20, 43]-=- with the toolbox of probability theory. Physical processes can be combined in sequence or in parallel, giving rise to circuits like the following ρ A A A ′ A ′ A ′′ a B B B ′ b . Here, A, A′, A′′, B,... |

73 |
Information processing in generalized probabilistic theories
- Barrett
(Show Context)
Citation Context ...er, it would be desirable to have an axiomatic foundation of quantum thermodynamics based on purely operational axioms. The problem can be addressed in the framework of general probabilistic theories =-=[29, 21, 8, 3, 13, 14, 7, 31, 30, 12]-=-. The first step in this direction is to prove an operational version of the spectral theorem from basic principles, showing that every state can be decomposed as a mixture of perfectly distinguishabl... |

59 |
Quantum measuring processes of continuous observables
- Ozawa
- 1984
(Show Context)
Citation Context ...IA⊗UB)ΨAB, where UB is a reversible channel on B. In quantum theory, the validity of Purification lies at the foundation of all dilation theorems, such Stinespring’s [46], Naimark’s [39], and Ozawa’s =-=[38]-=-. In the finite dimensional setting, these theorems have been reconstructed axiomatically in [13]. Finally, we introduce a new axiom, which we name Pure Sharpness. This axiom ensures that there exists... |

42 |
Entanglement and the foundations of statistical mechanics,
- Popescu, Short, et al.
- 2006
(Show Context)
Citation Context ...the fundamental level. In the concrete Hilbert space setting, the purified view of quantum thermodynamics was adopted in a number of works aimed at deriving the microcanonical and canonical ensembles =-=[34, 26, 41, 27, 25, 36, 9]-=-, an idea that has been recently explored also in hypothetical post-quantum scenarios [35, 37]. The paper is structured as follows: in section 2 we introduce the basic framework. The four axioms for d... |

39 |
Classical Mathematical Physics
- Thirring
- 1997
(Show Context)
Citation Context ...ction One of the possible routes to the foundations of quantum thermodynamics is provided by the theory of majorization, used to define an ordering among states according to their degree of mixedness =-=[48, 49, 50, 47, 24, 32, 28, 10]-=-. The viability of this approach heavily relies on the Hilbert space framework, for it is based on the fact that density operators can be diagonalized. Ideally, however, it would be desirable to have ... |

32 |
Informational Derivation of Quantum Theory
- Chiribella, D’Ariano, et al.
- 2011
(Show Context)
Citation Context ...nly sequential composition. Extending the axiom to parallel composition is important for our purposes, because it guarantees that the product of two pure states is pure. In the axiomatization of Ref. =-=[14]-=- this property was a consequence of the Local Tomography axiom, which, instead, is not assumed here. G. Chiribella & C. M. Scandolo 5 4 Consequences of the axioms 4.1 Consequences of Causality, Purity... |

30 |
Bedingt konvergente Reihen und konvexe Systeme, J. reine angew Math. 143
- Steinitz
- 1913
(Show Context)
Citation Context ...ation of ρ is a convex decomposition of ρ into perfectly distinguishable pure states. .The probabilities in such a convex decomposition will be called the eigenvalues of ρ . By Carathéodory’s theorem =-=[11, 45]-=-, we know that every diagonalization of ρ ∈ St1 (A) can have at most DA + 1 terms, where DA is the dimension of the vector space spanned by the state space St(A) (see Ref. [13]). Here we are not postu... |

27 |
Generalized no-broadcasting theorem,
- Barnum, Barrett, et al.
- 2007
(Show Context)
Citation Context ...er, it would be desirable to have an axiomatic foundation of quantum thermodynamics based on purely operational axioms. The problem can be addressed in the framework of general probabilistic theories =-=[29, 21, 8, 3, 13, 14, 7, 31, 30, 12]-=-. The first step in this direction is to prove an operational version of the spectral theorem from basic principles, showing that every state can be decomposed as a mixture of perfectly distinguishabl... |

26 |
Quantum Thermodynamics, Emergence of Thermodynamic Behavior within Composite Quantum Systems
- Gemmer, Michel, et al.
- 2004
(Show Context)
Citation Context ...the fundamental level. In the concrete Hilbert space setting, the purified view of quantum thermodynamics was adopted in a number of works aimed at deriving the microcanonical and canonical ensembles =-=[34, 26, 41, 27, 25, 36, 9]-=-, an idea that has been recently explored also in hypothetical post-quantum scenarios [35, 37]. The paper is structured as follows: in section 2 we introduce the basic framework. The four axioms for d... |

24 | Probabilistic Theories: What Is Special about Quantum Mechanics?,
- D’Ariano
- 2009
(Show Context)
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23 | Categorical Quantum Mechanics - Abramsky, Coecke - 2008 |

20 | Quantum picturalism.
- Coecke
- 2010
(Show Context)
Citation Context ...he specific variant of Refs. [13, 14, 12], known as the framework of operational-probabilistic theories (OPTs). OPTs arise from the marriage of the graphical language of symmetric monoidal categories =-=[1, 2, 19, 20, 43]-=- with the toolbox of probability theory. Physical processes can be combined in sequence or in parallel, giving rise to circuits like the following ρ A A A ′ A ′ A ′′ a B B B ′ b . Here, A, A′, A′′, B,... |

18 | Reformulating and Reconstructing Quantum Theory,
- Hardy
- 1104
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14 |
Probabilistic theories with purification
- Chiribella, DAriano, et al.
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13 | Ensemble steering, weak self-duality and the structure of probabilistic theories,
- Barnum, Gaebler, et al.
- 912
(Show Context)
Citation Context ...ng 10 Operational axioms for diagonalizing states Proposition 3 of Ref. [6] we obtain that the state space is self-dual—a property that plays an important role in the reconstruction of quantum theory =-=[5]-=-. Another important application of our result is the axiomatic reconstruction of (quantum) thermodynamics. In a previous work [17], we defined a set of thermodynamic transformations in the operational... |

12 |
Information processing in convex operational theories
- Barnum, Wilce
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12 | Foliable operational structures for general probabilistic theories. Arxiv preprint arXiv:0912.4740
- Hardy
- 2009
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11 |
Entropy and information causality in general probabilistic theories
- Barnum, Barrett, et al.
(Show Context)
Citation Context ...ique invariant state χA. Also, transitivity implies that the set of pure states is compact for every system (see corollary 32 of Ref. [13]). This property is generally a non-trivial property—cf. Ref. =-=[4]-=- for a counterexample of a state space with a non-closed set of pure states. Finally, combining Purification with Purity Preservation one obtains the following properties [14]: Proposition 4. For ever... |

11 |
Quantum approach to a derivation of the second law of thermodynamics
- Gemmer, Otte, et al.
- 1927
(Show Context)
Citation Context ...the fundamental level. In the concrete Hilbert space setting, the purified view of quantum thermodynamics was adopted in a number of works aimed at deriving the microcanonical and canonical ensembles =-=[34, 26, 41, 27, 25, 36, 9]-=-, an idea that has been recently explored also in hypothetical post-quantum scenarios [35, 37]. The paper is structured as follows: in section 2 we introduce the basic framework. The four axioms for d... |

11 |
Endlich-dimensionale dichtematrizen Univ
- Uhlmann
- 1973
(Show Context)
Citation Context ...ction One of the possible routes to the foundations of quantum thermodynamics is provided by the theory of majorization, used to define an ordering among states according to their degree of mixedness =-=[48, 49, 50, 47, 24, 32, 28, 10]-=-. The viability of this approach heavily relies on the Hilbert space framework, for it is based on the fact that density operators can be diagonalized. Ideally, however, it would be desirable to have ... |

9 |
Entropy in general physical theories
- Short, Wehner
- 2010
(Show Context)
Citation Context ...antitative measures of mixedness [42, 18]. Such an application contributes also to the difficult problem of finding axioms that imply well-behaved notions of entropy in general probabilistic theories =-=[4, 44, 33]-=- and, therefore, to the development of an axiomatic approach to information theory—specifically including data compression and transmission over noisy channels. Acknowledgements We acknowledge P Perin... |

7 |
The second laws of quantum thermodynamics.
- Brandao, Horodecki, et al.
- 2015
(Show Context)
Citation Context ...ction One of the possible routes to the foundations of quantum thermodynamics is provided by the theory of majorization, used to define an ordering among states according to their degree of mixedness =-=[48, 49, 50, 47, 24, 32, 28, 10]-=-. The viability of this approach heavily relies on the Hilbert space framework, for it is based on the fact that density operators can be diagonalized. Ideally, however, it would be desirable to have ... |

7 |
Fundamental limitations for quantum and nanoscale thermodynamics.
- Horodecki, Oppenheim
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7 |
Concentration of measure for quantum states with a fixed expectation value
- Müller, Gross, et al.
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6 | Quantum Theory, Namely the Pure and Reversible Theory of Information - Chiribella, D’Ariano, et al. |

6 | Categories for the practising physicist
- Coecke, Paquette
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5 |
The resource theory of informational nonequilibrium in thermodynamics. Phys
- Gour, Müller, et al.
- 2015
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5 |
Unifying typical entanglement and coin tossing: on randomization in probabilistic theories
- Müller, Dahlsten, et al.
- 2012
(Show Context)
Citation Context ...pted in a number of works aimed at deriving the microcanonical and canonical ensembles [34, 26, 41, 27, 25, 36, 9], an idea that has been recently explored also in hypothetical post-quantum scenarios =-=[35, 37]-=-. The paper is structured as follows: in section 2 we introduce the basic framework. The four axioms for diagonalizing states are presented in section 3 and their consequences are examined in section ... |

5 |
The black hole information problem beyond quantum theory
- Mueller, Oppenheim, et al.
- 2012
(Show Context)
Citation Context ...pted in a number of works aimed at deriving the microcanonical and canonical ensembles [34, 26, 41, 27, 25, 36, 9], an idea that has been recently explored also in hypothetical post-quantum scenarios =-=[35, 37]-=-. The paper is structured as follows: in section 2 we introduce the basic framework. The four axioms for diagonalizing states are presented in section 3 and their consequences are examined in section ... |

3 |
Higher-order interference and single-system postulates characterizing quantum theory
- Barnum, Mueller, et al.
(Show Context)
Citation Context ... and Pure Sharpness. Our result has several applications: first of all, it allows one to import all the known consequences of diagonalization in the axiomatic context, such as those presented in Ref. =-=[6]-=-, where diagonalization is assumed as Axiom 1. For example, combining our axioms with Hardy’s Permutability axiom we obtain that all set of perfectly distinguishable pure states of the same cardinalit... |

3 |
H.: Distinguishability measures and entropies for general probabilistic theories
- Kimura, Nuida, et al.
- 2010
(Show Context)
Citation Context ...antitative measures of mixedness [42, 18]. Such an application contributes also to the difficult problem of finding axioms that imply well-behaved notions of entropy in general probabilistic theories =-=[4, 44, 33]-=- and, therefore, to the development of an axiomatic approach to information theory—specifically including data compression and transmission over noisy channels. Acknowledgements We acknowledge P Perin... |

3 |
Satze uber dichtematrizen
- Uhlmann
- 1971
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2 |
2014): Dilation of states and processes in operational-probabilistic theories
- Chiribella
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1 |
Equivalence of statistical mechanical ensembles for non-critical quantum systems. arXiv preprint arXiv:1502.03263
- Brandão, Cramer
- 2015
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1 | Conservation of information and the foundations of quantum mechanics
- Chiribella, Scandolo
(Show Context)
Citation Context ...e corollary 3 of Ref. [13]): Proposition 1. In a causal theory, if {a}i∈X is an observation-test, then ∑i∈X ai = Tr. The second axiom in our list is Purity Preservation: Axiom 2 (Purity Preservation1 =-=[21, 14, 16]-=-). Sequential and parallel compositions of pure transformations are pure transformations. We consider Purity Preservation as a most fundamental requirement. Considering the theory as an algorithm to m... |

1 |
Entanglement and thermodynamics in general probabilistic theories. Master’s thesis, Università degli Studi di
- Scandolo
- 2014
(Show Context)
Citation Context ...y that plays an important role in the reconstruction of quantum theory [5]. Another important application of our result is the axiomatic reconstruction of (quantum) thermodynamics. In a previous work =-=[17]-=-, we defined a set of thermodynamic transformations in the operational framework, establishing a duality between the resource theory of entanglement and a resource theory of purity defined in terms of... |

1 |
Towards an axiomatic foundation of (quantum) thermodynamics
- Chiribella, Scandolo
(Show Context)
Citation Context ...the diagonalization theorem is the formulation of a majorization criterion capable to detect whether a thermodynamic transition is possible or not, and to establish quantitative measures of mixedness =-=[42, 18]-=-. Such an application contributes also to the difficult problem of finding axioms that imply well-behaved notions of entropy in general probabilistic theories [4, 44, 33] and, therefore, to the develo... |

1 |
Fermionic computation is non-local tomographic and violates monogamy of entanglement
- D’Ariano, Manessi, et al.
(Show Context)
Citation Context ...e derive the diagonalizability of states from a strictly weaker set of axioms, which is compatible with quantum theory on real Hilbert spaces and, potentially, other generalizations of quantum theory =-=[22, 23]-=-. Our list includes three of the six axioms of Ref. [14] (Causality, Purity Preservation, and Purification) plus one new axiom, which we name Pure Sharpness. Pure Sharpness stipulates that every physi... |

1 |
The Feynman problem and fermionic entanglement: Fermionic theory versus qubit theory
- D’Ariano, Manessi, et al.
(Show Context)
Citation Context ...e derive the diagonalizability of states from a strictly weaker set of axioms, which is compatible with quantum theory on real Hilbert spaces and, potentially, other generalizations of quantum theory =-=[22, 23]-=-. Our list includes three of the six axioms of Ref. [14] (Causality, Purity Preservation, and Purification) plus one new axiom, which we name Pure Sharpness. Pure Sharpness stipulates that every physi... |

1 |
A quantitative Landauer’s principle. arXiv preprint arXiv:1211.1037
- Faist, Dupuis, et al.
- 2012
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1 |
Average quantal behavior and thermodynamic isolation
- Lubkin, Lubkin
- 1993
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