THE INTERFEROMETRIC POWER OF QUANTUM STATES GERARDO ADESSO

Transcription

1 THE INTERFEROMETRIC POWER OF QUANTUM STATES GERARDO ADESSO

2 IDENTIFYING AND EXPLORING THE QUANTUM-CLASSICAL BORDER Quantum Classical FOCUSING ON CORRELATIONS AMONG COMPOSITE SYSTEMS

3 OUTLINE Quantum correlations Quantum metrology Interferometric power Qubit states Gaussian states Summary

4 CORRELATIONS Classical correlations A B Quantum correlations

5 CORRELATIONS A B Pure bipartite states: entanglement = nonlocality nonclassicality (one kind of quantum correlations) Mixed bipartite states: A hierarchy of different quantum correlations

6 MANY SHADES OF QUANTUMNESS Partially device-independent quantum information processing Black-box quantum metrology nonlocal steerable entangled discordant classical Fully device-independent quantum information processing Teleportation, dense coding,

7 (TWO) SHADES OF QUANTUMNESS entangled discordant (with respect to subsystem A) ρ AB p k τ A k ν B k k the state is not separable i.e. it cannot be created by LOCC ρ AB p k k k A ν B k k the state is not classical-quantum i.e. it is not invariant under any local measurement on party A

8 DISCORDANT VS CLASSICAL Ollivier, Zurek, PRL 2001; Henderson, Vedral JPA 2001; Horodecki et al PRA 2005; Groisman et al. arxiv 2007; Piani et al. PRL 2008; Piani et al. PRL 2011; Girolami et al. PRL 2013; 2014; review: Modi et al. RMP 2012 etc. If there is at least one local measurement I can perform without affecting my state A classical-quantum (or classically correlated) B otherwise A with quantum discord B

9 QUANTUM METROLOGY exploits quantum mechanical features to improve the available precision in estimating physical parameters See e.g. S. Huelga et al. Phys. Rev. Lett. 1997; B. Escher, R. de Matos Filho, L. Davidovich, Nature Phys For a review, see V. Giovannetti, S. Lloyd, L. Maccone, Nature Photon. 2011

10 PHASE ESTIMATION TASK Given the generator H A Find the phase φ A C D ρ AB E φ B

11 ν times PHASE ESTIMATION Quantum Cramer-Rao Bound Var φ 1/[ν F ρ AB ; H A ] Quantum Fisher Information F ρ AB ; H A measures the precision What is the resource? coherence in the eigenbasis of H A W. K. Wootters, Phys. Rev. D 1981; S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 1994

12 BLACK BOX QUANTUM METROLOGY let the phase generator be unknown a priori

13 BLACK BOX PHASE ESTIMATION TASK Only the spectrum of H A is initially known Find the phase φ A C D ρ AB E φ B

14 BLACK BOX PHASE ESTIMATION TASK The generator H A is revealed after ρ AB is prepared Find the phase φ A C D ρ AB E φ B

15 WHICH PASSE-PARTOUT INPUT PROBE STATES GUARANTEE A PASS?

16 WORST-CASE SCENARIO FIGURE OF MERIT How useful the probe state ρ AB is for estimation Guaranteed precision for any possible phase generator ρ AB P ρ AB = 1 4 inf H A F(ρ AB ; H A ) INTERFEROMETRIC POWER

17 QUANTUM DISCORD IS THE RESOURCE FIGURE OF MERIT The task requires coherence in the eigenbases of all H A s The interferometric power is a measure of discord P ρ AB = 1 4 inf H A F(ρ AB ; H A ) ρ AB P is invariant under local unitaries and nonincreasing under local operations on B It vanishes iff ρ is classically correlated, ρ AB = i p i i i A τ i B It reduces to an entanglement monotone for pure states It is analytically computable if A is a qubit

18 Fidelity 2 CLASSES OF STATES classically correlated K. Modi et al. Phys. Rev. X 2011 CBPF, RJ with Nuclear Magnetic Resonance with quantum discord Cl A 1 H B C C AB Q AB p Cl Cl

19 3 DIRECTIONS 1. σ z A CBPF, RJ with Nuclear Magnetic Resonance 2. (σ x A + σ y A )/ 2 Cl A 1 H B 13 C 3. σ x A Cl Cl

20 SCHEME Alice H Charlie -y -y G z ρ AB G z -y H φ -y Classical probesbob H -y -y G z G z -y -y Discordant probes

21 RESULTS Discordant probes Classical probes x Interferometric power Direction 1 Direction 2 Direction 3 (worst case)

22 DISCORD-TYPE QUANTUM CORRELATIONS = MINIMUM LOCAL COHERENCE = GUARANTEED PRECISION

23 OPTICAL INTERFEROMETRY GEO600, LIGO Collaboration

24 BLACK BOX OPTICAL INTERFEROMETRY TASK Unknown generator (with fixed harmonic spectrum), P A φ = exp i φ na Find the phase φ

25 BLACK BOX OPTICAL INTERFEROMETRY FOCUS Gaussian probes ρ AB Gaussian unitaries V A 0

26 GAUSSIAN STATES are states whose Wigner distribution is a Gaussian function in phase space q p 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 A covariance matrix (second moments) σ of elements σ jk σ jk = R j R k + R k R j ρ 2 R j ρ R k ρ

27 GAUSSIAN STATES Very natural: ground and thermal states of all physical systems in the harmonic approximation regime Relevant theoretical testbeds for the study of structural properties of entanglement and 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

28 GAUSSIAN INTERFEROMETRIC POWER Defined as for discrete systems, but with optimization restricted to Gaussian generators with harmonic spectrum Same properties: it is a measure of discord-type correlations Computable in closed form for two-mode Gaussian states

29 METROLOGICAL SCALING Gaussian interferometric power versus mean number of photons n A in the probe arm A SEPARABLE ENTANGLED

30 GAUSSIAN IP VS ENTANGLEMENT Gaussian interferometric power normalized by n A versus entanglement (log. negativity)

31 SUMMARY The most general forms of quantum correlations ( discord ) manifest as coherence in all local bases They can be measured via a faithful, operational and computable quantifier, named interferometric power Quantum correlations even beyond entanglement guarantee a metrological precision in phase estimation, as opposed to classically correlated probes Phys.Org May 2014 New Scientist Sept 2014 Nature Physics July 2014

32 APPENDIX: CLOSED FORMULAE Qubit-qudit state ρ AB (2 x d)-dim Gaussian state with covariance matrix σ AB (two-mode) σ AB = α γ γ T β Phys. Rev. Lett. 112, (2014) Phys. Rev. A 90, (2014)

33 THANK YOU D. Girolami, T. Tufarelli, G. Adesso; Characterizing Nonclassical Correlations via Local Quantum Uncertainty; Phys. Rev. Lett. 110, (2013) D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, G. Adesso; Quantum Discord Determines the Interferometric Power of Quantum States; Phys. Rev. Lett. 112, (2014) G. Adesso; Gaussian interferometric power; Phys. Rev. A 90, (2014)