586. WE-Heraeus-Seminar “Quantum Correlations beyond Entanglement”
Coherence, Discord, and Entanglement:
Activating one resource into another and beyond
Gerardo Adesso
School of Mathematical Sciences
The University of Nottingham
United Kingdom
Our research
WHERE
IS
Classical
Quantum
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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Beyond entanglement
Building on more
than two decades
of knowledge in the
theory of quantum
entanglement
THIS TALK
Quantum
correlations
To characterise
coherence and
general quantum
correlations beyond
entanglement
Quantum
coherence
nonlocal
steerable
entangled
discordant
classical
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Entanglement 𝑬
Discord 𝑫
Coherence 𝑪
bipartition 𝐴|𝐵
with respect to a fixed
reference basis 𝑖
The main players
with respect to a directional
with respect to an arbitrary
bipartition 𝐴: 𝐵
(D1) 𝐷𝐴|𝐵 𝜒𝐴𝐵 = 0 for
quantum-classical states
𝜒𝐴𝐵 ∈ 𝒞: 𝜒𝐴𝐵 =
𝑘
𝑘 𝑝𝑘 𝜌𝐴 ⊗ 𝑘 𝑘 𝐵
(C1) 𝐶 𝜏𝐴 = 0 for
incoherent states 𝜏𝐴 ∈ ℐ:
𝜏𝐴 = 𝑖 𝑝𝑖 𝑖 𝑖 𝐴
(E2) 𝐸𝐴:𝐵 is nonincreasing
under LOCC on 𝐴 and 𝐵,
i.e. acting as Λ𝐿𝑂𝐶𝐶 𝜌 =
†
𝐿
𝜌𝐿
𝑗 𝑗 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 = 𝑗 𝐾𝑗 𝜌𝐾𝑗† , with
𝐾𝑗 ℐ𝐾𝑗† ⊆ ℐ
(E2b) 𝐸𝐴:𝐵 is
nonincreasing on average
under selective LOCC, i.e.
𝐸𝐴:𝐵 𝜌 ≥ 𝑗 𝑝𝑗 𝐸𝐴:𝐵 𝜌𝑗
with 𝜌𝑗 = 𝐿𝑗 𝜌𝐿𝑗† /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 ≥ 𝑗 𝑝𝑗 𝐶 𝜌𝑗
with 𝜌𝑗 = 𝐾𝑗 𝜌𝐾𝑗† /𝑝𝑗
(E1) 𝐸𝐴:𝐵 𝜎𝐴𝐵 = 0 for
separable states 𝜎𝐴𝐵 ∈ 𝒮:
𝜎𝐴𝐵 = 𝑘 𝑝𝑘 𝜌𝐴𝑘 ⊗ 𝜈𝐵𝑘
[Baumgratz et al PRL 2014]
(D3) 𝐷𝐴|𝐵 reduces to
(E3) 𝐸𝐴:𝐵 is convex
(C3) 𝐶 is convex
(optional)
𝐸𝐴:𝐵 Activating
on pure
(optional)
Adesso
Coherence, Discord, andsome
Entanglement:
one states
resource into another
and beyond
586. WEH
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|.
output
ancilla
• Non-classicality of a light beam is equivalent
to entanglement produced across a beam splitter
with an ancillary mode initially in the vacuum
input system
mode (state 𝜌)
Asboth et al PRL 2005; see also N. Killoran’s talk
• Entanglement is created if and only if
the input mode is non-classical
output
system
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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Discord & Entanglement
Let us model a local measurement (Von Neumann)
S
A
M
B
system
apparatus
𝜌A:B
B
0M
Adesso
M
UB
C-NOT
A
𝜌AB:M
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Discord & Entanglement
Theorem. The output premeasurement state
𝜌𝑨𝑩: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
B
0M
Adesso
M
UB
C-NOT
A
𝜌A:B
Piani et al. PRL 2011
Streltsov et al. PRL 2011
Piani & Adesso PRA(R) 2012
𝜌AB:M
Entanglement
with the
apparatus
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Discord & Entanglement
The connection can be made quantitative
Piani et al. PRL 2011
Streltsov et al. PRL 2011
Piani & Adesso PRA(R) 2012
Ent-based Discord (𝜌) : 𝐷𝐸 𝜌𝐴𝐵 = inf 𝐸 𝜌𝐴𝐵:𝑀
{𝑈𝐵 }
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
A
M
Adesso
C-NOT
B
Discord
Entanglement
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Examples
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)
…
…
Piani et al. PRL 2011
Piani & Adesso PRA(R) 2012
A
B
M
C-NOT
Entanglement E
The correspondence is hierarchical
𝐷𝐸 𝜌𝐴𝐵 ≥ 𝐸 𝜌𝐴𝐵 for every monotone 𝐸
“quantum correlations go beyond entanglement”
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Experimental activation
Adesso, D’Ambrosio, Nagali, Piani, Sciarrino PRL 2014
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Experimental activation
Adesso 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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Demonstration of the ‘iff’
Adesso et al. PRL 2014
Entanglement (negativity) between AB and M in the pre-measurement state
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 1: Quantitative activation
𝜌
out
𝜌
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
in
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.
Adesso et al. PRL 2014
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
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
𝜌𝑺 has nonzero coherence
Quantum
coherence
Entanglement
creation
Streltsov, Singh, Dhar, Bera, Adesso arXiv 2015
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 2: Coherence & Entanglement
The connection can be made quantitative
Streltsov et al. arXiv 2015
Entanglement-based Coherence (𝜌𝑆 ) :
𝐶𝐸 𝜌𝑆 = lim sup 𝐸𝐴:𝐵 Λ𝑆𝐴
ℐ 𝜌𝑆 ⊗ 0 0
dim𝐴 →∞
Λ𝑆𝐴
ℐ
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
Coherence
The correspondence is faithful
𝐶𝐸 is a (convex) coherence monotone for
every (convex) entanglement monotone 𝐸
Adesso
Entanglement
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Link 2: Examples
Streltsov et al. arXiv 2015
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)
…
…
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Summary
WHERE
IS
ENTANGLEMENT
Classical
Quantum
DISCORD
THE
BORDER?
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
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
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
Relative entropy of
quantumness
[Baumgratz et al PRL 2014]
[Modi et al PRL 2010]
L1-norm of coherence
Negativity of
quantumness
[Baumgratz et al PRL 2014]
[Piani et al PRL 2011]
Wigner-Yanase skew
information
Local quantum
uncertainty
[Girolami PRL 2014]
[Girolami et al PRL 2013]
…
…
Adesso
This is why discord-type
correlations guarantee
a metrological precision
in interferometry for
any possible generator
Girolami et al PRL 2014
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
References
WHERE
IS
ENTANGLEMENT
Classical
Quantum
DISCORD
THE
BORDER?
COHERENCE
• Link 1: D vs E
• Link 2: C vs E
• M. Piani et al., Phys. Rev. Lett.
• T. Baumgratz et al., Phys. Rev.
106, 220403 (2011)
Lett. 113, 140401 (2014)
• A. Streltsov et al., Phys. Rev.
• A. Streltsov et al.,
Lett. 106, 160401 (2011)
arXiv:1502.05876 (2015)
• M. Piani and G. Adesso, Phys. Rev. • Link 3: D vs C
A 85, 040301(R) (2012)
• D. Girolami et al., Phys. Rev.
• G. Adesso et al., Phys. Rev. Lett.
Lett. 112, 210401 (2014)
112, 140501 (2014)
• T. Bromley et al.,
arXiv:1412.7161 (2014)
Adesso
Coherence, Discord, and Entanglement: Activating one resource into another and beyond
586. WEH
Thank you
586. WE-Heraeus-Seminar “Quantum Correlations beyond Entanglement”
Coherence, Discord, and Entanglement:
Activating one resource into another and beyond
Gerardo Adesso
School of Mathematical Sciences
The University of Nottingham
United Kingdom