Entanglement of indistinguishable particles

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

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)

Entanglement of Identical Particles From particle entanglement to mode entanglement

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

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

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

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 -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: 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

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

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

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

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,,,

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

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?

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)

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

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

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

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

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)

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

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

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))

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

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

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

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

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

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

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

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

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

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

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 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 ]

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

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 ]

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

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

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

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

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

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

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))

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

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

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

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)

(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))

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

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]

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 ]

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 ]

Conclusions Summary

Conclusions Summary Identical particles: mode-dependent entanglement

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

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

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

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

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