GENERALIZED INFORMATION ENTROPIES IN QUANTUM PHYSICS

Transcription

1 GENERALIZED INFORMATION ENTROPIES IN QUANTUM PHYSICS Mariela Portesi 1, S. Zozor 2, G.M. Bosyk 1, F. Holik 1, P.W. Lamberti 3 1 IFLP & Dpto de Fisica, CONICET & Univ. Nac. de La Plata, Argentina 2 GIPSA-Lab, CNRS, Saint Martin d Hères, France 3 FaMAF, CONICET & Univ. Nac. de Córdoba, Argentina Gent, België July 12th, 2016 MaxEnt 2016

2 OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

3 OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS

4 Motivation During last decades, a growing field of research is centered on the study of processing, transmission and storage of quantum information. This entails the use of quantum measures of information (or quantum entropies, H). There are some available proposals: von Neumann, quantum versions of Renyi, Tsallis and Kaniadakis entropies,... These are nontrivially related and share some properties. For classical systems, an ample family of generalized info measures was advanced by Salicru et al, in terms of two functions. These comprise Shannon, Renyi, HC-V-D-LN-CR-T (or Tsallis ) and other already known cases.

5 Motivation During last decades, a growing field of research is centered on the study of processing, transmission and storage of quantum information. This entails the use of quantum measures of information (or quantum entropies, H). There are some available proposals: von Neumann, quantum versions of Renyi, Tsallis and Kaniadakis entropies,... These are nontrivially related and share some properties. For classical systems, an ample family of generalized info measures was advanced by Salicru et al, in terms of two functions. These comprise Shannon, Renyi, HC-V-D-LN-CR-T (or Tsallis ) and other already known cases.

6 Motivation During last decades, a growing field of research is centered on the study of processing, transmission and storage of quantum information. This entails the use of quantum measures of information (or quantum entropies, H). There are some available proposals: von Neumann, quantum versions of Renyi, Tsallis and Kaniadakis entropies,... These are nontrivially related and share some properties. For classical systems, an ample family of generalized info measures was advanced by Salicru et al, in terms of two functions. These comprise Shannon, Renyi, HC-V-D-LN-CR-T (or Tsallis ) and other already known cases.

7 Motivation During last decades, a growing field of research is centered on the study of processing, transmission and storage of quantum information. This entails the use of quantum measures of information (or quantum entropies, H). There are some available proposals: von Neumann, quantum versions of Renyi, Tsallis and Kaniadakis entropies,... These are nontrivially related and share some properties. For classical systems, an ample family of generalized info measures was advanced by Salicru et al, in terms of two functions. These comprise Shannon, Renyi, HC-V-D-LN-CR-T (or Tsallis ) and other already known cases.

8 Motivation During last decades, a growing field of research is centered on the study of processing, transmission and storage of quantum information. This entails the use of quantum measures of information (or quantum entropies, H). There are some available proposals: von Neumann, quantum versions of Renyi, Tsallis and Kaniadakis entropies,... These are nontrivially related and share some properties. For classical systems, an ample family of generalized info measures was advanced by Salicru et al, in terms of two functions. These comprise Shannon, Renyi, HC-V-D-LN-CR-T (or Tsallis ) and other already known cases.

9 GOALS: to define a very general family of quantum information measures and to study their (common) properties in order to provide with tools that can be implemented in the treatment of quantum information processes.

10 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS

11 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Definition of (h, φ)-entropies Let p = [p 1 p N ] t [0, 1] N with N i=1 p i = 1 ( N ) H (h,φ) (p) = h φ(p i ) where h : R R and φ : [0, 1] R, with φ(0) = 0 and h(φ(1)) = 0, and either (h increasing, φ concave) or (h decreasing, φ convex) i=1 Asymptotic distribution of (h, φ)-entropies, M.Salicrú, M.L. Menéndez, D. Morales, and L. Pardo, Comm. in Stat.: Th. Meth. (1993)

12 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Name Entropic functionals Entropy Shannon h(x) = x, φ(x) = x ln x H(p) = i p i ln p i Rényi HC/D/V/ LN/CR/T Unified h(x) = ln(x) 1 α, φ(x) = x α R α (p) = 1 1 α ln ( i p α i ) h(x) = x 1 1 α, φ(x) = x α T α (p) = i p α i 1 1 α h(x) = x s 1 (1 r)s, φ(x) = x r E s r (p) = ( i p r i ) s 1 (1 r)s Kaniadakis h(x) = x, φ(x) = x κ+1 x κ+1 2κ S κ (p) = i p κ+1 i p κ+1 i 2κ

13 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Some basic properties of (h, φ)-entropies For every pair of entropic functionals (h, φ) : Invariance under permutation of the p i Expansibility: H (h,φ) ([p 1 p N 0] t ) = H (h,φ) ([p 1 p N ] t )

14 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Majorization of probability distributions Let p and p be arranged with components in decreasing order p p (p is majorized by p ) when n p i n i=1 i=1 p i n = 1,..., N 1 and Majorization is a relation of partial order. For any p: N p i = i=1 N i=1 p i [ ] 1 N 1 t N [ 1 p 0 ] 1 t 0 0 p [1 0 0] t p 0

15 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Majorization of probability distributions Let p and p be arranged with components in decreasing order p p (p is majorized by p ) when n p i n i=1 i=1 p i n = 1,..., N 1 and Majorization is a relation of partial order. For any p: N p i = i=1 N i=1 p i [ ] 1 N 1 t N [ 1 p 0 ] 1 t 0 0 p [1 0 0] t p 0

16 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition Properties COMPOSITE QUANTUM SYSTEMS SUMMAR Properties of (h, φ)-entropies related with Majorization For every pair of entropic functionals (h, φ) : Schur-concavity : p p H (h,φ) (p) H (h,φ) (p ), with equality iff p = p Entropy bounds : ( ( )) 1 0 H (h,φ) (p) h p 0 φ p 0 (complete certainty) ( ( 1 h N φ N )) (uniform distribution)

17 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS

18 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Definition of quantum (h, φ)-entropies Let ρ on H N, with ρ 0, Tr ρ = 1 and Hermitian. H (h,φ) (ρ) = h (Tr φ(ρ)) where (same as before) h : R R and φ : [0, 1] R, with φ(0) = 0 and h(φ(1)) = 0, and either (h increasing, φ concave) or (h decreasing, φ convex)

19 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Classical Quantum link In diagonal form ρ = N λ i e i e i i=1 where { e i } N i=1 is an orthonormal basis, and λ i 0 with N i=1 λ i = 1 are the eigenvalues Then H (h,φ) (ρ) = H (h,φ) (λ)

20 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Classical Quantum link In diagonal form ρ = N λ i e i e i i=1 where { e i } N i=1 is an orthonormal basis, and λ i 0 with N i=1 λ i = 1 are the eigenvalues Then H (h,φ) (ρ) = H (h,φ) (λ)

21 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Properties of quantum (h, φ)-entropies derived from Majorization By definition ρ ρ means that λ λ For every pair of entropic functionals (h, φ) : Schur-concavity : ρ ρ H (h,φ) (ρ) H (h,φ) (ρ ), with equality iff ρ = UρU or ρ = Uρ U for any isometric operator U (i.e., U U = I) Entropy bounds : 0 H (h,φ) (ρ) h ( ( 1 rank ρ φ )) ( ( 1 h N φ N )) rank ρ (pure state Ψ Ψ ) ( ρ = 1 N I N ) Concavity: provided h is a concave function, H (h,φ) is concave, i.e. 0 ω 1 : H (h,φ) (ωρ+(1 ω)ρ ) ω H (h,φ) (ρ)+(1 ω) H (h,φ) (ρ )

22 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Properties of quantum (h, φ)-entropies derived from Majorization By definition ρ ρ means that λ λ For every pair of entropic functionals (h, φ) : Schur-concavity : ρ ρ H (h,φ) (ρ) H (h,φ) (ρ ), with equality iff ρ = UρU or ρ = Uρ U for any isometric operator U (i.e., U U = I) Entropy bounds : 0 H (h,φ) (ρ) h ( ( 1 rank ρ φ )) ( ( 1 h N φ N )) rank ρ (pure state Ψ Ψ ) ( ρ = 1 N I N ) Concavity: provided h is a concave function, H (h,φ) is concave, i.e. 0 ω 1 : H (h,φ) (ωρ+(1 ω)ρ ) ω H (h,φ) (ρ)+(1 ω) H (h,φ) (ρ )

23 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Definition BasicCOMPOSITE properties QUANTUM SYSTEMS SUMMAR Specific properties of quantum (h, φ)-entropies Invariance under any unitary transformation ρ UρU (e.g. time evolution) : H (h,φ) (UρU ) = H (h,φ) (ρ) Increase under the operation of a bistochastic map (e.g. a measurement or more general q.op.): Let E(ρ) = K k=1 A kρa k with the completeness relations K k=1 A k A k = I = K k=1 A ka k. Then the quantum operation ρ E(ρ) degrades the information: H (h,φ) (E(ρ)) H (h,φ) (ρ)

24 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Additivity laws for COMPOSITE bipartite systems QUANTUM An application SYSTEMSto entangleme SUMMAR OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS

25 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Additivity laws for COMPOSITE bipartite systems QUANTUM An application SYSTEMSto entangleme SUMMAR Additivity: If the functional eqs (i) φ(ab) = φ(a)b + aφ(b) and h(x + y) = h(x) + h(y), or (ii) φ(ab) = φ(a)φ(b) and h(xy) = h(x) + h(y), are fulfilled then H (h,φ) (ρ A ρ B ) = H (h,φ) (ρ A ) + H (h,φ) (ρ B ) (e.g. von Neumann, Renyi) Subadditivity: H (h,φ) (ρ AB ) H (h,φ) (ρ A ρ B ) φ(x) = x ln x Entropy of subsystems of a bipartite pure state ρ AB = Ψ Ψ : H (h,φ) (ρ A ) = H (h,φ) (ρ B )

26 MOTIVATION CLASSICAL (h, φ)-entropies QUANTUM (h, φ)-entropies Additivity laws for COMPOSITE bipartite systems QUANTUM An application SYSTEMSto entangleme SUMMAR Entropy of a separable state: Let ρ AB sep = i ω iρ A i ρ B i, ω i 0 and i ω i = 1 H (h,φ) (ρ AB sep) max { H (h,φ) (ρ A ), H (h,φ) (ρ B ) Entanglement criteria: Werner states: ρ AB = ω Ψ Ψ + (1 ω) I 4, Ψ = We choose h(x) = f (x) 1 α and φ(x) = x α }

27 OUTLINE 1 MOTIVATION 2 CLASSICAL (h, φ)-entropies Definition Properties 3 QUANTUM (h, φ)-entropies Definition Basic properties 4 COMPOSITE QUANTUM SYSTEMS Additivity laws for bipartite systems An application to entanglement detection 5 SUMMARY AND CONCLUSIONS

28 Summary and Conclusions We proposed an extension to quantum systems, of a general class of entropies defined in terms of two functions, that go beyond the trace-form entropies. The extension is based on the family of (h, φ)-entropies introduced by Salicru et al, which encompass in a unified expression most of the available measures of information. Particular cases, choosing adequately h and φ, are: SMI, Renyi and Tsallis measures, and many others, trace or not trace-form ones.

29 Summary and Conclusions We proposed an extension to quantum systems, of a general class of entropies defined in terms of two functions, that go beyond the trace-form entropies. The extension is based on the family of (h, φ)-entropies introduced by Salicru et al, which encompass in a unified expression most of the available measures of information. Particular cases, choosing adequately h and φ, are: SMI, Renyi and Tsallis measures, and many others, trace or not trace-form ones.

30 We studied various properties of the generalized quantum entropies, showing that most of them arise as consequences of the majorization property between density operators. Among other properties, the Schur-concavity was particularly analyzed in the QM context. We provided implications on the entropy behaviour with the action of quantum operations over the system. Other quantum generalized quantities, like divergences, can be given. A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

31 We studied various properties of the generalized quantum entropies, showing that most of them arise as consequences of the majorization property between density operators. Among other properties, the Schur-concavity was particularly analyzed in the QM context. We provided implications on the entropy behaviour with the action of quantum operations over the system. Other quantum generalized quantities, like divergences, can be given. A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

32 We studied various properties of the generalized quantum entropies, showing that most of them arise as consequences of the majorization property between density operators. Among other properties, the Schur-concavity was particularly analyzed in the QM context. We provided implications on the entropy behaviour with the action of quantum operations over the system. Other quantum generalized quantities, like divergences, can be given. A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

33 We studied various properties of the generalized quantum entropies, showing that most of them arise as consequences of the majorization property between density operators. Among other properties, the Schur-concavity was particularly analyzed in the QM context. We provided implications on the entropy behaviour with the action of quantum operations over the system. Other quantum generalized quantities, like divergences, can be given. A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

34 We studied various properties of the generalized quantum entropies, showing that most of them arise as consequences of the majorization property between density operators. Among other properties, the Schur-concavity was particularly analyzed in the QM context. We provided implications on the entropy behaviour with the action of quantum operations over the system. Other quantum generalized quantities, like divergences, can be given. A family of generalized quantum entropies: definition and properties, G.M. Bosyk, S. Zozor, F. Holik, MP, and P.W. Lamberti, QINP (2016)

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