Superqubits. Leron Borsten Imperial College, London

Transcription

1 Superqubits Leron Borsten Imperial College, London in collaboration with: K. Brádler, D. Dahanayake, M. J. Duff, H. Ebrahim, and W. Rubens Phys. Rev. A 80, (2009) arxiv: June - 3 July 2013 INTERNATIONAL SCHOOL OF SUBNUCLEAR PHYSICS CCSEM, ERICE

2 Introduction Superqubits Super entanglement classification

3 Entanglement and information Introduction In providing nonlocal resources, quantum mechanics distinguishes itself from the classical world. Perhaps the best known example of a nonlocal resource is the EPR-pair, introduced by Einstein, Podolsky and Rosen [1935] 1 2 ( )

4 Entanglement and information Introduction In providing nonlocal resources, quantum mechanics distinguishes itself from the classical world. Perhaps the best known example of a nonlocal resource is the EPR-pair, introduced by Einstein, Podolsky and Rosen [1935] 1 2 ( ) Bell famously showed that this quantum state can be used to violate an inequality required to be satisfied by any local hidden variable model [Bell:1964].

5 Entanglement and information Introduction In providing nonlocal resources, quantum mechanics distinguishes itself from the classical world. Perhaps the best known example of a nonlocal resource is the EPR-pair, introduced by Einstein, Podolsky and Rosen [1935] 1 2 ( ) Bell famously showed that this quantum state can be used to violate an inequality required to be satisfied by any local hidden variable model [Bell:1964]. This fundamental shift is our understanding of reality has more recently precipitated a generalisation of information theory: qubits 0 or 1 a A A = a a 1 1 C a A1 A 2...A n A 1 A 2... A n C 2 C 2... C 2 Entanglement classification is an essential open problem

6 Introduction Supersymmetric quantum information Here we propose a supersymmetric generalization of the qubit, the superqubit Ψ = A a A + a A single superqubit forms a 3-dimensional representation of OSp(1 2) consisting of two commuting bosonic components and one anticommuting fermionic component.

7 Introduction Supersymmetric quantum information Here we propose a supersymmetric generalization of the qubit, the superqubit Ψ = A a A + a A single superqubit forms a 3-dimensional representation of OSp(1 2) consisting of two commuting bosonic components and one anticommuting fermionic component. Superqubit entanglement Super-Bell and super-ghz states are characterized, respectively, by nonvanishing superdeterminant (distinct from the Berezinian) and superhyperdeterminant

8 Entanglement: Non-local games The 3-player game [Greenberger, Horne, Zeilinger: 1989; Mermin: 1990; Watrous et al: 2004] Players (Alice, Bob, Charlie... ): act cooperatively in order to win. Referee: coordinates the game, question set Q and answer set A

9 Entanglement: Non-local games The 3-player game [Greenberger, Horne, Zeilinger: 1989; Mermin: 1990; Watrous et al: 2004] Players (Alice, Bob, Charlie... ): act cooperatively in order to win. Referee: coordinates the game, question set Q and answer set A r, s, t 0,1 Alice r t Referee s Charlie Bob

10 Entanglement: Non-local games The 3-player game [Greenberger, Horne, Zeilinger: 1989; Mermin: 1990; Watrous et al: 2004] Players (Alice, Bob, Charlie... ): act cooperatively in order to win. Referee: coordinates the game, question set Q and answer set A r, s, t 0,1 a, b, c 0,1 c Alice r a t Referee s Charlie b Bob

11 Entanglement: Non-local games The 3-player game: rules for winning The referee ensures that rst {000, 110, 101, 011} and the players are aware of this. The players win if r s t = a b c, where and respectively denote disjunction and addition mod 2: rs a b c

12 The 3-player game: Deterministic strategy What is the best possible classical deterministic strategy? A deterministic strategy amounts to specifying three functions, one for each player, from the question set Q to the answer set A a : Q A; b : Q A; c : Q A; r a(r), s b(s), t c(t),

13 The 3-player game: Deterministic strategy What is the best possible classical deterministic strategy? A deterministic strategy amounts to specifying three functions, one for each player, from the question set Q to the answer set A a : Q A; b : Q A; c : Q A; r a(r), s b(s), t c(t), The condition that the players win may then be written as, a(0) b(0) c(0) = 0, a(1) b(1) c(0) = 1, a(1) b(0) c(1) = 1, a(0) b(1) c(1) = 1. Add mod 2 contradiction best one can do is win 75% of the time

14 The 3-player game: Deterministic strategy What is the best possible classical deterministic strategy? A deterministic strategy amounts to specifying three functions, one for each player, from the question set Q to the answer set A a : Q A; b : Q A; c : Q A; r a(r), s b(s), t c(t), The condition that the players win may then be written as, a(0) b(0) c(0) = 0, a(1) b(1) c(0) = 1, a(1) b(0) c(1) = 1, a(0) b(1) c(1) = 1. Add mod 2 contradiction best one can do is win 75% of the time In this language the contradiction with local realism exposed by Mermin translates into the existence of a local strategy utilising the GHZ state that wins the game with p win > 0.75.

15 The 3-player game: Quantum strategy Alice, Bob and Charlie share an entangled GHZ state: Ψ = 1 2 ( )

16 The 3-player game: Quantum strategy Alice, Bob and Charlie share an entangled GHZ state: Ψ = 1 2 ( ) If r, s, t = 0 then Alice, Bob, Charlie measure in the computational basis { 0, 1 } If r, s, t = 1 then Alice, Bob, Charlie measure in the Hadamard basis { 1 2 ( ), 1 2 ( 0 1 )} Alice r = 0 0, 1 r = 1 0 ± 1 t = 0 0, 1 t = 1 0 ± 1 Charlie Referee s = 0 0, 1 s = 1 0 ± 1 Bob

17 Winning probability The 3-player game: Quantum strategy 1. rst = 000: All measure in the computational basis Ψ = 1 2 ( ) a b c = 0. Always win.

18 Winning probability The 3-player game: Quantum strategy 1. rst = 000: All measure in the computational basis Ψ = 1 2 ( ) a b c = 0. Always win. 2. rst = {011, 101, 110}: Two measure in the Hadamard basis 1 H H Ψ = 1 2 ( ) a b c = 1. Always win.

19 Winning probability The 3-player game: Quantum strategy 1. rst = 000: All measure in the computational basis Ψ = 1 2 ( ) a b c = 0. Always win. 2. rst = {011, 101, 110}: Two measure in the Hadamard basis 1 H H Ψ = 1 2 ( ) a b c = 1. Always win. Conclusion With an entangled resource we can win the game with certainty: At the tables quantum players beat their poor classical cousins! [Pan et al:2000]

20 Comparing entanglement properties Different kinds of entanglement Using the totally entangled state GHZ state Ψ GHZ = 1 2 ( ) With a single run we can rule out local realist hidden model theories However, playing the same game this totally entangled state Ψ W = 1 2 ( ) we only win with p win (W ) = 7/8. Cannot win with certainty [LB:2013]

21 Comparing entanglement properties Different kinds of entanglement Using the totally entangled state GHZ state Ψ GHZ = 1 2 ( ) With a single run we can rule out local realist hidden model theories However, playing the same game this totally entangled state Ψ W = 1 2 ( ) we only win with p win (W ) = 7/8. Cannot win with certainty [LB:2013] How are we to mathematically distinguish the properties of W and GHZ?

22 Entanglement classes Measuring and classifying entanglement Two n-qubit states have the same entanglement iff they are related by the equivalence group of Stochastic Local Operations and Classical Communication (SLOCC) [SL(2, C)] n gauge group of n-qubit entanglement The space of entanglement classes: [Bennett et al:1999, Dur et al:2000] C 2 C 2... C 2 SL 1 (2, C) SL 2 (2, C)... SL n(2, C)

23 2-qubit entanglement classification ψ = a AB AB

24 2-qubit entanglement classification ψ = a AB AB Just two entanglement classes separated by the SL A (2, C) SL A (2, C)-invariant det a AB 1. Separable A-B: det a AB = 0 2. Entangled EPR: det a AB 0

25 3-qubit entanglement classification ψ = a ABC ABC N 3 N Susy Genuine W GHZ Tripartite Entangled N 2a N 2b N 2c 1 4 Susy A BC B CA C AB Bipartite N Susy Degenerate A B C Separable Unentangled N 0 1 Susy Null Null There are 6 entanglement classes [Dur-Vidal:2000] SLOCC Freudenthal Triple Systems and Groups of type E 7 : HA/AH row/column the magic square, cf. MJ Duff s lectures [Ferrara-Gunaydin:1997;LB-Dahanayake-Duff-Ebrahim-Rubens:2008, 2009]

26 3-qubit entanglement classification Six entanglement classes are separated by [SL(2, C)] 3 -covariants [LB-Dahanayake-Duff-Ebrahim-Rubens:2009]: 1. Separable A-B-C: γ A = γ B = γ C = 0 (γ A ) A1 A 2 := a BC A 1 a A2 BC, (γ B ) B1 B 2 := a A C B 1 a AB2 C, (γ C ) C1 C 2 := a AB C 1 a ABC2,

27 3-qubit entanglement classification Six entanglement classes are separated by [SL(2, C)] 3 -covariants [LB-Dahanayake-Duff-Ebrahim-Rubens:2009]: 1. Separable A-B-C: γ A = γ B = γ C = 0 (γ A ) A1 A 2 := a BC A 1 a A2 BC, (γ B ) B1 B 2 := a A C B 1 a AB2 C, (γ C ) C1 C 2 := a AB C 1 a ABC2, 2. Three biseparable A-EPR: γ A 0 and γ B = γ C = 0

28 3-qubit entanglement classification Six entanglement classes are separated by [SL(2, C)] 3 -covariants [LB-Dahanayake-Duff-Ebrahim-Rubens:2009]: 1. Separable A-B-C: γ A = γ B = γ C = 0 (γ A ) A1 A 2 := a BC A 1 a A2 BC, (γ B ) B1 B 2 := a A C B 1 a AB2 C, (γ C ) C1 C 2 := a AB C 1 a ABC2, 2. Three biseparable A-EPR: γ A 0 and γ B = γ C = 0 3. Totally entangled W states: T ABC 0 and Det a ABC = 0 T ABC := (γ A ) AA a A BC and Det a is Cayley s Hyperdeterminant [Cayley:1845] Det a ABC := det γ A = det γ B = det γ C

29 3-qubit entanglement classification Six entanglement classes are separated by [SL(2, C)] 3 -covariants [LB-Dahanayake-Duff-Ebrahim-Rubens:2009]: 1. Separable A-B-C: γ A = γ B = γ C = 0 (γ A ) A1 A 2 := a BC A 1 a A2 BC, (γ B ) B1 B 2 := a A C B 1 a AB2 C, (γ C ) C1 C 2 := a AB C 1 a ABC2, 2. Three biseparable A-EPR: γ A 0 and γ B = γ C = 0 3. Totally entangled W states: T ABC 0 and Det a ABC = 0 T ABC := (γ A ) AA a A BC and Det a is Cayley s Hyperdeterminant [Cayley:1845] Det a ABC := det γ A = det γ B = det γ C 4. Totally entangled GHZ states: Det a ABC 0

30 3-qubit entanglement classification Six entanglement classes are separated by [SL(2, C)] 3 -covariants [LB-Dahanayake-Duff-Ebrahim-Rubens:2009]: 1. Separable A-B-C: γ A = γ B = γ C = 0 (γ A ) A1 A 2 := a BC A 1 a A2 BC, (γ B ) B1 B 2 := a A C B 1 a AB2 C, (γ C ) C1 C 2 := a AB C 1 a ABC2, 2. Three biseparable A-EPR: γ A 0 and γ B = γ C = 0 3. Totally entangled W states: T ABC 0 and Det a ABC = 0 T ABC := (γ A ) AA a A BC and Det a is Cayley s Hyperdeterminant [Cayley:1845] Det a ABC := det γ A = det γ B = det γ C 4. Totally entangled GHZ states: Det a ABC 0 Det a = 0 no strategy that wins the 3-player game with certainty [LB:2013]

31 Introduction Superqubits Super entanglement classification

32 The superqubit The superqubit [LB-Dahanayake-Duff-Rubens: 2009]: Ψ = X a X = A a A + a where a A (A = 0, 1) and a are commuting/anticommuting supernumbers

33 The superqubit The superqubit [LB-Dahanayake-Duff-Rubens: 2009]: Ψ = X a X = A a A + a where a A (A = 0, 1) and a are commuting/anticommuting supernumbers Even element of a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000]: : H H ; ψ ( ψ ) := ψ. 1. sends pure bosonic (fermionic) supervectors in H into bosonic (fermionic) supervectors in H. 2. is linear ( ψ + φ ) = ψ + φ. 3. For pure even/odd α and ψ ( ψ α) = ( ) αψ α # ψ, (α ψ ) = ( ) ψ+αψ ψ α #, where # is the superconjugate.

34 Superqubit norm The even Grassmann valued norm Ψ Ψ = δ A 1A 2 a # A a A2 a # 1 a, is invariant under the supergroup of local unitaries SU(2) UOSp(2 1) [Berezin-Tolstoy:1981] A normalized superqubit Ψ may be regarded as an element of the projective space [Landi:1999] S 2 2 = UOSp(1 2)/ U(0 1) known as the supersphere Ψ = Z(η, α, β) 0, where Z UOsp(2 1) Z(η, α, β) = S(η)U(α, β) = 4 ηη# η η # η# ηη# 0 0 α β # η ηη# 0 β α #

35 2-superqubit Superqubits Ψ = AB a AB + A a A + B a B + a a AB a A a B a Figure : The 3 3 square supermatrix

36 3-superqubit Superqubits Ψ = ABC a ABC + AB a AB + A C a A C + BC a BC + A a A + B a B + C a C + a a BC a C a B a a ABC a A C a A a AB Figure : The cubic superhypermatrix

37 Introduction Superqubits Super entanglement classification

38 Super entanglement classification The super SLOCC group SL(2, C) OSp(2 1) [Berezin-Tolstoy:1981]

39 Super entanglement classification The super SLOCC group SL(2, C) OSp(2 1) [Berezin-Tolstoy:1981] 2 superqubits: the superdeterminant SL A (2, C) SL B (2, C) OSp A (2 1) OSp B (2 1) OSp A (2 1) OSp B (2 1)-invariant superdeterminant sdet a XY = 1 2 (aab a AB a A a A a B a B a a ) = (a 00 a 11 a 01 a 10 + a 0 a 1 + a 0 a 1 ) 1 2 a 2,

40 Super entanglement classification The super SLOCC group SL(2, C) OSp(2 1) [Berezin-Tolstoy:1981] 2 superqubits: the superdeterminant SL A (2, C) SL B (2, C) OSp A (2 1) OSp B (2 1) OSp A (2 1) OSp B (2 1)-invariant superdeterminant sdet a XY = 1 2 (aab a AB a A a A a B a B a a ) = (a 00 a 11 a 01 a 10 + a 0 a 1 + a 0 a 1 ) 1 2 a 2, Not the Berezinian, but quite natural: det a AB = 1 2 tr[(aε)t εa] sdet a XY = 1 2 str[(ae)st Ea]

41 3 superqubits: the superhyperdeterminant Recall, for 3 qubits we had [SL(2, C)] 3 -covariants a ABC γ A A 1 A 2 γ B B 1 B 2 γ C C 1 C 2 T ABC Det a ABC

42 3 superqubits: the superhyperdeterminant Need [OSp(2 1)] 3 -supercovariants: 1. Quadratic (5, 1, 1) ( Γ A γa1 A X 1 X 2 := 2 γ A1 γ A2 γ ) = ( ) γa1 A 2 γ A1, γ A2 0 γ A1 A 2 := a A1 BC a A2 BC a A1 B a A2 B a A1 C a A2 C a A1 a A2 γ A1 := a A1 BC a BC + a A1 B a B + a A1 C a C a A1 a 2. Cubic triple product (3, 3, 3) T XYZ = Γ A XX ax YZ. 3. Quartic superhyperdeterminant (1, 1, 1) sdet a XYZ = sdet Γ A = sdet Γ B = sdet Γ C This is the unique quartic invariant first found in [Castellani-Grassi-Sommovigo:2010] in the context of N = 2 sugra.

43 3-superqubit entanglment Summary All 3-qubit entanglement covariants may supersymmetrized in a very nature manner: a ABC a XYZ γ A Γ X γ B Γ Y γ C Γ Z T ABC T XYZ Det a ABC sdet a XYZ such that cf. sdet a XYZ = sdet Γ X = sdet Γ Y = sdet Γ Z. Det a ABC = det Γ A = det Γ B = det Γ C. Cayley s Hyperdeterminant may be supersymmtrized! The Superhyperdeterminant is invariant under OSp(2 1) OSp(2 1) OSp(2 1)

44 Conclusions Here we proposed a supersymmetric generalization of the qubit, the superqubit. Not HEP susy: Not a representation of the super-poincaré group It is a non-trivial consistent extension of conventional quantum mechanics based on a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000]

45 Conclusions Here we proposed a supersymmetric generalization of the qubit, the superqubit. Not HEP susy: Not a representation of the super-poincaré group It is a non-trivial consistent extension of conventional quantum mechanics based on a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000] Extend the n-qubit SLOCC equivalence group [SL(2, C)] n and the LOCC equivalence group [SU(2)] n to the supergroups [OSp(1 2)] n and [UOSp(1 2)] n, respectively.

46 Conclusions Here we proposed a supersymmetric generalization of the qubit, the superqubit. Not HEP susy: Not a representation of the super-poincaré group It is a non-trivial consistent extension of conventional quantum mechanics based on a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000] Extend the n-qubit SLOCC equivalence group [SL(2, C)] n and the LOCC equivalence group [SU(2)] n to the supergroups [OSp(1 2)] n and [UOSp(1 2)] n, respectively. For n = 2 and n = 3 we introduce the appropriate supersymmetric generalizations of the conventional entanglement measures.

47 Conclusions Here we proposed a supersymmetric generalization of the qubit, the superqubit. Not HEP susy: Not a representation of the super-poincaré group It is a non-trivial consistent extension of conventional quantum mechanics based on a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000] Extend the n-qubit SLOCC equivalence group [SL(2, C)] n and the LOCC equivalence group [SU(2)] n to the supergroups [OSp(1 2)] n and [UOSp(1 2)] n, respectively. For n = 2 and n = 3 we introduce the appropriate supersymmetric generalizations of the conventional entanglement measures. In particular, super-bell and super-ghz states are characterized, respectively, by nonvanishing superdeterminant (distinct from the Berezinian) and superhyperdeterminant

48 Conclusions Here we proposed a supersymmetric generalization of the qubit, the superqubit. Not HEP susy: Not a representation of the super-poincaré group It is a non-trivial consistent extension of conventional quantum mechanics based on a super Hilbert space [DeWitt:1984, Rogers:1980, Rudolph:2000] Extend the n-qubit SLOCC equivalence group [SL(2, C)] n and the LOCC equivalence group [SU(2)] n to the supergroups [OSp(1 2)] n and [UOSp(1 2)] n, respectively. For n = 2 and n = 3 we introduce the appropriate supersymmetric generalizations of the conventional entanglement measures. In particular, super-bell and super-ghz states are characterized, respectively, by nonvanishing superdeterminant (distinct from the Berezinian) and superhyperdeterminant This mathematical construction seems a very natural one. From a physical point of view, it makes contact with various condensed-matter systems: supersymmetric t-j [Wiegmann:1988, Sarkar:1991, Mavromatos:1999] quantum Hall effect [Hasebe], supersymmetric valence-bond solid states [Hasebe-Totsuka].

49 Superqubits and nonlocality Bell inequaltity S(LHVM) 2, S(EPR) = 2 2

50 Superqubits and nonlocality Bell inequaltity S(LHVM) 2, S(EPR) = 2 2 It was later realised by Tsirelson [1980] that not only does the EPR-pair violate the Bell inequality, it violates it maximally: there is no quantum system that can do better. Is the Tsirelson Bound a fundamental limit of Nature? Can there at least in principle be a theory that crosses Tsirelson?

51 Superqubits and nonlocality Bell inequaltity S(LHVM) 2, S(EPR) = 2 2 It was later realised by Tsirelson [1980] that not only does the EPR-pair violate the Bell inequality, it violates it maximally: there is no quantum system that can do better. Is the Tsirelson Bound a fundamental limit of Nature? Can there at least in principle be a theory that crosses Tsirelson? In principle Nature could do better S(no-sulu) 4 without violating the no superluminal signaling principle [Popescu-Rohrlich:1994]

52 Superqubits and nonlocality Can superqubits break Tsirlson s bound? [LB-Brádler-Duff:2012] Super resource: ( Γ AB = X X 2)( 1 2 ( ) + p θ A 1 + q ) θ B Local superunitaries: Born rule: Z ia Z jb = S(2r i θ A )U(α i, β i ) S(2s j θ B )U(γ j, δ j ). p G (ϕ, ψ) = ϕ ψ ( ϕ ψ ) #. The rationale behind this definition is clear: for ordinary (non-grassmann) states we recover the usual Born rule. Grassmann norm: τ R df = n e θ i θ # i τd 2n θ, Beats Tsirelson s bound - but at a cost: extended probabilities! i

53 Super SLOCC algebras Table : The action of the osp(1 2) generators on the superqubit fields. Generator Field acted upon a A3 a P A1 A 2 ε (A1 A 3 a A2 ) 0 2Q A1 ε A1 A 3 a a A1 Table : The action of the osp(1 2) osp(1 2) generators on the 2-superqubit fields. Field acted upon Generator Bosons Fermions a A3 B 3 a a A3 a B3 P A1 A 2 ε (A1 A 3 a A2 )B 3 0 ε (A1 A 3 a A2 ) 0 P B1 B 2 ε (B1 B 3 a A3 B 2 ) 0 0 ε (B1 B 3 a B2 ) 2Q A1 ε A1 A 3 a B3 a A1 ε A1 A 3 a a A1 B 3 2Q B1 ε B1 B 3 a A3 a B1 a A3 B 1 ε B1 B 3 a

54 Super SLOCC algebras Table : The action of the osp(1 2) osp(1 2) osp(1 2) generators on the 3-superqubit fields. Generator a A3 B 3 C 3 Bosons acted upon a A3 a B3 a C3 P A1 A 2 ε (A1 A 3 a A2 )B 3 C 3 ε (A1 A 3 a A2 ) 0 0 P B1 B 2 ε (B1 B 3 a A3 B 3 )C 2 0 ε (B1 B 3 A2 ) a 0 P C1 C 2 ε (C1 C 3 a A3 B 3 C 2 ) 0 0 ε (C1 C 3 a C2 ) 2Q A1 ε A1 A 3 a B3 C 3 ε A1 A 3 a a A1 B 3 a A1 C 3 2Q B1 ε B1 B 3 a A3 C 3 a A3 B 1 ε B1 B 3 a a B1 C 3 2Q C1 ε C1 C 3 a A3 B 3 a A3 C 1 a B3 C 1 ε C1 C 3 a Fermions acted upon a A3 B 3 a A3 C 3 a B3 C 3 a P A1 A 2 ε (A1 A 3 a A2 )B 3 ε (A1 A 3 a A2 ) C P B1 B 2 ε (B1 B 3 a A3 B 3 ) 0 ε (B1 B 3 a B3 )C 2 0 P C1 C 2 0 ε (C1 C 3 a A3 C 2 ) ε (C1 C 3 a B3 C 2 ) 0 2Q A1 ε A1 A 3 a B3 ε A1 A 3 a C3 a A1 B 3 C 3 a A1 2Q B1 ε B1 B 3 a A3 a A3 B 1 C 3 ε B1 B 3 a C3 a B1 2Q C1 a A3 B 3 C 1 ε C1 C 3 a A3 ε C1 C 3 a B3 a C1