The Fermionic Quantum Theory

Transcription

1 The Fermionic Quantum Theory CEQIP, Znojmo, May 2014 Authors: Alessandro Tosini Giacomo Mauro D Ariano Paolo Perinotti Franco Manessi Fermionic systems in computation and physics Fermionic Quantum theory Parity SSR in the Wick sense Consequences of parity SSR Violation of local tomography Work supported by: Violation of entanglement monogamy

2 Physics Computation Simulating Physics with Computers Richard P. Feynman Department of Physics, California Institute of Technology, Pasadena, California Received May 7, 1981 The question is, if we wrote a Hamiltonian which involved only these operators, locally coupled to corresponding operators on the other space-time points, could we imitate every quantum mechanical system which is discrete and has a finite number of degrees of freedom? I know, almost certainly, that we could do that for any quantum mechanical system which involves Bose particles. I'm not sure whether Fermi particles could be described by such a system. So I leave that open. Well, that's an example of what I meant by a general quantum mechanical simulator. I'm not sure that it's sufficient, because I'm not sure that it takes care of Fermi particles.

3 Physics Computation Relativity (spin-statistics) Fermionic anticommuting fields Parity superselection rule Haag, R., Local quantum physics, volume 2, Springer Berlin, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

4 Physics Computation Relativity (spin-statistics) Fermionic anticommuting fields Parity superselection rule Computation: local fermionic modes (LFM) Haag, R., Local quantum physics, volume 2, Springer Berlin, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1, Bravyi-Kitaev: Universal fermionic computation Fermionic computation is equivalent to Quantum computation S. B. Bravyi and A. Y. Kitaev, Annals of Physics 298, 210 (2002)

5 Physics Computation Relativity (spin-statistics) Fermionic anticommuting fields Parity superselection rule Computation: local fermionic modes (LFM) Haag, R., Local quantum physics, volume 2, Springer Berlin, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1, Bravyi-Kitaev:? Universal fermionic computation Fermionic computation is equivalent to Quantum computation S. B. Bravyi and A. Y. Kitaev, Annals of Physics 298, 210 (2002)

6 Jordan-Wigner isomorphism:

7 Jordan-Wigner isomorphism: N-LFMs = N-qubits

8 Jordan-Wigner isomorphism: N-LFMs = N-qubits Fermionic algebra {ϕ i, ϕ j } =0 {ϕ i, ϕ i } = δ iji Represent on Fock space F N C 2N s 1,...,s N := (ϕ 1 )s1 (ϕ N )s N 000 0

9 Jordan-Wigner isomorphism: N-LFMs = N-qubits Fermionic algebra {ϕ i, ϕ j } =0 {ϕ i, ϕ i } = δ iji Represent on Fock space F N C 2N s 1,...,s N := (ϕ 1 )s1 (ϕ N )s N J(ϕ i ):=σ z 1 σ z i 1 σ i I i+1 I N Local fermionic operations into nonlocal quantum operations

10 Jordan-Wigner isomorphism: N-LFMs = N-qubits Fermionic algebra {ϕ i, ϕ j } =0 {ϕ i, ϕ i } = δ iji Represent on Fock space F N C 2N s 1,...,s N := (ϕ 1 )s1 (ϕ N )s N J(ϕ i ):=σ z 1 σ z i 1 σ i I i+1 I N Local fermionic operations into nonlocal quantum operations On the top impose the Parity SSR

11 Jordan-Wigner isomorphism: N-LFMs = N-qubits Fermionic algebra {ϕ i, ϕ j } =0 {ϕ i, ϕ i } = δ iji Represent on Fock space F N C 2N s 1,...,s N := (ϕ 1 )s1 (ϕ N )s N J(ϕ i ):=σ z 1 σ z i 1 σ i I i+1 I N Local fermionic operations into nonlocal quantum operations On the top impose the Parity SSR What do we map? Where does SSR come from?

12 Fermionic Quantum Theory

13 Fermionic Quantum Theory States, effects and maps in terms of ϕ i and ϕ i

14 Fermionic Quantum Theory States, effects and maps in terms of ϕ i and ϕ i States of N-LFMs: density matrixes on the Fock space F N C 2N ρ = st ρ st N ϕ i s i ϕ i ϕ i ϕ i t i ρ st C i=1

15 Fermionic Quantum Theory States, effects and maps in terms of ϕ i and ϕ i States of N-LFMs: density matrixes on the Fock space F N C 2N ρ = st ρ st N ϕ i s i ϕ i ϕ i ϕ i t i ρ st C i=1 Prob(ρ,a):=Tr[aρ]

16 Fermionic Quantum Theory States, effects and maps in terms of ϕ i and ϕ i States of N-LFMs: density matrixes on the Fock space F N C 2N ρ = st ρ st N ϕ i s i ϕ i ϕ i ϕ i t i ρ st C i=1 Prob(ρ,a):=Tr[aρ] Transformations linear Hermitian preserving maps: KRAUS FORM T (ρ) = i s i K i ρk i s i = ±1

17 Assumptions on maps

18 Assumptions on maps Maps with single Kraus ϕ i, ϕ i or ϕ i + ϕ i are maps of the theory

19 Assumptions on maps ϕ i ϕ i ϕ i + ϕ i Maps with single Kraus, or are maps of the theory Maps with Kraus operators involving only, with are LOCAL on the LFMs in χ ϕ i ϕ i i χ. 1 LOCAL in the operational sense T.. N } χ =

20 Derivation of parity SSR Proposition. No map can have Kraus that are combination of even and odd products of fields.

21 Derivation of parity SSR Proposition. No map can have Kraus that are combination of even and odd products of fields. Proof. T (ρ) = i s i K i ρk i if K i = E i + O i for some i E i O i even number of field operators odd number of field operators

22 Derivation of parity SSR Proposition. No map can have Kraus that are combination of even and odd products of fields. Proof. T (ρ) = i s i K i ρk i if K i = E i + O i for some i E i O i even number of field operators odd number of field operators then ρ N T M N a =0 ρ St(NM), a Eff(NM)

23 Derivation of parity SSR Corollary. States must be combination of even products of field operators. F = F e F o pρe 0 ρ = 0 (1 p)ρ o ρ = p e ρ e + p o ρ o p o + p e =1

24 Derivation of parity SSR Corollary. States must be combination of even products of field operators. F = F e F o pρe 0 ρ = 0 (1 p)ρ o ρ = p e ρ e + p o ρ o p o + p e =1 Set of States of N-LFMs St R (N F )=Herm(C 2N 1 ) Herm(C 2N 1 ) = N-1 qubit state spaces

25 Parity SSRs is not conservation of Parity Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

26 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

27 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

28 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 α 00 + β 10 Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

29 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 α 00 + β 10 Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

30 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 α 00 + β 10 E.g: 1-LFM ψ = 0, 1 α 0 + β 1 Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

31 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 α 00 + β 10 E.g: 1-LFM ψ = 0, 1 α 0 + β 1 Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

32 Parity SSRs is not conservation of Parity SSR is an inhibition to the superposition principle parity is conserved F = F e F o never superimposed E.g: 2-LFMs ψ = α 10 + β 01, α 00 + β 11 α 00 + β 10 E.g: 1-LFM ψ = 0, 1 α 0 + β 1 1-LFM = calssical bit Wick, G. C., Wightman, A. S., and Wigner, E. P., 1952, Phys. Rev. 88, Wick, G. C., Wightman, A. S., and Wigner, E. P., 1970, Phys. Rev. D1,

33 Fermions violate local tomography L. Hardy and W. K. Wootters, Foundations of Physics 42, 454 (2012)

34 Fermions violate local tomography Local tomography: ρ A B A A a = σ ρ B B = b σ A B a b D AB = D A D B L. Hardy and W. K. Wootters, Foundations of Physics 42, 454 (2012)

35 Fermions violate local tomography Local tomography: ρ A B A A a = σ ρ B B = b σ A B a b D AB = D A D B Bilocal tomography: I need local and bilocal effects for state tomography D AB >D A D B D ABC f(d A,D B,D C,D AB,D AC,D BC ) L. Hardy and W. K. Wootters, Foundations of Physics 42, 454 (2012)

36 Fermions violate local tomography

37 Fermions violate local tomography Quantum theory N-qubits St R (N Q ) = Herm(C 2N ) D NQ =2 2N local tomographic

38 Fermions violate local tomography Quantum theory N-qubits Fermionic Quantum theory N-LFMs St R (N Q ) = Herm(C 2N ) St R (N F )=Herm(C 2N 1 ) Herm(C 2N 1 ) D NQ =2 2N D NF =2 2N 1 = 1 2 D N Q local tomographic bilocal tomographic

39 Fermionic entanglement

40 Fermionic entanglement Oper. prob. theory: 1) provide a notion of entanglement 2) amount of entanglement quantified in operational terms

41 Fermionic entanglement Oper. prob. theory: 1) provide a notion of entanglement 2) amount of entanglement quantified in operational terms 1)Proposition. Non-separability as the unique notion of fermionic entanglement

42 Fermionic entanglement Oper. prob. theory: 1) provide a notion of entanglement 2) amount of entanglement quantified in operational terms 1)Proposition. Non-separability as the unique notion of fermionic entanglement 2)Entanglement cost: resource under LOCC operations Ψ res N N of resource states (ebits) LOCC ρ M M copies of ρ

43 Quantum entanglement of formation W. K. Wootters, Quantum Information & Computation 1, 27 (2001) S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997)

44 Quantum entanglement of formation Asymptotical operational meaning: Ψ res N LOCC D ρ ρ M E(ρ) = lim M N(M) M W. K. Wootters, Quantum Information & Computation 1, 27 (2001) S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997)

45 Quantum entanglement of formation Asymptotical operational meaning: Ψ res N LOCC D ρ ρ M E(ρ) = lim M N(M) M E( Ψ) =S(Tr A ΨΨ )) E(ρ) :=min p i E( Ψ i ) D ρ i pure states mixed states W. K. Wootters, Quantum Information & Computation 1, 27 (2001) S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997)

46 Quantum entanglement of formation Asymptotical operational meaning: Ψ res N LOCC D ρ ρ M E(ρ) = lim M N(M) M E( Ψ) =S(Tr A ΨΨ )) E(ρ) :=min p i E( Ψ i ) D ρ i pure states mixed states E(ρ) =0 E(ρ) =1 ρ ρ separable maximally entangled W. K. Wootters, Quantum Information & Computation 1, 27 (2001) S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997)

47 Fermionic resource states Ψ res N LOCC ρ M *J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett. 111, (2013)

48 Fermionic resource states Ψ res N LOCC ρ M Quantum: 2-qubits Maximally entangled state unique bipartite state LOCC convertible to any other 1 2 ( ) *J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett. 111, (2013)

49 Fermionic resource states Ψ res N LOCC ρ M Quantum: 2-qubits Maximally entangled state unique bipartite state LOCC convertible to any other 1 2 ( ) Fermionic: 2-LFMs Maximally entangled sets* not unique bipartite state LOCC convertible to any other {α 00 + β 11, α, β > 0} *J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett. 111, (2013)

50 Fermionic entanglement of formation M.-C. Banũls, J. I. Cirac and M. M. Wolf, Phys. Rev. A 76, (Aug 2007)

51 Fermionic entanglement of formation Ψ res N D F ρ locc F E F (ρ) = lim ρ M M N(M) M M.-C. Banũls, J. I. Cirac and M. M. Wolf, Phys. Rev. A 76, (Aug 2007)

52 Fermionic entanglement of formation Ψ res N D F ρ locc F E F (ρ) = lim ρ M M N(M) M Proposition. Ẽ F (ρ) E F (ρ) Ẽ F (ρ) =p e E(ρ e )+p o E(ρ o ) p e ρ e + p o ρ o M.-C. Banũls, J. I. Cirac and M. M. Wolf, Phys. Rev. A 76, (Aug 2007)

53 Maximally entangled mixed state G. M. D Ariano, F. Manessi, P. Perinotti and A. Tosini, arxiv preprint arxiv: (2013)

54 Maximally entangled mixed state ρ = 1 2 ρ e ρ o Ψ e = 1 2 ( ) Ψ o 1 2 ( ) G. M. D Ariano, F. Manessi, P. Perinotti and A. Tosini, arxiv preprint arxiv: (2013)

55 Maximally entangled mixed state ρ = 1 2 ρ e ρ o Ψ e = 1 2 ( ) Ψ o 1 2 ( ) In Fermionic QT has entanglement of formation 1 E F (ρ )= 1 2 E(ρ e)+ 1 2 E(ρ o)=1 G. M. D Ariano, F. Manessi, P. Perinotti and A. Tosini, arxiv preprint arxiv: (2013)

56 Maximally entangled mixed state ρ = 1 2 ρ e ρ o Ψ e = 1 2 ( ) Ψ o 1 2 ( ) In Fermionic QT has entanglement of formation 1 E F (ρ )= 1 2 E(ρ e)+ 1 2 E(ρ o)=1 In QT has entanglement of formation 0 ρ = ± = 1 2 ( 0 ± 1) E(ρ )=0 G. M. D Ariano, F. Manessi, P. Perinotti and A. Tosini, arxiv preprint arxiv: (2013)

57 Maximally entangled mixed state ρ = 1 2 ρ e ρ o Ψ e = 1 2 ( ) Ψ o 1 2 ( ) In Fermionic QT has entanglement of formation 1 E F (ρ )= 1 2 E(ρ e)+ 1 2 E(ρ o)=1 In QT has entanglement of formation 0 ρ = ± = 1 2 ( 0 ± 1) E(ρ )=0 G. M. D Ariano, F. Manessi, P. Perinotti and A. Tosini, arxiv preprint arxiv: (2013)

58 Fermionic entanglement is not monogamous V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000)

59 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob 3-qubits: Ψ ABC V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie

60 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie

61 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C E(ρ AB )+E(ρ AC ) 1 V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie

62 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C E(ρ AB )+E(ρ AC ) 1 1 V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie

63 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C E(ρ AB )+E(ρ AC ) V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie

64 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C E(ρ AB )+E(ρ AC ) V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) The fermionic entanglement is not monogamous 3-LFMs: Ψ ABC = 1 2 ( ) Alice Charlie Charlie Bob

65 Fermionic entanglement is not monogamous Quantum entanglement is monogamous Alice Bob Max. Ent 3-qubits: Ψ ABC = Ψ AB Ψ C E(ρ AB )+E(ρ AC ) V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, (2000) Charlie The fermionic entanglement is not monogamous 3-LFMs: Alice ρ Bob Ψ ABC = 1 2 ( ) ρ AB = ρ AC = ρ BC = ρ E F (ρ )=1 ρ Charlie ρ

66 Conclusions Wick parity SSR for fermionic systems derived in an operational context Fermionic Quantum theory differs from Quantum theory: it does not satisfy local tomography fermionic entanglement is not monogamous Possible applications of these two features? Computation: model alternative to the one based on qubits, e.g. cryptography in a Fermionic scenario. Physics: black hole information P. Hayden and J. Preskill, J. High Energy Phys. 09 (2007) 120 E. Verlinde and H. Verlinde, arxiv preprint arxiv: (2013) R. Bousso, Phys. Rev. D (2013)

67 Conclusions Wick parity SSR for fermionic systems derived in an operational context Fermionic Quantum theory differs from Quantum theory: it does not satisfy local tomography fermionic entanglement is not monogamous Possible applications of these two features? Computation: model alternative to the one based on qubits, e.g. cryptography in a Fermionic scenario. Physics: black hole information P. Hayden and J. Preskill, J. High Energy Phys. 09 (2007) 120 E. Verlinde and H. Verlinde, arxiv preprint arxiv: (2013) R. Bousso, Phys. Rev. D (2013) Thanks!