Quantification of Gaussian quantum steering. Gerardo Adesso

Transcription

1 Quantification of Gaussian quantum steering Gerardo Adesso

2 Outline Quantum steering Continuous variable systems Gaussian entanglement Gaussian steering Applications

3 Steering timeline

4 EPR paradox (1935) 0 q As a consequence of two different measurements performed upon the first system, the second system may be left in states with two different [kinds of] wavefunctions COOL STORY, BRO The theory allows a system to be steered into one or the other type of state at the experimenter s mercy in spite of his having no access to it. [ ] Since I can predict either [q] or [p] without interfering with [the second] system, [it] must know both answers; which is an amazing knowledge.

5 Reid s criterion (1989) 0 q ρ AB Bob can measure q B or p B, obtaining respective outcomes Q B, P B Alice wants to guess those outcomes by measuring her system Alice measures q A with outcome Q A and infers Q est B Q A e.g. by linear inference, Q est B Q A = g q Q A The inferred variance on Bob s outcome given Alice s estimate is 2 Q B = Q B Q est B Q 2 A and it can be minimised by finding Δ inf the optimal g q Alice can equivalently estimate P B by measuring p A Criterion: If the Heisenberg-type condition Δ 2 inf Q B Δ 2 inf P B 1 is violated, then ρ AB is A B steerable

6 Wiseman et al. (2007) For a bipartite state ρ AB, given measurement operators a and b on Alice s and Bob s parties, with outcomes A and B, if for all pairs a and b the joint statistics of the measurement results cannot have arisen from correlations between a random local hidden variable for Alice and a random local hidden variable for Bob, p a, b A, B; ρ AB λ p a A, λ p b B, λ p λ ρ AB is Bell nonlocal have arisen from correlations between a random local hidden variable for Alice and a random pure state measured by Bob, p a, b A, B; ρ AB λ p a A, λ p b B; τ λ B p λ ρ AB is A B steerable have arisen from correlations between a random pure state measured by Alice and a random pure state measured by Bob, p a, b A, B; ρ AB λ p a A; σ A λ p b B; τ B λ p λ ρ AB is entangled

7 Hierarchy of correlations nonlocal steerable entangled discordant classical

8 Continuous variable systems Prototype: a quantised scalar field (e.g. the electromagnetic field), modelled as a collection of harmonic oscillators Each oscillator: a mode of the field, a k, a k = 1 Natural units: ħ = c = 1 a k = 1 2 q k + i p k, a k = 1 2 q k i p k q k = 1 2 a k + a k, p k = 1 i 2 a k a k

9 Continuous variable systems We introduce a vector of canonical operators: R = q 1, p 1, q 2, p 2,, q N, p N Canonical commutation relations: q j, p k = i δ jk q j, q k = 0, p j, p k = 0 Introducing the N-mode symplectic matrix Ω N = Ω N, with Ω = 0 1, then we can write the commutation relations 1 0 compactly: R j, R k = i Ω N jk

10 Phase space description Introduce vectors ξ, κ R 2N of phase-space coordinates Weyl displacement operator Characteristic function Wigner function Properties W is real ( ρ is Hermitian) W can be negative (it is a quasi-probability distribution) W is normalised: dξ R 2N W ρ ξ = 1 ( tr ρ = 1 ) State purity: μ = tr ρ 2 = 2π N dξ R 2N W ρ ξ 2

11 Gaussian states are states whose Wigner distribution is a Gaussian function in phase space W ρ ξ = exp ξ R T σ 1 (ξ R) π N det σ Completely specified by: A vector of means R (first moments): R = R ρ = q 1, p 1,, q N, p N [irrelevant: can be set to 0] A covariance matrix (second moments) σ of elements σ jk = R j R k + R k R j ρ 2 R j ρ R k ρ

12 Gaussian states Very natural: ground and thermal states of all physical systems in the harmonic approximation regime Relevant theoretical testbeds to study the structural properties of quantum correlations, thanks to the symplectic formalism Preferred resources for experimental unconditional implementations of continuous variable protocols Crucial role and remarkable control in quantum optics coherent states squeezed states thermal states

13 Gaussian toolbox Adesso et al, J. Phys. A: Math. Gen. 40, 7821 (2007); Open Syst. Inf. Dyn. 21, (2014)

14 Two-mode Gaussian states σ AB = α γ γ T β σ σ AB is the (4x4) covariance matrix of the two-mode state ρ AB α is the (2x2) reduced covariance matrix of mode A β is the (2x2) reduced covariance matrix γ is the (2x2) block encoding intermodal correlations By local unitary operations (symplectic on the covariance matrix) sf we can transform any σ AB into a standard form σ AB S A S B σ AB S A S B a 0 c 0 = σ AB = 0 a 0 d c 0 0 d b 0 0 b T

15 Example Two-mode squeezed state ( Twin Beam ) Beam Splitter 50:50 EPR correlations: ζ = 1 2 Δ2 q A q B + Δ 2 p A + p B = e 2r Approaches the EPR state for large squeezing r momentumsqueezed (r) positionsqueezed (r) σ AB r = cosh 2r 0 0 cosh (2r) sinh (2r) 0 0 sinh (2r) sinh (2r) 0 0 sinh (2r) cosh (2r) 0 0 cosh (2r)

16 Entanglement (pure states) Entropy of Entanglement : E ρ AB = S ρ A = S ρ B where ρ A(B) is the marginal state of mode A(B), obtained by partial trace over mode B(A) Von Neumann entropy of a Gaussian state: S ρ = N k=1 ν k +1 2 log ν k+1 2 For a two-mode squeezed state: ν k 1 2 log ν k 1 2 E ρ AB = cosh 2 r log cosh 2 r sinh 2 r log sinh 2 r α σ AB r = cosh 2r 0 0 cosh (2r) sinh (2r) 0 0 sinh (2r) sinh (2r) 0 0 sinh (2r) cosh (2r) 0 0 cosh (2r)

17 Entanglement (pure states) Renyi-2 Entropy of Entanglement : E 2 ρ AB = S 2 ρ A = S 2 ρ B where ρ A(B) is the marginal state of mode A(B), obtained by partial trace over mode B(A) Renyi-2 entropy of a Gaussian state: S 2 ρ = 1 log det σ = N log ν 2 k=1 k For a two-mode squeezed state: E 2 ρ AB = log cosh 2r α σ AB r = cosh 2r 0 0 cosh (2r) sinh (2r) 0 0 sinh (2r) sinh (2r) 0 0 sinh (2r) cosh (2r) 0 0 cosh (2r)

18 Entanglement (mixed states) Gaussian convex roof: E ρ AB = inf {p i, ψab i } p i E( ψ AB i ) i where the minimisation is restricted to decompositions into Gaussian states ψ AB (for any chosen E for pure states) i For two-mode Gaussian states with cov. matrix σ AB E σ AB = inf E(ς pure AB ) Examples: pure ς AB σab Gaussian entanglement of formation Gaussian Renyi-2 entanglement

19 Entanglement (mixed states) PPT criterion Transposition = time reversal (Simon 2000) Partial transposition (say w.r.t. B) = flipping the momentum operator in mode B For two-mode Gaussian states with cov. matrix σ AB PPT is necessary and sufficient for separability σ AB = a 0 0 a c 0 0 d c 0 0 d b 0 0 b partial T B σ AB = a 0 0 a c 0 0 d c 0 0 d b 0 0 b

20 Entanglement (mixed states) Bona fide vs PPT criterion Physical PPT Density matrix ρ AB 0 T ρ B AB 0 Covariance matrix Symplectic spectrum σ AB + i Ω A Ω B 0 σ AB + i Ω A Ω B 0 σ AB + i Ω A Ω B 0 σ AB + i Ω A ( Ω B ) 0 ν k 1 k Logarithmic negativity E N σ AB = 0, if ν k 1 k log ν k k: ν k <1 ν k 1 k

21 Gaussian steering Bona fide vs PPT vs steering Density matrix Covariance matrix Symplectic spectrum Physical PPT A B non-steerable ρ AB 0 σ AB + i Ω A Ω B 0 T ρ B AB 0 σ AB + i Ω A Ω B 0 ν k 1 k ν k 1 k??? no simple criterion σ AB + i 0 A Ω B 0 Violation of this condition is necessary and sufficient for steerability of all bipartite Gaussian states by Gaussian measurements (Wiseman et al 2007)

22 Then: Gaussian steering Recall: σ AB = σ AB + i 0 A Ω B 0 α γ γ T β α > 0 (always true) σ B + i Ω B 0 where we defined the Schur complement σ B = β γ T α 1 γ with symplectic spectrum ν k Criterion: the bipartite Gaussian state σ AB is A B steerable by Gaussian measurements if and only if the Schur covariance matrix σ B is not bona fide

23 Gaussian steering: measure Covariance matrix Symplectic spectrum Physical PPT A B non-steerable σ AB + i Ω A Ω B 0 ν k 1 k (k = 1,, n A + n B ) σ AB + i Ω A Ω B 0 ν k 1 k (k = 1,, n A + n B ) σ B + i Ω B 0 ν k 1 k (k = 1,, n B ) Gaussian steerability: 0, if ν k 1 k G A B σ AB = log ν k k: ν k <1

24 Gaussian steering: properties 0, if ν k 1 k G A B σ AB = log ν Computable Convex k: ν k <1 Additive on tensor product states Nonincreasing under local operations on A Equal to entanglement on pure states (measured by Renyi-2 entropy) Smaller than entanglement on mixed states (i.e. respects the hierarchy of correlations!)

25 Gaussian steering: 2 modes Gaussian steerability: σ AB = α γ γ T β G A B σ AB = max {0, log det σ B } = max 0, 1 det α log 2 det σ AB = max 0, S 2 α S 2 (σ AB ) The degree of steerability takes a form of Renyi-2 coherent information This formula is valid for more general (n+1)-mode Gaussian states

26 Classifying 2-mode states Local purities μ A, μ B ; global purity ratio η = μ Aμ B μ μ μ A μ B μ B 2004 μ A

27 Classifying 2-mode states Local purities μ A, μ B ; global purity ratio η = μ Aμ B μ μ A μ B μ μ B 2014 μ A

28 Steering vs entanglement

29 Wiseman vs Reid (2-modes) Wiseman s condition: det σ B 1 Necessary and sufficient for all Gaussian states Specific to Gaussian measurements Reid s criterion: Δ 2 inf Q B Δ 2 inf P B 1 Does not detect all Gaussian states Can be applied to arbitrary states and measurements

30 Wiseman vs Reid (2-modes) Wiseman s condition: det σ B 1 Reid s criterion: Δ 2 inf Q B Δ 2 inf P B 1 We have proven: min Δ 2 inf Q B Δ 2 inf P B = det σ B {rotated quadratures} Our measure thus also quantifies the degree of violation of an optimised Reid s criterion for the EPR paradox!

31 Applications: secure quantum cryptography One-sided device independent quantum key distribution Alice and Bob share a two-mode Gaussian state (with covariance matrix in standard form) They adopt a reverse reconciliation protocol Secure key rate K K max 0, G A B σ AB + log 2 1

32 Applications: Peres conjecture Peres: PPT states cannot be nonlocal (1999) Pusey: PPT states cannot even be steerable (2013) Both conjectures are now disproven (Vertesi & Brunner Nat. Commun. 2014) But they hold in the Gaussian domain! Physical PPT A B non-steerable Covariance matrix σ AB + i Ω A Ω B 0 σ AB + i Ω A Ω B 0 σ AB + i 0 A Ω B 0 NO-GO: Steering bound entangled Gaussian states by Gaussian measurements is impossible

33 Conclusions We discussed the concept of EPR steering, and how to deal with it in continuous variable systems We proposed the first measure of steering for Gaussian states It is computable, respects the hierarchy of correlations, and has notable properties and applications We proved Peres conjecture for Gaussian states and measurements

34 Future directions Multipartite steering and monogamy (He & Reid PRL 2013; Reid PRA 2013; Nature Phys. 2014) Steering in non-gaussian states (our Gaussian steerability gives a lower bound) Applications to subchannel discrimination in the Gaussian setting Role of steering in existing or novel quantum communication protocols

35 Thank you Quantification of Gaussian quantum steering Ioannis Kogias, Antony R. Lee, Sammy Ragy, and Gerardo Adesso arxiv: EPR-Steering measure for two-mode continuous variable states Ioannis Kogias and Gerardo Adesso arxiv: