coherence, discord, and entanglement - quantum correlations...

Coherence, Discord, and Entanglement: Activating one resource into another and beyond

Gerardo Adesso

School of Mathematical Sciences

The University of Nottingham

United Kingdom

586. WE-Heraeus-Seminar “Quantum Correlations beyond Entanglement”

Quantum Classical

Identifying quantumness by its most essential and genuine signatures in general composite systems Providing novel operational interpretations and satisfactory measures for quantum resources

Our research

586. WEH Coherence, Discord, and Entanglement: Activating one resource into another and beyond Adesso

WHERE IS

THE BORDER?

Building on more than two decades

of knowledge in the theory of quantum

entanglement

Beyond entanglement

To characterise coherence and

general quantum correlations beyond

entanglement


classical

discordant

entangled

steerable

nonlocal

Quantum correlations

Quantum coherence

THIS TALK

The main players

• The fact that a state cannot be created by local operations and classical communication

What is entanglement?

• The fact that a state is sensitive to any local measurement or dynamics on one subsystem

What is discord?

• The fact that a state is in a superposition of a fixed set of states forming a reference basis

What is coherence?


The main players


Entanglement 𝑬 Discord 𝑫 Coherence 𝑪 with respect to an arbitrary bipartition 𝐴: 𝐵

with respect to a directional bipartition 𝐴|𝐵

with respect to a fixed reference basis 𝑖

(E1) 𝐸𝐴:𝐵 𝜎𝐴𝐵 = 0 for separable states 𝜎𝐴𝐵 ∈ 𝒮: 𝜎𝐴𝐵 = 𝑝𝑘𝜌𝐴

𝑘 ⊗ 𝜈𝐵𝑘

𝑘

(D1) 𝐷𝐴|𝐵 𝜒𝐴𝐵 = 0 for quantum-classical states 𝜒𝐴𝐵 ∈ 𝒞: 𝜒𝐴𝐵 = 𝑝𝑘𝜌𝐴

𝑘 ⊗ 𝑘 𝑘 𝐵𝑘

(C1) 𝐶 𝜏𝐴 = 0 for incoherent states 𝜏𝐴 ∈ ℐ: 𝜏𝐴 = 𝑝𝑖 𝑖 𝑖 𝐴𝑖 [Baumgratz et al PRL 2014]

(E2) 𝐸𝐴:𝐵 is nonincreasing under LOCC on 𝐴 and 𝐵, i.e. acting as Λ𝐿𝑂𝐶𝐶 𝜌 = 𝐿𝑗𝜌𝐿𝑗

†𝑗 , with 𝐿𝑗 being

the Kraus operators (their properties omitted here)

(D2) 𝐷𝐴|𝐵 is nonincreasing under any quantum channel on 𝐴

(C2) 𝐶 is nonincreasing under incoherent operations i.e. acting as Λℐ 𝜌 = 𝐾𝑗𝜌𝐾𝑗

†𝑗 , with

𝐾𝑗ℐ𝐾𝑗† ⊆ ℐ

(E2b) 𝐸𝐴:𝐵 is nonincreasing on average under selective LOCC, i.e. 𝐸𝐴:𝐵 𝜌 ≥ 𝑝𝑗𝐸𝐴:𝐵 𝜌𝑗𝑗 with 𝜌𝑗 = 𝐿𝑗𝜌𝐿𝑗

†/𝑝𝑗

(D2b) 𝐷𝐴|𝐵 is nonincreasing under commutativity-preserving operations Λ𝐶𝑂𝑃 on 𝐵, i.e. Λ𝐶𝑂𝑃 𝜎𝐵 , Λ𝐶𝑂𝑃 𝜏𝐵 =0 if 𝜎𝐵, 𝜏𝐵 = 0

(C2b) 𝐶 is nonincreasing on average under selective incoherent operations, i.e. 𝐶 𝜌 ≥ 𝑝𝑗𝐶 𝜌𝑗𝑗 with 𝜌𝑗 = 𝐾𝑗𝜌𝐾𝑗

†/𝑝𝑗

(E3) 𝐸𝐴:𝐵 is convex (optional)

(D3) 𝐷𝐴|𝐵 reduces to some 𝐸𝐴:𝐵 on pure states

(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 = ∫ 𝑑2𝛼 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 • Entanglement is created if and only if the input mode is non-classical • The non-classicality is an “entanglement potential”:

the created entanglement can be used to quantify non-classicality

input ancillary mode (vacuum)

input system mode (state 𝜌)

output system

output ancilla

BEAM SPLITTER

50:50

Asboth et al PRL 2005; see also N. Killoran’s talk

Let us model a local measurement (Von Neumann)

system apparatus

S M

A B

Link 1: Discord & Entanglement


0 M

𝜌A:B UB

C-N

OT A

B

M

𝜌 AB:M

Theorem. The output premeasurement state 𝜌 𝑨𝑩:𝑴 is entangled for all choices of the 𝑈𝐵 IF AND ONLY IF the initial system state 𝜌𝑨𝑩

has nonzero discord (with respect to B)

Measurement sensitivity

Entanglement with the

apparatus


Link 1: Discord & Entanglement Piani et al. PRL 2011

Streltsov et al. PRL 2011 Piani & Adesso PRA(R) 2012

0 M

𝜌A:B UB

C-N

OT A

B

M

𝜌 AB:M

The connection can be made quantitative

Ent-based Discord (𝜌) : 𝐷𝐸 𝜌𝐴𝐵 = inf{𝑈𝐵}

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 𝐷𝐸 in the system

Piani et al. PRL 2011 Streltsov et al. PRL 2011

Piani & Adesso PRA(R) 2012

Discord

Entanglement

Link 1: Discord & Entanglement


C-N

OT

A

B

M

The correspondence is hierarchical

𝐷𝐸 𝜌𝐴𝐵 ≥ 𝐸 𝜌𝐴𝐵 for every monotone 𝐸

Entanglement E Discord DE

Relative entropy of entanglement, distillable entanglement

-Relative entropy of quantumness (already defined within a geometric approach)

Negativity -Negativity of quantumness (equal to the trace-distance discord if B is a qubit)

… …

“quantum correlations go beyond entanglement”

Piani et al. PRL 2011 Piani & Adesso PRA(R) 2012

Link 1: Examples


C-N

OT

A

B

M

Adesso, D’Ambrosio, Nagali, Piani, Sciarrino PRL 2014

Link 1: Experimental activation


Input states of the system

• 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

A: polarization of photon 1 B: polarization of photon 2 M: path of photon 2

Adesso et al. PRL 2014

Link 1: Experimental activation


Entanglement (negativity) between AB and M in the pre-measurement state


Link 1: Demonstration of the ‘iff’


The minimum negativity in the output premeasurement state, minimized over the UB’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.

𝜌

𝜌 in

out


Link 1: Quantitative activation


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, Adesso arXiv 2015



• 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

𝜌𝑺 has nonzero coherence

Quantum coherence

Entanglement creation

Streltsov, Singh, Dhar, Bera, Adesso arXiv 2015

The connection can be made quantitative

Entanglement-based Coherence (𝜌𝑆) :

𝐶𝐸 𝜌𝑆 = limdim𝐴→∞

supΛℐ

𝑆𝐴𝐸𝐴:𝐵 Λℐ

𝑆𝐴 𝜌𝑆 ⊗ 0 0 𝐴

The maximum entanglement 𝐸 between the system 𝑆 and the initially incoherent ancilla 𝐴 created by incoherent operations quantifies the initial coherence 𝐶𝐸 in the system state 𝜌𝑆

Streltsov et al. arXiv 2015

Coherence

Entanglement



The correspondence is faithful

𝐶𝐸 is a (convex) coherence monotone for every (convex) entanglement monotone 𝐸

Entanglement E Coherence CE

Relative entropy of entanglement, distillable entanglement

-Relative entropy of coherence (already defined within a geometric approach)

Geometric measure of entanglement (1-Fidelity)

-Geometric measure of coherence (now proven to be a full coherence monotone, and computable for an arbitrary single-qubit state)

… …

Streltsov et al. arXiv 2015


Link 2: Examples

Quantum Classical

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

Summary


WHERE IS

THE BORDER?

ENTANGLEMENT

DISCORD COHERENCE

Link 3: Discord & Coherence


See talk by Tom Bromley

Discord 𝐷𝐴|𝐵 can be interpreted and quantified in terms of bipartite

coherence 𝐶 minimized over all local bases for subsystem 𝐵

Coherence C Discord D

Relative entropy of coherence [Baumgratz et al PRL 2014]

Relative entropy of quantumness [Modi et al PRL 2010]

L1-norm of coherence [Baumgratz et al PRL 2014]

Negativity of quantumness [Piani et al PRL 2011]

Wigner-Yanase skew information [Girolami PRL 2014]

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

Quantum Classical

References


WHERE

IS

THE

BORDER?

ENTANGLEMENT

DISCORD COHERENCE

• 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. Adesso, Phys. Rev.

A 85, 040301(R) (2012) • G. Adesso et al., Phys. Rev. Lett.

112, 140501 (2014)

• 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)

Coherence, Discord, and Entanglement: Activating one resource into another and beyond

Gerardo Adesso

School of Mathematical Sciences

The University of Nottingham

United Kingdom

Thank you

586. WE-Heraeus-Seminar “Quantum Correlations beyond Entanglement”

coherence, discord, and entanglement - quantum correlations...

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