COMPOSITIONAL THERMODYNAMICS

Transcription

1 COMPOSITIONL THERMODYNMICS Giulio Chiribella Department of Computer Science, The University of Hong Kong, CIFR-zrieli Global Scholars Program Carlo Maria Scandolo, Department of Computer Science, The University of Oxford Compositionally Workshop, Simons Institute for the Theory of Computing December

2 THERMODYNMICS Thermodynamics provides a general paradigm that can be applied to different physical theories (e.g. classical mechanics, relativistic mechanics, quantum mechanics). Is this paradigm universal? Which conditions must a physical theory satisfy in order to allow for a sensible thermodynamics?

3 FROM DYNMICS TO THERMODYNMICS Statistical mechanics reduces thermodynamical to dynamical + probabilistic notions the initial state of the system is chosen at random according to a probability distribution e.g. or thermodynamical quantities are expectation values

4 { IN THE CLSSICL WORLD: TENSION T THE FOUNDTIONS In the classical world of Newton and Laplace, there is no place for probability at the fundamental level. Why should measured quantities depend on the expectations of an agent? ttempted explanations: -ergodic theory, -symmetries -max ent principle -... single-system, non-compositional approaches

5 ENTERS COMPOSITIONLITY: THE QUNTUM CSE In quantum theory, probability has a different status: mixed states can be modelled as marginals of pure states i B B =Tr B [ ih ]] This property is called purification

6 PURIFICTION S FOUNDTION Statistical ensembles from pure states of the system + its environment Popescu-Short-Winter 2006, Gemmer-Michel-Mahler 2006,... Idea of this work: bstract from quantum mechanics, build an axiomatic foundation for statistical mechanics in general theories.

7 WHT IS COMPOSITIONL HERE? 1) The framework: general physical theories built on symmetric monoidal categories, inspired by bramsky and Coecke s Categorical Quantum Mechanics. 2) The axioms: specify properties of composition, how physical systems are combined together

8 PLN OF THE TLK 1) The framework 2) The axioms 3) The results

9 THE FRMEWORK: OPERTIONL-PROBBILISTIC THEORIES Chiribella, D riano, Perinotti, Probabilistic Theories with Purification, Phys. Rev. 81, (2010)

10 FRMEWORKS FOR GENERL PHYSICL THEORIES Single-system (non-compositional) Mackey Ludwig Gudder Piron Holevo Composite systems Hardy 2001 bramsky-coecke 2004 D riano 2006 Barrett 2006 Wilce-Barnum 2007 CDP 2009, Hardy 2009

11 OPERTIONL STRUCTURE In a nutshell: Strict symmetric monoidal category. Objects = physical systems Morphisms = (non-deterministic) physical processes

12 SYSTEMS ND TESTS -Systems:, B, C,..., I = trivial system (nothing) -Tests: a test represents a (non-deterministic) physical process B {C i } i2x : input system B : output system input-output arrow i: outcome, in some outcome set X : possible transformation, graphically represented as C i C i B

13 PREPRTIONS ND MESUREMENTS Special cases of tests: trivial input: preparation ρ i : state ρ i B trivial output: (demolition) measurement a i : effect a i

14 SEQUENTIL COMPOSITION -Sequential composition of tests: = B C C {C i } i2x {D j } j2y {D j C i } (i,j)2x Y...induces sequential composition of processes: C i B D C j = D C j C i

15 IDENTITY Identity test on system = doing nothing on system It is a test with a single outcome {I } where I is the identity process, defined by the relations C B = i B I C i 8B, 8C i C D j = C D j I 8C, 8D j

16 PRLLEL COMPOSITION -Composite systems: B ( I = I = ) -Parallel composition of tests: {C i } i2x = {C i D j } (i,j)2x Y B {D j } j2y B B B...induces parallel composition of processes (same diagram, without brackets)

17 CIRCUITS OPERTIONL THEORY: a theory of devices that can be mounted to form circuits. j0 B C C i1 C C i2 in D j1 D jn input-output arrow

18 PROBBILISTIC STRUCTURE

19 PROBBILITY SSIGNMENT Preparation + measurement = probability distribution ρ i a j = p(a j,ρ i ) { p(a j,ρ i ) 0 p(a j,ρ i )=1 i X j Y

20 INDEPENDENT EXPERIMENTS Experiments performed in parallel are statistically independent: ρ i a j = p(a j,ρ i )p(b l,σ k ) σ B k b l e.g. the roll of two dice

21 OPERTIONL-PROBBILISTIC THEORIES

22 OPERTIONL-PROBBILISTIC THEORIES (OPTS) Operational-probabilistic theory = operational structure + probabilistic structure Examples: -classical theory -quantum theory -quantum theory on real Hilbert spaces -...

23 QUOTIENT THEORIES B C B i and D j are statistically equivalent iff k C i R B M l = k D j R B M l 8R, 8 k, 8M l The quotient yields a new OPT: the quotient theory

24 ORDERED VECTOR SPCE STRUCTURE Theorem: In the quotient theory the processes of type! B span a real ordered vector space the sequential composition of two processes is linear in both arguments X X j x i C i! 0 Transf R (! B) 1 y j D X j = i,j x i y j (C i D j )

25 FINITRY SSUMPTION In this talk: restrict our attention to finite systems, i.e. systems for which

26 KEY NOTIONS

27 REVERSIBLE PROCESSES T B is reversible iff exists B S such that T B S = B S T B = B

28 PURE PROCESSES = X x T B is pure iff T B T B x implies B / 8x T x B T special case: pure states X is pure iff = x implies / 8x

29 XIOMS

30 XIOM 1: CUSLITY Operations performed in the future cannot affect the probability of outcomes of experiments done in the present Equivalently: there is only one way to discard a system Tr e.g. in QT: ρ Tr = Tr[ρ] := Tr [ B ]

31 XIOM 2: PURITY PRESERVTION Purity Preservation: the composition of pure transformations is pure T pure, S pure =) T S pure Special case: the product of two pure states is a pure state.

32 XIOM3: PURE SHRPNESS Pure Sharpness: for every system, there exist at least one pure effect that happens with probability 1 on some state. a such that a is pure a = 1 for some state

33 XIOM 4: PURIFICTION Existence: For every state of there is a system B and a pure state such that ρ of B ρ = B Tr Uniqueness: two purifications of the same state are equivalent up to a reversible transformation Ψ B Tr = Ψ B Tr = Ψ B = Ψ B U B

34 PURIFICTION: EQUIVLENT FORMULTION Theorem: For every process there exist environments E and E a pure state of E, and a reversible process from E to E such that C = U E E ϕ 0 Tr This simulation is unique up to reversible processes on the environment.

35 EXMPLES OF THEORIES STISFYING THE XIOMS quantum theory real vector space quantum theory other variants of quantum theory classical theory (!) suitably extended with non-classical systems classical bit non-classical bit

36 RESULTS Chiribella and Scandolo, Entanglement as an axiomatic foundation for statistical mechanics arxiv:

37 THE DIGONLIZTION THEOREM Thm: In a theory satisfying the 4 axioms, every state can be diagonalized, i.e. decomposed as (the spectrum ) The spectrum is uniquely defined, the diagonalization is canonical. Corollary: There exist distinguishable states.

38 THE STTE-EFFECT DULITY Thm: For every system, there exists a linear, order-preserving isomorphism between and For every pure normalized state, one has = 1 (cf. Hardy s Sharpness xiom)

39 OBSERVBLES Observables := elements of Thm. Every observable H, can be diagonalized as

40 FUNCTIONL CLCULUS Given an observable and a function one can define a new observable Formally, we have all the tools required to define the Gibbs state: But what is the interpretation of this state?

41 PURITY

42 THE RRE PRDIGM n agent has partial control on the parameters of a reversible dynamics: Prepare, Discard U x p x s a result, she implements a Random Reversible (RaRe) transformation R = X x p x U x RaRe transformations = monoidal subcategory of the category of transformations

43 THE PURITY PREORDER Definition: purer than iff RaRe cf. group majorization by Masanes & Müller Proposition. The pure states are maxima, the invariant state is the minimum.

44 PURITY MONOTONES ND ENTROPIES Definition: is a purity monotone if purer than implies Definition: family of functions is an entropy if is a purity monotone

45 EXMPLE Shannon/von Neumann entropy:

46 ENTROPY/ENERGY TRDEOFF

47 CHRCTERIZTION OF THE GIBBS STTE Proposition: In every theory satisfying the 4 axioms, the Gibbs state is the state with maximum S/vN entropy among the states with given expectation value of H.

48 LNDUER S PRINCIPLE Suppose that the system interacts reversibly with an environment in the Gibbs state S S E U E In every theory satisfying the 4 axioms, one has

49 CONCLUSIONS

50 4 axioms for thermodynamics Causality Purity Preservation Pure Sharpness Purification Obtain diagonalization, entropies, Gibbs states, and Landauer s principle. Conjecture: every physical theory admitting a thermodynamical description can be extended to a theory that satisfies the 4 axioms.

51 REFERENCES FOR THIS TLK Chiribella, D riano, Perinotti, Probabilistic Theories with Purification Phys. Rev. 81, (2010) Informational Derivation of Quantum Theory Phys. Rev. 84, (2011) Quantum from Principles, in Chiribella and Spekkens (eds), Quantum Theory: information-theoretic foundations and foils, Springer Verlag (2016) Chiribella and Scandolo, Entanglement and thermodynamics in general probabilistic theories New J. Phys (2015) Operational axioms for diagonalizing states EPTCS (2015) Entanglement as an axiomatic foundation for statistical mechanics arxiv: Purity in microcanonical thermodynamics: a tale of three resource theories arxiv: