Entanglement of indistinguishable particles
|
|
- Adrian Barnett
- 6 years ago
- Views:
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
Quantum metrology from a quantum information science perspective
1 / 41 Quantum metrology from a quantum information science perspective Géza Tóth 1 Theoretical Physics, University of the Basque Country UPV/EHU, Bilbao, Spain 2 IKERBASQUE, Basque Foundation for Science,
More informationQuantum Fisher information and entanglement
1 / 70 Quantum Fisher information and entanglement G. Tóth 1,2,3 1 Theoretical Physics, University of the Basque Country (UPV/EHU), Bilbao, Spain 2 IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
More informationEntanglement detection close to multi-qubit Dicke states in photonic experiments (review)
Entanglement detection close to multi-qubit Dicke states in photonic experiments (review) G. Tóth 1,2,3 1 Theoretical Physics, University of the Basque Country UPV/EHU, Bilbao, Spain 2, Bilbao, Spain 3
More informationExtremal properties of the variance and the quantum Fisher information; Phys. Rev. A 87, (2013).
1 / 24 Extremal properties of the variance and the quantum Fisher information; Phys. Rev. A 87, 032324 (2013). G. Tóth 1,2,3 and D. Petz 4,5 1 Theoretical Physics, University of the Basque Country UPV/EHU,
More informationarxiv: v2 [quant-ph] 8 Jul 2014
TOPICAL REVIEW Quantum metrology from a quantum information science perspective arxiv:1405.4878v2 [quant-ph] 8 Jul 2014 Géza Tóth 1,2,3, Iagoba Apellaniz 1 1 Department of Theoretical Physics, University
More informationQuantum entanglement and its detection with few measurements
Quantum entanglement and its detection with few measurements Géza Tóth ICFO, Barcelona Universidad Complutense, 21 November 2007 1 / 32 Outline 1 Introduction 2 Bipartite quantum entanglement 3 Many-body
More informationarxiv: v2 [quant-ph] 16 Sep 2014
Optimal Quantum-Enhanced Interferometry Matthias D. Lang 1 and Carlton M. Caves 1, 1 Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico, 87131-0001, USA Centre
More informationQuantum estimation for quantum technology
Quantum estimation for quantum technology Matteo G A Paris Applied Quantum Mechanics group Dipartimento di Fisica @ UniMI CNISM - Udr Milano IQIS 2008 -- CAMERINO Quantum estimation for quantum technology
More informationarxiv: v4 [quant-ph] 21 Oct 2014
REVIEW ARTICLE Quantum metrology from a quantum information science perspective arxiv:1405.4878v4 [quant-ph] 21 Oct 2014 Géza Tóth 1,2,3, Iagoba Apellaniz 1 1 Department of Theoretical Physics, University
More informationIntroduction to entanglement theory & Detection of multipartite entanglement close to symmetric Dicke states
Introduction to entanglement theory & Detection of multipartite entanglement close to symmetric Dicke states G. Tóth 1,2,3 Collaboration: Entanglement th.: G. Vitagliano 1, I. Apellaniz 1, I.L. Egusquiza
More informationQuantum metrology with Dicke squeezed states
PAPER OPEN ACCESS Quantum metrology with Dicke squeezed states To cite this article: 2014 New J. Phys. 16 103037 View the article online for updates and enhancements. Related content - Spin squeezing,
More informationPhysics Reports 509 (2011) Contents lists available at SciVerse ScienceDirect. Physics Reports
Physics Reports 509 (011) 89 165 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Quantum spin squeezing Jian Ma a,b,, Xiaoguang Wang
More informationTowards quantum metrology with N00N states enabled by ensemble-cavity interaction. Massachusetts Institute of Technology
Towards quantum metrology with N00N states enabled by ensemble-cavity interaction Hao Zhang Monika Schleier-Smith Robert McConnell Jiazhong Hu Vladan Vuletic Massachusetts Institute of Technology MIT-Harvard
More informationNonlinear Quantum Interferometry with Bose Condensed Atoms
ACQAO Regional Workshop 0 onlinear Quantum Interferometry with Bose Condensed Atoms Chaohong Lee State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun
More informationEstimation of Optimal Singlet Fraction (OSF) and Entanglement Negativity (EN)
Estimation of Optimal Singlet Fraction (OSF) and Entanglement Negativity (EN) Satyabrata Adhikari Delhi Technological University satyabrata@dtu.ac.in December 4, 2018 Satyabrata Adhikari (DTU) Estimation
More informationA complete criterion for convex-gaussian states detection
A complete criterion for convex-gaussian states detection Anna Vershynina Institute for Quantum Information, RWTH Aachen, Germany joint work with B. Terhal NSF/CBMS conference Quantum Spin Systems The
More informationTHE INTERFEROMETRIC POWER OF QUANTUM STATES GERARDO ADESSO
THE INTERFEROMETRIC POWER OF QUANTUM STATES GERARDO ADESSO IDENTIFYING AND EXPLORING THE QUANTUM-CLASSICAL BORDER Quantum Classical FOCUSING ON CORRELATIONS AMONG COMPOSITE SYSTEMS OUTLINE Quantum correlations
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationWitnessing Genuine Many-qubit Entanglement with only Two Local Measurement Settings
Witnessing Genuine Many-qubit Entanglement with only Two Local Measurement Settings Géza Tóth (MPQ) Collaborator: Otfried Gühne (Innsbruck) uant-ph/0405165 uant-ph/0310039 Innbruck, 23 June 2004 Outline
More informationIMPROVED QUANTUM MAGNETOMETRY
(to appear in Physical Review X) IMPROVED QUANTUM MAGNETOMETRY BEYOND THE STANDARD QUANTUM LIMIT Janek Kołodyński ICFO - Institute of Photonic Sciences, Castelldefels (Barcelona), Spain Faculty of Physics,
More informationClassical and quantum simulation of dissipative quantum many-body systems
0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 Classical and quantum simulation of dissipative quantum many-body systems
More informationEstimating entanglement in a class of N-qudit states
Estimating entanglement in a class of N-qudit states Sumiyoshi Abe 1,2,3 1 Physics Division, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China 2 Department of Physical
More informationUniversality of the Heisenberg limit for phase estimation
Universality of the Heisenberg limit for phase estimation Marcin Zwierz, with Michael Hall, Dominic Berry and Howard Wiseman Centre for Quantum Dynamics Griffith University Australia CEQIP 2013 Workshop
More informationBOGOLIUBOV TRANSFORMATIONS AND ENTANGLEMENT OF TWO FERMIONS
BOGOLIUBOV TRANSFORMATIONS AND ENTANGLEMENT OF TWO FERMIONS P. Caban, K. Podlaski, J. Rembieliński, K. A. Smoliński and Z. Walczak Department of Theoretical Physics, University of Lodz Pomorska 149/153,
More informationQuantum Fisher Information. Shunlong Luo Beijing, Aug , 2006
Quantum Fisher Information Shunlong Luo luosl@amt.ac.cn Beijing, Aug. 10-12, 2006 Outline 1. Classical Fisher Information 2. Quantum Fisher Information, Logarithm 3. Quantum Fisher Information, Square
More informationPermutationally invariant quantum tomography
/ 4 Permutationally invariant quantum tomography G. Tóth,2,3, W. Wieczorek 4,5, D. Gross 6, R. Krischek 4,5, C. Schwemmer 4,5, and H. Weinfurter 4,5 Theoretical Physics, The University of the Basque Country,
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More informationNon-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives
Non-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives Heinz-Peter Breuer Universität Freiburg QCCC Workshop, Aschau, October 2007 Contents Quantum Markov processes Non-Markovian
More informationA tutorial on non-markovian quantum processes. Kavan Modi Monash University Melbourne, Australia
A tutorial on non-markovian quantum processes Kavan Modi Monash University Melbourne, Australia Quantum Information Science http://monqis.physics.monash.edu Postdoc Felix Pollock PhD Students Francesco
More informationΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ αβγδεζηθικλμνξοπρςστυφχψω +<=>± ħ
CHAPTER 1. SECOND QUANTIZATION In Chapter 1, F&W explain the basic theory: Review of Section 1: H = ij c i < i T j > c j + ij kl c i c j < ij V kl > c l c k for fermions / for bosons [ c i, c j ] = [ c
More informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 03/19/2018 Postulates of Quantum Mechanics The postulates (axioms) of quantum mechanics
More informationarxiv: v2 [quant-ph] 2 Aug 2013
Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer arxiv:1307.8009v [quant-ph] Aug 013 Xiaoxing Jing, Jing Liu, Wei Zhong, and Xiaoguang Wang Zhejiang Institute
More informationFinite temperature form factors in the free Majorana theory
Finite temperature form factors in the free Majorana theory Benjamin Doyon Rudolf Peierls Centre for Theoretical Physics, Oxford University, UK supported by EPSRC postdoctoral fellowship hep-th/0506105
More informationQuantum Memory with Atomic Ensembles
Lecture Note 5 Quantum Memory with Atomic Ensembles 04.06.2008 Difficulties in Long-distance Quantum Communication Problems leads Solutions Absorption (exponentially) Decoherence Photon loss Degrading
More informationEntanglement witnesses
1 / 45 Entanglement witnesses Géza Tóth 1 Theoretical Physics, University of the Basque Country UPV/EHU, Bilbao, Spain 2 IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 3 Wigner Research Centre
More informationEntanglement of Identical Particles
Entanglement of Identical Particles Amílcar de Queiroz IF - UnB (Brasília - Brasil) DFT - UZ (Zaragoza - España) 27 de maio de 2014 Martes Cuántico Zaragoza 1 / 46 1 Identical Particles 2 Statement of
More informationQuantum Computing Lecture 3. Principles of Quantum Mechanics. Anuj Dawar
Quantum Computing Lecture 3 Principles of Quantum Mechanics Anuj Dawar What is Quantum Mechanics? Quantum Mechanics is a framework for the development of physical theories. It is not itself a physical
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationBose Description of Pauli Spin Operators and Related Coherent States
Commun. Theor. Phys. (Beijing, China) 43 (5) pp. 7 c International Academic Publishers Vol. 43, No., January 5, 5 Bose Description of Pauli Spin Operators and Related Coherent States JIANG Nian-Quan,,
More informationQuantum mechanics in one hour
Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might
More informationS.K. Saikin May 22, Lecture 13
S.K. Saikin May, 007 13 Decoherence I Lecture 13 A physical qubit is never isolated from its environment completely. As a trivial example, as in the case of a solid state qubit implementation, the physical
More informationarxiv: v1 [quant-ph] 19 Oct 2016
Hamiltonian extensions in quantum metrology Julien Mathieu Elias Fraïsse 1 and Daniel Braun 1 1 Eberhard-Karls-Universität Tübingen, Institut für Theoretische Physik, 72076 Tübingen, Germany Abstract arxiv:1610.05974v1
More informationMany-Body Coherence in Quantum Thermodynamics
Many-Body Coherence in Quantum Thermodynamics [] H. Kwon, H. Jeong, D. Jennings, B. Yadin, and M. S. Kim Contents I. Introduction II. Resource theory of quantum thermodynamics 1. Thermal operation and
More informationQuantum Metric and Entanglement on Spin Networks
Quantum Metric and Entanglement on Spin Networks Fabio Maria Mele Dipartimento di Fisica Ettore Pancini Universitá degli Studi di Napoli Federico II COST Training School Quantum Spacetime and Physics Models
More informationTutorial: Statistical distance and Fisher information
Tutorial: Statistical distance and Fisher information Pieter Kok Department of Materials, Oxford University, Parks Road, Oxford OX1 3PH, UK Statistical distance We wish to construct a space of probability
More informationRepresentations of angular momentum
Representations of angular momentum Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 September 26, 2008 Sourendu Gupta (TIFR Graduate School) Representations of angular momentum QM I 1 / 15 Outline
More informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More informationSingle-Mode Displacement Sensor
Single-Mode Displacement Sensor Barbara Terhal JARA Institute for Quantum Information RWTH Aachen University B.M. Terhal and D. Weigand Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation,
More informationEmergence of the classical world from quantum physics: Schrödinger cats, entanglement, and decoherence
Emergence of the classical world from quantum physics: Schrödinger cats, entanglement, and decoherence Luiz Davidovich Instituto de Física Universidade Federal do Rio de Janeiro Outline of the talk! Decoherence
More informationarxiv: v1 [quant-ph] 16 Jan 2009
Bayesian estimation in homodyne interferometry arxiv:0901.2585v1 [quant-ph] 16 Jan 2009 Stefano Olivares CNISM, UdR Milano Università, I-20133 Milano, Italy Dipartimento di Fisica, Università di Milano,
More informationShunlong Luo. Academy of Mathematics and Systems Science Chinese Academy of Sciences
Superadditivity of Fisher Information: Classical vs. Quantum Shunlong Luo Academy of Mathematics and Systems Science Chinese Academy of Sciences luosl@amt.ac.cn Information Geometry and its Applications
More informationarxiv: v1 [quant-ph] 31 Oct 2011
EPJ manuscript No. (will be inserted by the editor) Optimal quantum estimation of the coupling constant of Jaynes-Cummings interaction Marco G. Genoni 1,a and Carmen Invernizzi 1 QOLS, Blackett Laboratory,
More informationPostulates of Quantum Mechanics
EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 1: Stern-Gerlach experiment Postulates of Quantum Mechanics AStern-Gerlach(SG)deviceisabletoseparateparticlesaccordingtotheirspinalonga
More informationSusana F. Huelga. Dephasing Assisted Transport: Quantum Networks and Biomolecules. University of Hertfordshire. Collaboration: Imperial College London
IQIS2008, Camerino (Italy), October 26th 2008 Dephasing Assisted Transport: Quantum Networks and Biomolecules Susana F. Huelga University of Hertfordshire Collaboration: Imperial College London Work supported
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationQuantum Parameter Estimation: From Experimental Design to Constructive Algorithm
Commun. Theor. Phys. 68 (017 641 646 Vol. 68, No. 5, November 1, 017 Quantum Parameter Estimation: From Experimental Design to Constructive Algorithm Le Yang ( 杨乐, 1, Xi Chen ( 陈希, 1 Ming Zhang ( 张明, 1,
More informationarxiv:quant-ph/ v1 14 Mar 2001
Optimal quantum estimation of the coupling between two bosonic modes G. Mauro D Ariano a Matteo G. A. Paris b Paolo Perinotti c arxiv:quant-ph/0103080v1 14 Mar 001 a Sezione INFN, Universitá di Pavia,
More informationChapter 5. Density matrix formalism
Chapter 5 Density matrix formalism In chap we formulated quantum mechanics for isolated systems. In practice systems interect with their environnement and we need a description that takes this feature
More informationENTANGLEMENT TRANSFORMATION AT ABSORBING AND AMPLIFYING DIELECTRIC FOUR-PORT DEVICES
acta physica slovaca vol. 50 No. 3, 351 358 June 2000 ENTANGLEMENT TRANSFORMATION AT ABSORBING AND AMPLIFYING DIELECTRIC FOUR-PORT DEVICES S. Scheel 1, L. Knöll, T. Opatrný, D.-G.Welsch Theoretisch-Physikalisches
More informationOpen Quantum Systems. Sabrina Maniscalco. Turku Centre for Quantum Physics, University of Turku Centre for Quantum Engineering, Aalto University
Open Quantum Systems Sabrina Maniscalco Turku Centre for Quantum Physics, University of Turku Centre for Quantum Engineering, Aalto University Turku Quantum Technologies GOAL at least 3 useful concepts
More informationMultipartite entanglement in fermionic systems via a geometric
Multipartite entanglement in fermionic systems via a geometric measure Department of Physics University of Pune Pune - 411007 International Workshop on Quantum Information HRI Allahabad February 2012 In
More informationCoherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WE-Heraeus-Seminar Quantum Correlations beyond Entanglement Coherence, Discord, and Entanglement: Activating one resource into another and beyond Gerardo School of Mathematical Sciences The University
More informationHomework assignment 3: due Thursday, 10/26/2017
Homework assignment 3: due Thursday, 10/6/017 Physics 6315: Quantum Mechanics 1, Fall 017 Problem 1 (0 points The spin Hilbert space is defined by three non-commuting observables, S x, S y, S z. These
More informationLecture 19 (Nov. 15, 2017)
Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an
More informationQuantum Entanglement- Fundamental Aspects
Quantum Entanglement- Fundamental Aspects Debasis Sarkar Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata- 700009, India Abstract Entanglement is one of the most useful
More informationQuantization of the Spins
Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.
More information1 Quantum field theory and Green s function
1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More informationd 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)
LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will
More informationarxiv:quant-ph/ v5 10 Feb 2003
Quantum entanglement of identical particles Yu Shi Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom and Theory of
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationMany-Body Fermion Density Matrix: Operator-Based Truncation Scheme
Many-Body Fermion Density Matrix: Operator-Based Truncation Scheme SIEW-ANN CHEONG and C. L. HENLEY, LASSP, Cornell U March 25, 2004 Support: NSF grants DMR-9981744, DMR-0079992 The Big Picture GOAL Ground
More informationQuantum-limited measurements: One physicist's crooked path from quantum optics to quantum information
Quantum-limited measurements: One physicist's crooked path from quantum optics to quantum information II. I. Introduction Squeezed states and optical interferometry III. Ramsey interferometry and cat states
More informationConsider a system of n ODEs. parameter t, periodic with period T, Let Φ t to be the fundamental matrix of this system, satisfying the following:
Consider a system of n ODEs d ψ t = A t ψ t, dt ψ: R M n 1 C where A: R M n n C is a continuous, (n n) matrix-valued function of real parameter t, periodic with period T, t R k Z A t + kt = A t. Let Φ
More information0.5 atoms improve the clock signal of 10,000 atoms
0.5 atoms improve the clock signal of 10,000 atoms I. Kruse 1, J. Peise 1, K. Lange 1, B. Lücke 1, L. Pezzè 2, W. Ertmer 1, L. Santos 3, A. Smerzi 2, C. Klempt 1 1 Institut für Quantenoptik, Leibniz Universität
More informationUniversity of New Mexico
Quantum State Reconstruction via Continuous Measurement Ivan H. Deutsch, Andrew Silberfarb University of New Mexico Poul Jessen, Greg Smith University of Arizona Information Physics Group http://info.phys.unm.edu
More informationQuantum Linear Systems Theory
RMIT 2011 1 Quantum Linear Systems Theory Ian R. Petersen School of Engineering and Information Technology, University of New South Wales @ the Australian Defence Force Academy RMIT 2011 2 Acknowledgments
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More information4. Two-level systems. 4.1 Generalities
4. Two-level systems 4.1 Generalities 4. Rotations and angular momentum 4..1 Classical rotations 4.. QM angular momentum as generator of rotations 4..3 Example of Two-Level System: Neutron Interferometry
More informationQuantum superpositions and correlations in coupled atomic-molecular BECs
Quantum superpositions and correlations in coupled atomic-molecular BECs Karén Kheruntsyan and Peter Drummond Department of Physics, University of Queensland, Brisbane, AUSTRALIA Quantum superpositions
More informationUltimate bounds for quantum and Sub-Rayleigh imaging
Advances in Optical Metrology --- 14 June 2016 Ultimate boun for quantum and Sub-Rayleigh imaging Cosmo Lupo & Stefano Pirandola University of York arxiv:1604.07367 Introduction Quantum imaging for metrology
More informationPath Entanglement. Liat Dovrat. Quantum Optics Seminar
Path Entanglement Liat Dovrat Quantum Optics Seminar March 2008 Lecture Outline Path entangled states. Generation of path entangled states. Characteristics of the entangled state: Super Resolution Beating
More informationApplication of Structural Physical Approximation to Partial Transpose in Teleportation. Satyabrata Adhikari Delhi Technological University (DTU)
Application of Structural Physical Approximation to Partial Transpose in Teleportation Satyabrata Adhikari Delhi Technological University (DTU) Singlet fraction and its usefulness in Teleportation Singlet
More informationLie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains
.. Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains Vladimir M. Stojanović Condensed Matter Theory Group HARVARD UNIVERSITY September 16, 2014 V. M. Stojanović (Harvard)
More informationHong-Ou-Mandel effect with matter waves
Hong-Ou-Mandel effect with matter waves R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, DB, C. I. Westbrook Laboratoire Charles Fabry, Institut d Optique, CNRS, Univ Paris-Sud Progresses in quantum information
More informationQuantum noise studies of ultracold atoms
Quantum noise studies of ultracold atoms Eugene Demler Harvard University Collaborators: Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Mikhail Lukin, Anatoli Polkovnikov Funded by NSF,
More informationValid lower bound for all estimators in quantum parameter estimation
PAPER OPEN ACCESS Valid lower bound for all estimators in quantum parameter estimation To cite this article: Jing Liu and Haidong Yuan 06 New J. Phys. 8 093009 View the article online for updates and enhancements.
More informationCarlton M. Caves University of New Mexico
Quantum metrology: dynamics vs. entanglement I. Introduction II. Ramsey interferometry and cat states III. Quantum and classical resources IV. Quantum information perspective V. Beyond the Heisenberg limit
More informationb) (5 points) Give a simple quantum circuit that transforms the state
C/CS/Phy191 Midterm Quiz Solutions October 0, 009 1 (5 points) Short answer questions: a) (5 points) Let f be a function from n bits to 1 bit You have a quantum circuit U f for computing f If you wish
More informationarxiv: v3 [quant-ph] 17 Nov 2014
REE From EOF Eylee Jung 1 and DaeKil Park 1, 1 Department of Electronic Engineering, Kyungnam University, Changwon 631-701, Korea Department of Physics, Kyungnam University, Changwon 631-701, Korea arxiv:1404.7708v3
More informationInterferometria atomica con un Condensato di Bose-Einstein in un potenziale a doppia buca
Scuola di Scienze Matematiche Fisiche e Naturali Corso di Laurea Specialistica in Scienze Fisiche e Astrofisiche Interferometria atomica con un Condensato di Bose-Einstein in un potenziale a doppia buca
More informationLearning about order from noise
Learning about order from noise Quantum noise studies of ultracold atoms Eugene Demler Harvard University Collaborators: Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Mikhail Lukin, Anatoli
More informationSingle-Particle Interference Can Witness Bipartite Entanglement
Single-Particle Interference Can Witness ipartite Entanglement Torsten Scholak 1 3 Florian Mintert 2 3 Cord. Müller 1 1 2 3 March 13, 2008 Introduction Proposal Quantum Optics Scenario Motivation Definitions
More informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
More informationMESOSCOPIC QUANTUM OPTICS
MESOSCOPIC QUANTUM OPTICS by Yoshihisa Yamamoto Ata Imamoglu A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore Preface xi 1 Basic Concepts
More informationQuantum Mechanics: Fundamentals
Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationQuantum Fisher Information: Theory and Applications
Quantum Fisher Information: Theory and Applications Volkan Erol Okan University Computer Engineering Department, 34959 Istanbul, Turkey volkan.erol@gmail.com Abstract Entanglement is at the heart of quantum
More informationMatrix Product States
Matrix Product States Ian McCulloch University of Queensland Centre for Engineered Quantum Systems 28 August 2017 Hilbert space (Hilbert) space is big. Really big. You just won t believe how vastly, hugely,
More informationBose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation
Bose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation Robert Seiringer IST Austria Mathematical Horizons for Quantum Physics IMS Singapore, September 18, 2013 R. Seiringer Bose Gases,
More informationPHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions
PHYS851 Quantum Mechanics I, Fall 009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular
More information