Entanglement of indistinguishable particles

Transcription

1 Entanglement of indistinguishable particles Fabio Benatti Dipartimento di Fisica, Università di Trieste QISM Innsbruck -5 September 01

2 Outline 1 Introduction Entanglement: distinguishable vs identical qubits 3 Quantum metrology: cold atom interferometry 4 Summary F.B., R. Floreanini, U. Marzolino: Ann. Phys. 35 (010) F.B., R. Floreanini, U. Marzolino: JPB 44 (011) G. Argentieri, F.B., R. Floreanini, U. Marzolino: IJQI 9 (011) F.B., R. Floreanini, U. Marzolino: Ann. Phys. 37 (01) F.B., R. Floreanini, U. Marzolino: PRA 85 (01)

3 Entanglement of Identical Particles From particle entanglement to mode entanglement

4 Entanglement of Identical Particles From particle entanglement to mode entanglement Identical versus Indistinguishable qubits Single qubit states out of the vacuum 0 : a 0 = 1, b 0 = Two qubits: H = C C i, j, i, j = 1, Two -mode Bosons: Hilbert space H () symm = C 3 1, 1 = (a ) 0,, = (b ) 0 1, +, 1 = a b 0

5 Spatial modes a 0 = 1 : one Boson in the left well b 0 = : one Boson in the right well Figure: Double-Well Potential Figure: Left and Right localized states

6 N -mode Bosons: pseudo-spins Total Boson number: a a + b b = N

7 N -mode Bosons: pseudo-spins Total Boson number: a a + b b = N Schwinger representation: J x = a b + a b, J y = a b a b i, J z = a a b b Pseudo total angular angular momentum: [J x, J y ] = ij z

8 N -mode Bosons: pseudo-spins Total Boson number: a a + b b = N Schwinger representation: J x = a b + a b, J y = a b a b i, J z = a a b b Pseudo total angular angular momentum: [J x, J y ] = ij z N standard qubits: N J = j k = Spin Squeezing Inequalities (SSI) k=1 satisfied by fully separable states

9 N -mode Bosons: pseudo-spins Total Boson number: a a + b b = N Schwinger representation: J x = a b + a b, J y = a b a b i, J z = a a b b Pseudo total angular angular momentum: [J x, J y ] = ij z N standard qubits: N J = j k = Spin Squeezing Inequalities (SSI) k=1 satisfied by fully separable states N -mode Bosons: single particle angular moment not accessible Mode-separable states may violate some SSI

10 Mode richer structure From Spatial modes to Energy modes Bogolubov transformation: c = a + b, d = a b Single particle energy eigenstates g = d 0 = 1 e = c 0 = 1 + Figure: Ground and first excited states

11 N distinguishable qubits: separability N distinguishable qubits: natural tensor product structure of single particle Hilbert spaces: N j=1 of single particle algebras: N j=1 M ( ) fully separable states: ρ = k λ k N j=1 Ψk j Ψk j

12 N distinguishable qubits: separability N distinguishable qubits: natural tensor product structure of single particle Hilbert spaces: N j=1 of single particle algebras: N j=1 M ( ) fully separable states: ρ = k λ k N j=1 Ψk j Ψk j Local operators: tensor products of single qubit operators Total spin operator All rotations are local: J = N j=1 σ j, σ j = (σ jx, σ jy, σ jz ) e iθ J n = N e i θ σj n = e i θ σ1 n e i θ σ n e i θ σ N n j=1

13 Separability: identical Bosons No natural tensor product structure Only projections onto symmetrized vector states are allowed 1, +, 1 two-mode Bosonic vector states: 1, 1,,,

14 Separability: identical Bosons No natural tensor product structure Only projections onto symmetrized vector states are allowed 1, +, 1 two-mode Bosonic vector states: 1, 1,,, Bosonic states are not obtainable by symmetrizing density matrices of distinguishable qubits symmetric states ρ ρ not allowed in general: asym ρ ρ asym = Det(ρ) 0, asym = 1,, 1

15 Separability: identical Bosons No natural tensor product structure Only projections onto symmetrized vector states are allowed 1, +, 1 two-mode Bosonic vector states: 1, 1,,, Bosonic states are not obtainable by symmetrizing density matrices of distinguishable qubits symmetric states ρ ρ not allowed in general: asym ρ ρ asym = Det(ρ) 0, asym = 1,, 1 How to proceed?

16 Separability: identical Bosons No natural tensor product structure Only projections onto symmetrized vector states are allowed 1, +, 1 two-mode Bosonic vector states: 1, 1,,, Bosonic states are not obtainable by symmetrizing density matrices of distinguishable qubits symmetric states ρ ρ not allowed in general: asym ρ ρ asym = Det(ρ) 0, asym = 1,, 1 How to proceed? Associate locality with commutativity in a second quantized context Zanardi: PRA 65 (00), Narnhofer: PLA 310 (004) Barnum et al., PRL 9 (004)

17 Locality: commuting sub-algebras Commuting sub-algebras qubits: local (single-particle) algebras commute [ ] A 1, 1 B = 0, A B M (C) M (C)

18 Locality: commuting sub-algebras Commuting sub-algebras qubits: local (single-particle) algebras commute [ ] A 1, 1 B = 0, A B M (C) M (C) -mode Bosons: single particle Hilbert space C { 1, } creation and annihilation operators: a, a ; b, b [a, a ] = [b, b ] = 1, [a, b] = 0 Commuting sub-algebras: A = {a, a }, B = {b, b }, [A, B] = 0

19 Indistinguishable particles: local operators Definition An observable X is (A, B) local iff X = AB A A, B B

20 Indistinguishable particles: local operators Definition An observable X is (A, B) local iff X = AB A A, B B Example pseudo angular momentum operators: J x = a b + a b (A, B)-non-local rotations:, J y = a b a b i, J z = a a b b (A, B)-local rotation: e i θ Jx = e i θ (a b+a b ), e i θ Jy = e θ (a b a b) e i θ Jz = e i θ a a e i θ b b

21 Bogolubov transformations c = a + b, d = a b turns (A, B)-local rotations into (C, D)-non-local rotations e i θ Jz = e i θ (c d + d c)

22 Separability of Bosonic states Definition: Bosonic separable states States ρ on the -mode Boson algebra are (A, B)-separable iff Tr(ρ AB) = ( )( ) p i Tr(ρ (a) i A) Tr(ρ (b) i B), i for all A A, B B, ρ (a,b) i states on the -mode Boson algebra

23 Separability of Bosonic states Definition: Bosonic separable states States ρ on the -mode Boson algebra are (A, B)-separable iff Tr(ρ AB) = ( )( ) p i Tr(ρ (a) i A) Tr(ρ (b) i B), i for all A A, B B, ρ (a,b) i Example Fock number states: A A, B B: states on the -mode Boson algebra a a n a, n b = n a n a, n b, b b n a, n b = n b n a, n b n a, n b AB n a, n b = n a, n b A n a, n b n a, n b B n a, n b

24 Theorem ρ is a (A, B) separable state for N -mode Bosons iff ρ sep N A,B = p k k, N k A,B k, N k k=0 (F.B., R. Floreanini, U. Marzolino: Ann. Phys. 35 (010))

25 Example Fock number states: k, N k A,B = (a ) k (b ) (N k) k!(n k)! 0 (A, B)-local and separable

26 Example Fock number states: k, N k A,B = (a ) k (b ) (N k) k!(n k)! 0 (A, B)-local and separable Bogulobov rotate the nodes: k, N k A,B = ) k ) N k ( 1 ) N (c d (c + d 0 k!(n k)! (C, D) non-local and entangled

27 Negativity and Entanglement Witnessing Partial transposition on the first mode N N ρ = ρ kl k, N k l, N l ρ T1 = ρ kl l, N k k, N l k,l=0 k,l=0

28 Negativity and Entanglement Witnessing Partial transposition on the first mode ρ = N ρ kl k, N k l, N l ρ T1 = k,l=0 N ρ kl l, N k k, N l k,l=0 ) ( ) ρ Negativity: N(ρ) = ρ T1 1 1, ρ T1 1 = Tr( ρ T1 T 1

29 Negativity and Entanglement Witnessing Partial transposition on the first mode ρ = N ρ kl k, N k l, N l ρ T1 = k,l=0 N ρ kl l, N k k, N l k,l=0 ) ( ) ρ Negativity: N(ρ) = ρ T1 1 1, ρ T1 1 = Tr( ρ T1 T 1 (ρ T1 ) ρ T1 = k,l ρ kl k, N l k, N l ρ T1 1 = k,l ρ kl

30 Negativity and Entanglement Witnessing Partial transposition on the first mode ρ = N ρ kl k, N k l, N l ρ T1 = k,l=0 N ρ kl l, N k k, N l k,l=0 ) ( ) ρ Negativity: N(ρ) = ρ T1 1 1, ρ T1 1 = Tr( ρ T1 T 1 (ρ T1 ) ρ T1 = k,l ρ kl k, N l k, N l ρ T1 1 = k,l ρ kl N(ρ) = N = k l=0 ρ kl 0 iff ρkl = 0 for all k l

31 Negativity and Entanglement Witnessing Partial transposition on the first mode ρ = N ρ kl k, N k l, N l ρ T1 = k,l=0 N ρ kl l, N k k, N l k,l=0 ) ( ) ρ Negativity: N(ρ) = ρ T1 1 1, ρ T1 1 = Tr( ρ T1 T 1 (ρ T1 ) ρ T1 = k,l ρ kl k, N l k, N l ρ T1 1 = k,l ρ kl N(ρ) = N = k l=0 ρ kl 0 iff ρkl = 0 for all k l An exhaustive entanglement witness for two-mode Bosons N(ρ) = 0 iff ρ is (A, B)-separable

32 Spin Squeezing Inequalities SSI: N qubits vs N -mode Bosons Standard qubit entanglement condition Toth et al. PRA 79 (009) J n1 + J n1 N (N 1) J n3 >0 satisfied by (A, B)-separable states ρ = N p k k, N k (A,B) k, N k k=0 for suitable distributions p k. Standard qubit entanglement condition Korbicz et al. PRL 95 (005) N J n + J n < N 4 satisfied by k, N k (A,B), 0 < k < N, n = ẑ

33 Quantum metrology (Giovannetti et al. Science 306 (004), Phys. Rev. Lett. 96 (006)) Measuring rotation angles on N qubits Rotate an N-qubit density matrix: ρ ρ θ ρ θ = e iθj n 1 ρ e iθj n1 J = N j=1 σ j, σ j = (σ jx, σ jy, σ jz ), J n1 = n 1 J Measure J n, n n 1 and estimate θ with sensitivity δθ

34 Quantum metrology (Giovannetti et al. Science 306 (004), Phys. Rev. Lett. 96 (006)) Measuring rotation angles on N qubits Rotate an N-qubit density matrix: ρ ρ θ ρ θ = e iθj n 1 ρ e iθj n1 J = N j=1 σ j, σ j = (σ jx, σ jy, σ jz ), J n1 = n 1 J Measure J n, n n 1 and estimate θ with sensitivity δθ Shot Noise: Sub-shot Noise: δ θ = 1 N δ θ < 1 N Heisenberg Limit: δ θ = 1 N

35 Quantum Estimation and Fisher Information Quantum Fisher information (Paris, I.J.Q.I. 7 (009)) Rotation: ρ ρ θ = e iθj n ρ e iθj n Unbiased quantum estimator Ê: δ ρ θ = Tr ( ρ (Ê θ)) Quantum Fisher Information: ( F[ρ, J n ] = Tr ρ L ) θ=0, θ ρ θ = 1 ( ) [ ] ρ L + L ρ = i J n, ρ

36 Quantum Estimation and Fisher Information Quantum Fisher information (Paris, I.J.Q.I. 7 (009)) Rotation: ρ ρ θ = e iθj n ρ e iθj n Unbiased quantum estimator Ê: δ ρ θ = Tr ( ρ (Ê θ)) Quantum Fisher Information: ( F[ρ, J n ] = Tr ρ L ) θ=0, θ ρ θ = 1 ( ) [ ] ρ L + L ρ = i J n, ρ Quantum Cramer-Rao bound: δ ρ θ 1 F[ρ, J n ]

37 Quantum Estimation and Fisher Information Quantum Fisher information (Paris, I.J.Q.I. 7 (009)) Rotation: ρ ρ θ = e iθj n ρ e iθj n Unbiased quantum estimator Ê: δ ρ θ = Tr ( ρ (Ê θ)) Quantum Fisher Information: ( F[ρ, J n ] = Tr ρ L ) θ=0, θ ρ θ = 1 ( ) [ ] ρ L + L ρ = i J n, ρ Quantum Cramer-Rao bound: δ ρ θ 1 F[ρ, J n ] pure states: F[Ψ, J n ]= 4 Ψ J n

38 Quantum Estimation and Fisher Information Quantum Fisher information (Paris, I.J.Q.I. 7 (009)) Rotation: ρ ρ θ = e iθj n ρ e iθj n Unbiased quantum estimator Ê: δ ρ θ = Tr ( ρ (Ê θ)) Quantum Fisher Information: ( F[ρ, J n ] = Tr ρ L ) θ=0, θ ρ θ = 1 ( ) [ ] ρ L + L ρ = i J n, ρ Quantum Cramer-Rao bound: δ ρ θ 1 F[ρ, J n ] pure states: F[Ψ, J n ]= 4 Ψ J n convexity: ρ = j λ j Ψ j Ψ j : F[ρ, J n ] j λ j F[Ψ j, J n ]

39 QFI and Metrology QFI and separability: N standard qubits Fully separable N-qubit states: ρ N sep = k λ k Ψ N k Ψ N k, Ψ N k = N ψj k j=1

40 QFI and Metrology QFI and separability: N standard qubits Fully separable N-qubit states: ρ N sep = k λ k Ψ N k Ψ N k, Ψ N k = N ψj k j=1 From convexity (Pezzé, Smerzi, PRL (10)): F[ρ N sep, J n] j λ j F[Ψ N j, J n ] =4 j λ j J Ψ N n j

41 QFI and Metrology QFI and separability: N standard qubits Fully separable N-qubit states: ρ N sep = k λ k Ψ N k Ψ N k, Ψ N k = N ψj k j=1 From convexity (Pezzé, Smerzi, PRL (10)): F[ρ N sep, J n] j λ j F[Ψ N j, J n ] =4 j λ j J Ψ N n j Fully separable vector state Ψ FS : Ψ FS J n N 4 Fully separable mixed state ρ N FS : F[ρ N FS, J n ] N

42 Entanglement necessary to beat the shot-noise limit Quantum Cramer-Rao bound δ ρθ 1 F[ρ, J n ] In order to have necessarily 1 F[ρ, J n ] δ ρθ< 1 N F[ρ, J n ] > N

43 Cold atom interferometry Mach-Zehnder interferometry with ultracold atoms C. Gross et al., M.F. Riedel et al.: Nature 464 (010) N ultracold atoms trapped by a double-well potential as pseudo qubits via the Schwinger representation J x = a b + a b, J y = a b a b i, J z = a a b b If they were standard quits the locality of rotations would require entangled input states However, trapped cold atoms are Bosons: single angular momenta not accessible rotations not necessarily local

44 Beating the shot-noise limit with Bosons Quantum Fisher Information (A, B)-separable Fock states: k, N k A,B : n = (n x, n y, 0), ] F [ k, N k A,B, J n = 4 k,n k A,B J n = N ( k + 1) k > N k 0, N

45 Beating the shot-noise limit with Bosons Quantum Fisher Information (A, B)-separable Fock states: k, N k A,B : n = (n x, n y, 0), ] F [ k, N k A,B, J n = 4 k,n k A,B J n = N ( k + 1) k > N k 0, N Getting close to the Heisenberg limit F[ρ, J n ] = N : ] N/, N/ A,B = F [ N/, N/ A,B, J n = N + N (F.B., R. Floreanini, U. Marzolino: J. Phys. B 44 (011))

46 Non-locality from the apparatus Sub-shot noise with (A, B)-separable states: How?

47 Non-locality from the apparatus Sub-shot noise with (A, B)-separable states: How? The interferometer is (A, B)-non-local Take n = (0, 1, 0): ρ ρ θ = exp(i θ J y )ρ exp( i θ J y ) J y = a b a b i = exp(i θ J y ) (A, B) non-local

48 Non-locality from the apparatus Sub-shot noise with (A, B)-separable states: How? The interferometer is (A, B)-non-local Take n = (0, 1, 0): ρ ρ θ = exp(i θ J y )ρ exp( i θ J y ) J y = a b a b i = exp(i θ J y ) (A, B) non-local Theorem If the state (A, B)-separable and the apparatus (A, B)-local then [ ] F ρ sep (A,B), J A + J B = 0 FB, D. Braun: submitted to PRA

49 Entangling noise (A, B)-dephasing noise Lindblad master equation: ( t ρ(t) = γ J z ρ(t)j z 1 { }) Jz, ρ(t) Solution: mixture of (A, B)-local operations ρ(t) = 1 π + Exponential decay of (A, B)-entanglement: du e u /4 e i tγ/ u J z ρ e +i tγ/ u J z N A,B (ρ(t)) = k l e tγ(k l) / ρ kl e tγ/ N A,B (ρ)

50 (C, D)-entangling noise Initial (C, D)-separable state: State at time t > 0: ρ(t) = 1 π + (c ) N N! 0 = N, 0 C,D du e u /4 e i tγ/ u J z N, 0 C,D N, 0 e i tγ/ u J z e i tγ/ u J z N, 0 C,D = 1 ( ξt c + i (1 ξ t )d ) N 0 N! ) tγ ξ t = cos (u ρ(t) (C, D)-entangled: ρ(t) = N ρ kl (t) k, N k (C,D) k, N k k,l=0 (G. Argentieri, F.B., R. Floreanini, U. Marzolino: IJQI 9 (011))

51 QFI and dephasing Can a (C, D)-entangling environment increase QFI? NO

52 QFI and dephasing Can a (C, D)-entangling environment increase QFI? NO (C, D)-entangling irreversible time-evolution ρ(t) = 1 π + du e u /4 e i tγ/ u J z ρ e +i tγ/ u J z = t [ρ]

53 QFI and dephasing Can a (C, D)-entangling environment increase QFI? NO (C, D)-entangling irreversible time-evolution ρ(t) = 1 π + du e u /4 e i tγ/ u J z ρ e +i tγ/ u J z = t [ρ] Monotonicity under CPU maps ρ G[ρ]: F[G[ρ], J n ] F[ρ, J n ]

54 QFI and dephasing Can a (C, D)-entangling environment increase QFI? NO (C, D)-entangling irreversible time-evolution ρ(t) = 1 π + du e u /4 e i tγ/ u J z ρ e +i tγ/ u J z = t [ρ] Monotonicity under CPU maps ρ G[ρ]: F[G[ρ], J n ] F[ρ, J n ] From monotonicity and t = t s s : if 0 s t F[ρ(t), J n ] = F[ t s+s [ρ], J n ] F[ s [ρ], J n ] = F[ρ(s), J n ]

55 Conclusions Summary

56 Conclusions Summary Identical particles: mode-dependent entanglement

57 Conclusions Summary Identical particles: mode-dependent entanglement Algebraic formulation of mode-entanglement: commuting sub-algebras

58 Conclusions Summary Identical particles: mode-dependent entanglement Algebraic formulation of mode-entanglement: commuting sub-algebras Double-well interferometry with BECs: shot-noise limit beaten by mode separable states

59 Conclusions Summary Identical particles: mode-dependent entanglement Algebraic formulation of mode-entanglement: commuting sub-algebras Double-well interferometry with BECs: shot-noise limit beaten by mode separable states Physical origin: non-local action of the interferometer

60 Conclusions Summary Identical particles: mode-dependent entanglement Algebraic formulation of mode-entanglement: commuting sub-algebras Double-well interferometry with BECs: shot-noise limit beaten by mode separable states Physical origin: non-local action of the interferometer Noise can destroy as well as create mode-entanglement

61 Conclusions Summary Identical particles: mode-dependent entanglement Algebraic formulation of mode-entanglement: commuting sub-algebras Double-well interferometry with BECs: shot-noise limit beaten by mode separable states Physical origin: non-local action of the interferometer Noise can destroy as well as create mode-entanglement The noise-generated entanglement is not metrologically useful