Coherence, 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 of Nottingham United Kingdom

Our research WHERE IS Quantum Classical THE BORDER? Identifying quantumness by its most essential and genuine signatures in general composite systems Providing novel operational interpretations and satisfactory measures for quantum resources

Beyond entanglement Building on more than two decades of knowledge in the theory of quantum entanglement THIS TALK To characterise coherence and general quantum correlations beyond entanglement nonlocal steerable entangled discordant classical Quantum correlations Quantum coherence

The main players What is entanglement? The fact that a state cannot be created by local operations and classical communication What is discord? The fact that a state is sensitive to any local measurement or dynamics on one subsystem What is coherence? The fact that a state is in a superposition of a fixed set of states forming a reference basis

Entanglement E Discord D Coherence C The main players with respect to an arbitrary bipartition A: B (E1) E A:B σ AB = 0 for separable states σ AB S: σ AB = p k ρ k k A ν B k (E2) E A:B is nonincreasing under LOCC on A and B, i.e. acting as Λ LOCC ρ = j L j ρl j, with L j being the Kraus operators (their properties omitted here) (E2b) E A:B is nonincreasing on average under selective LOCC, i.e. E A:B ρ j p j E A:B ρ j with ρ j = L j ρl j /p j (E3) E A:B is convex (optional) with respect to a directional bipartition A B (D1) D A B χ AB = 0 for quantum-classical states χ AB C: χ AB = p k ρ k A k k k B (D2) D A B is nonincreasing under any quantum channel on A (D2b) D A B is nonincreasing under commutativity-preserving operations Λ COP on B, i.e. Λ COP σ B, Λ COP τ B = 0 if σ B, τ B = 0 (D3) D A B reduces to some E A:B on pure states with respect to a fixed reference basis i (C1) C τ A = 0 for incoherent states τ A I: τ A = i p i i i A [Baumgratz et al PRL 2014] (C2) C is nonincreasing under incoherent operations i.e. acting as Λ I ρ = j K j ρk j, with K j IK j I (C2b) C is nonincreasing on average under selective incoherent operations, i.e. C ρ j p j C ρ j with ρ j = K j ρk j /p j (C3) C is convex (optional)

Connections between resources Inspiration: non-classicality in quantum optics: the state ρ of a bosonic light mode is classical if and only if it is a mixture of Glauber coherent states, i.e., it has a positive and regular Glauber-Sudarshan P representation, ρ = d 2 α P α α α. Non-classicality of a light beam is equivalent to entanglement produced across a beam splitter with an ancillary mode initially in the vacuum output ancilla Asboth et al PRL 2005; see also N. Killoran s talk Entanglement is created if and only if the input mode is non-classical input system mode (state ρ) BEAM SPLITTER 50:50 input ancillary mode (vacuum) The non-classicality is an entanglement potential : the created entanglement can be used to quantify non-classicality output system

C-NOT Link 1: Discord & Entanglement Let us model a local measurement (Von Neumann) S M A B system apparatus ρ A:B A B U B ρ AB:M 0 M M

C-NOT Link 1: Discord & Entanglement Theorem. The output premeasurement state ρ AB:M is entangled for all choices of the U B IF AND ONLY IF the initial system state ρ AB has nonzero discord (with respect to B) Piani et al. PRL 2011 Streltsov et al. PRL 2011 Piani & PRA(R) 2012 ρ A:B A B U B ρ AB:M Measurement sensitivity 0 M M Entanglement with the apparatus

C-NOT Link 1: Discord & Entanglement The connection can be made quantitative Piani et al. PRL 2011 Streltsov et al. PRL 2011 Piani & PRA(R) 2012 Ent-based Discord (ρ) : D E ρ AB = inf {U B } E ρ AB:M The minimum entanglement E between the system AB and the apparatus M generated during a local (pre)measurement on subsystem B quantifies the initial discord D E in the system A Discord B M Entanglement

C-NOT Link 1: Examples Entanglement E Relative entropy of entanglement, distillable entanglement Negativity Discord D E -Relative entropy of quantumness (already defined within a geometric approach) -Negativity of quantumness (equal to the trace-distance discord if B is a qubit) Piani et al. PRL 2011 Piani & PRA(R) 2012 A B M The correspondence is hierarchical D E ρ AB E ρ AB for every monotone E quantum correlations go beyond entanglement

Link 1: Experimental activation, D Ambrosio, Nagali, Piani, Sciarrino PRL 2014

Link 1: Experimental activation et al. PRL 2014 Input states of the system A: polarization of photon 1 B: polarization of photon 2 M: path of photon 2 q = 0: the states are quantum-classical q > 0: the states are discordant (any measure of discord increases with q) q > 1/2: the states are also entangled

Link 1: Demonstration of the iff et al. PRL 2014 Entanglement (negativity) between AB and M in the pre-measurement state

Link 1: Quantitative activation ρ ρ in out The minimum negativity in the output premeasurement state, minimized over the U B s, is verified to coincide with the trace-distance discord in the input state, calculated numerically by a priori tomography before the activation step The output state of A,B,M, is found to possess genuine tripartite entanglement whenever the input state is not only discordant, but entangled as well. This is revealed by witness operators. et al. PRL 2014

Link 2: Coherence & Entanglement Bipartite coherence: a state is incoherent if it is diagonal in a local product basis Incoherent operations map the incoherent set into itself (note: CNOT is incoherent) Streltsov, Singh, Dhar, Bera, arxiv 2015

Link 2: Coherence & Entanglement Bipartite coherence: a state is incoherent if it is diagonal in a local product basis Incoherent operations map the incoherent set into itself (note: CNOT is incoherent) Theorem. Entanglement can be created by incoherent operations on a system S and an incoherent ancilla A IF AND ONLY IF the initial system state ρ S has nonzero coherence Quantum coherence Entanglement creation Streltsov, Singh, Dhar, Bera, arxiv 2015

Link 2: Coherence & Entanglement The connection can be made quantitative Streltsov et al. arxiv 2015 Entanglement-based Coherence (ρ S ) : C E ρ S = lim sup E A:B Λ SA dim A I ρ S 0 0 A Λ SA I The maximum entanglement E between the system S and the initially incoherent ancilla A created by incoherent operations quantifies the initial coherence C E in the system state ρ S The correspondence is faithful C E is a (convex) coherence monotone for every (convex) entanglement monotone E Coherence Entanglement

Link 2: Examples Streltsov et al. arxiv 2015 Entanglement E Relative entropy of entanglement, distillable entanglement Geometric measure of entanglement (1-Fidelity) Coherence C E -Relative entropy of coherence (already defined within a geometric approach) -Geometric measure of coherence (now proven to be a full coherence monotone, and computable for an arbitrary single-qubit state)

Summary DISCORD Quantum WHERE IS ENTANGLEMENT THE BORDER? Classical COHERENCE Discord can be quantified by the minimum entanglement created with an apparatus during a premeasurement Coherence can be quantified by the maximum entanglement created by incoherent operations

Link 3: Discord & Coherence See talk by Tom Bromley Discord D A B can be interpreted and quantified in terms of bipartite coherence C minimized over all local bases for subsystem B Coherence C Relative entropy of coherence [Baumgratz et al PRL 2014] L1-norm of coherence [Baumgratz et al PRL 2014] Wigner-Yanase skew information [Girolami PRL 2014] Discord D Relative entropy of quantumness [Modi et al PRL 2010] Negativity of quantumness [Piani et al PRL 2011] Local quantum uncertainty [Girolami et al PRL 2013] This is why discord-type correlations guarantee a metrological precision in interferometry for any possible generator Girolami et al PRL 2014

References WHERE IS ENTANGLEMENT Quantum Classical DISCORD Link 1: D vs E M. Piani et al., Phys. Rev. Lett. 106, 220403 (2011) A. Streltsov et al., Phys. Rev. Lett. 106, 160401 (2011) M. Piani and G., Phys. Rev. A 85, 040301(R) (2012) G. et al., Phys. Rev. Lett. 112, 140501 (2014) THE BORDER? COHERENCE Link 2: C vs E T. Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014) A. Streltsov et al., arxiv:1502.05876 (2015) Link 3: D vs C D. Girolami et al., Phys. Rev. Lett. 112, 210401 (2014) T. Bromley et al., arxiv:1412.7161 (2014)

Thank you 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 of Nottingham United Kingdom