Coherence and non-classicality in the multi-time statistics of a quantum Markovian process - Andrea Smirne - INFN Genova
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Three days in Quantum Mechanics
Coherence and non-classicality
in the multi-time statistics of a
quantum Markovian process
Andrea Smirne
Institute for Theoretical Physics
Genova, 7 June 2018
Outline • Introduction and motivation Is the presence of coherences in the quantum description of a physical phenomenon a synonym for its non-classicality? • General framework and main result Multi-time statistics due to sequential measurements on an open quantum system and precise link between quantum coherence and non-classicality • Non-Markovian multi-time statistics Simple examples showing how beyond the Markovian case there is no simple link between coherence and non-classicality • Outlook
Outline • Introduction and motivation • • •
Motivation: the role of quantum coherence
Quantum biology
Quantum thermodynamics
Quantum coherence plays a key role
in several physical phenomena: is this
a synonym for their non-classicality?
Resource theory of coherence
Noise assisted quantum(?) transport
L
X ✓ ◆
1
L[⇢] = mn Lmn ⇢L†mn L†mn Lmn , ⇢
mn
2
Lmn = |mihn| several sources of (white) noise
Plenio & Huelga, New J Phys 10, 113019 (2008)
Efficiency The interplay between the
Hamiltonian and the dissipative part
of the dynamics guarantees high
efficiency: noise assisted transport
Classical benchmark
12
Pauli (Förster) master equation:
2 72 classical hopping model
3
X
1 Ṗn (t) = (
53 4 mn Pn (t) nm Pm (t))
7 n
6
5
1.0
Full Lindblad equation
0.8
0.6 Pauli equation Pn (t) = hn|⇢(t)|ni
P1HtL
0.4 Data
⇢mn (t) = 0 for m 6= n
0.2
0.0 in the full equation
t @a.u.D
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Quantum coherences are needed in the quantum description, BUT…
does it mean that this is a non-classical phenomenon??
Quantum coherence non-classicality
The fact that QM predicts an event via
Affirmation
quantum coherence does not imply
that there is no classical description of the consequent
Classical
Quantum
• What is non-classical in quantum thermal machines ?
• Quantifying coherence as a resource vs
quantifying non-classicality
How can we certify or irrevocably exclude the existence of an alternative
classical description? Is there any link with quantum coherence?
Sequential measurements of one observable
• Leggett-Garg inequalities
Q dichotomic observable
• Leggett-Garg type inequalities
S. Huelga, T. Marshall, and E. Santos,
Phys. Rev. A 52 (1995)
Z. Zhou, S. Huelga, C.F. Li, G.C. Guo
Phys. Rev. Lett. 115 (2015)
• Witnesses of non-classicality
C.-M. Li, N. Lambert, Y.-N. Chen, G.-
Y. Chen and F. Nori Sc. Rep.2 (2012)
Outline • • General framework and main result • •
General framework
• We start from the quantum description of a multi-time statistics
Hilbert space
...
Projective measurements of the observable X
at instants t1 , t2 . . . tn
Joint probability distribution
Given this collection of probabilities as input, how can we
unambiguously state that there is or there is not an alternative
classical way to account for them?Kolmogorov consistency conditions
x1 x2 x3
Summing over all the possible intermediate
values we simply obtain the joint probability
referred to the initial and final values
X
P3 {x3 , t3 ; x2 , t2 ; x1 , t1 } = P2 {x3 , t3 ; x1 , t1 }
x2
t1 t2 t3
X
QX̂
n {xn , tn ; . . . xk+1 , tk+1 ; xk , tk ; xk 1 , tk 1 . . . x1 , t1 } = QX̂
n 1 {xn , tn ; . . . xk+1 , tk+1 ; xk , tk ; xk 1 , tk 1 . . . x1 , t1 }
xk
Kolmogorov extension theorem
There exists a classical stochastic process whose
joint probability distributions coincide with
X
• In the quantum realm it might not hold, as ⇢ 6= Px ⇢ ; role of coherences!
x
• For practical reasons, we speak about j-classicality to say that the
consistency conditions hold up to ; non-classical means not even 2-CLOpen quantum systems
• Any realistic description has to include the interaction with the environment,
i.e., the system under study has to be treated as an open (quantum) system
System: degrees of freedom we are interested in
Environment: usually very complex, it is averaged out
⇢S (t) = trE U (t)[⇢S (t0 ) ⌦ ⇢E (t0 )]U † (t) = ⇤S (t)[⇢S (t0 )]
Reduced dynamical maps: referred to HS only
• Gorini-Kossakowski-Lindblad-Sudarshan master equation
dX2
1 ✓ n o ◆
d † 1
⇢S = L⇢S = i [H, ⇢S ] + j L ⇢ L
j S j L†j Lj , ⇢S
dt j=1
2
semigroup of CPT maps
G. Lindblad Comm. Math. Phys. 48 (1976) 8t, s 0
V.Gorini, A. Kossakowski and E.C.G. Sudarshan, J. Math. Phys. 17 (1976)Quantum regression theorem
• Given an observable , the dynamical maps fix the one-time statistics
and that conditioned w.r.t. t0 = 0
What can we say about the higher order probability distributions?
• In general, we have to ’’go back’’ to the full unitary
• Under proper conditions (essentially, if S-E correlations are negligible)
Now only maps acting on the open system are involved:
the reduced dynamical maps define the whole hierarchy of multi-time probabilitiesMarkovianity of the multi-time statistics
• For a non-degenerate observable
Markov condition
• The whole hierarchy of probabilities can be reconstructed from
the initial condition and the transition probabilities
• We say that the statistics is j-Markovian if the QRT holds up to ;
it is non-Markovian if not even 2-M. [In the spirit of Lindblad Comm. Math. Phys. 65 (1979)]
Different from recent definitions,
which are referred to the dynamicsDynamics of quantum coherences
• Definition: coherence-generating-and-detecting (CGD) dynamics
The semigroup dynamics is CGD whenever there exist t, ⌧ 0 such that
Total dephasing
Applying dephasing at an intermediate time
changes the state transformation with
dephasing also at the initial and final times
• Equivalent formulation
h y |⇤(t) [| x ih x |] | z i ⇤h x̃ |⇤(⌧ )[| y ih z |]| x̃ i 6 0
=
coherences are generated The same coherences are
turned into populationsDynamics of quantum coherences - II
• Simple example, for a unitary dynamics
|1i
measurement of , eigenvectors
⇡
intermediate time t1 =
4
⇡
final time t2 =
2
Coherences are generated AND turned into populations
| 1i
• NCGD maps ( ) are connected with the resource theory of coherence
• Maps not creating coherence from incoherent states (MIO)
• Coherence non-activating set: coherence, even if present, is not a resource
Subsets of the set of NCGD maps; there are NCGD maps which are neither of the two
=Main result: one-to-one correspondence
Let be a system’s non-degenerate observable
and a j-Markovian statistics, then
the statistics is jCL for any initial diagonal state
if and only if the dynamics is NCGD
• KEY POINT: we want to connect a property of the dynamics, (N)CGD,
with a property of the whole hierarchy of probabilities, (N)Cl
Markovianity is what allows us to do that !
QX̂
n (xn , tn . . . ; x1 , t1 )
S
Classicality concerns the whole hierarchy
...
marginals
QX̂
2 (x2 , t2 ; x1 , t1 )
S
In general, this is not possible
Dynamical quantities ‘’live’’ hereMain result: one-to-one correspondence
Let be a system’s non-degenerate observable
and a j-Markovian statistics, then
the statistics is jCL for any initial diagonal state
if and only if the dynamics is NCGD
• KEY POINT: we want to connect a property of the dynamics, (N)CGD,
with a property of the whole hierarchy of probabilities, (N)Cl
Markovianity is what allows us to do that !
QX̂
n (xn , tn . . . ; x1 , t1 )
S
Classicality concerns the whole hierarchy
...
marginals
QX̂
2 (x2 , t2 ; x1 , t1 )
S
Markovianity
Dynamical quantities ‘’live’’ hereSketch of the proof
• Given a non-degenerate observable and a Lindblad dynamics
NCGD
time-homogeneous Chapman Kolmogorov equation!
• Every classical, Markovian time-homogenous process satisfies this C-K
Time-homogeneity of the statistics!
Due to Lindblad and 2-M
2CL + 2M C-K means 2CL + 2M NCGD
• j-M provides us with a notion of Markovianity which holds for any
(classical and nonclassical) statistics!
C-K + jM jCLLeggett-Garg-type inequality
• Introduced to provide an easily-detectable counterpart of the Leggett-Garg
inequality S. Huelga, T. Marshall, and E. Santos, Phys. Rev. A 52 (1995)
Given a dichotomic observable X and its correlation function
under the assumptions of macroscopic realism and stationarity
LGt inequality
Measurements only at the initial and at different final times: easy to access
• Stationarity was shown to be related with Markovianity and, actually, one
can show that C-K implies the LGt inequality
Given a Lindblad dynamics, the LGt inequality can be violated only if the
dynamics is CGD: LGt inequality as a witness of quantum coherence
• As a corollary of our main result
Given a 2M statistics, the LGt inequality can be violated only if the
hierarchy is non-classical: LGt inequality as a witness of non-classicalityOutline • • • Non-Markovian multi-time statistics •
The model
• System: two-level system. Environment: continuous degree of freedom
• Global unitary evolution ˆz |`i = `|`i ` = ±1
Example: polarization and momentum d.o.f. of a photon produced in SPDC
• Initial product state with pure environmental state
• Reduced dynamics: pure dephasingSemigroup dynamics, non-Markovian statistics
• Given a Lorentzian distribution
1.0
0.8
0.6
0.4
0.2
1 2 3 4 5 6
the reduced dynamics is the semigroup fixed by the Lindblad equation
G.Lindblad, preprint, Stocholm (1980)
• But the multi-time statistics is non-Markovian! L.Accardi, A.Frigerio, and J.Lewis,
Publ RIMS Kyoto Univ. 18 (1982)
ˆx |±i = ±1|±i
Strong difference,
even qualitative!Non-classicality without quantum coherence
• Beyond Markov, the hypothesis of our Theorem does not apply, in fact…
• The statistics is non-classical: 2-time Kolmogorov does not hold
• For the dynamics is NCGD (with respect to ˆx)
|1i |1i |1i
| i | i |+i
| i |+i
|+i
| 1i | 1i | 1i
Quantum coherence is not even generated!!
• Properties of the 2-time probabilities cannot be inferred from the dynamics
⇢(0) = /2 2M and 2CL, but not 3M and 3CLQuantum coherence without non-classicality
• Distribution given by the combination of two Gaussians
• The statistics is non-Markovian [and the dynamics non-semigroup]
• There are (couples of) instants of time where the dynamics generates
coherence and turns it into population, but the statistics is classical!!
Quantum coherence only
Amount of CGD (without info about higher orders)
cannot be used as a witness of
Violation of the non-classicality
2-t KolmogorovSummary
• We derived a one-to-one correspondence between
x1 x2 x3
Non-classicality of the
multi-time statistics, in the
... sense of Kolmogorov
t1 t2 t3
Capability of the dynamics to generate and
detect quantum coherence
• Crucial role of the Markovianity of the multi-time statistics (in the sense of QRT)
QX̂
n (xn , tn . . . ; x1 , t1 )
S |1i |1i |1i
...
| i | i |+i
| i |+i
|+i
QX̂
2 (x2 , t2 ; x1 , t1 )
S
| 1i | 1i | 1iOutlook
• Application to specific physical systems (spin-boson model) and, in
particular, to quantum thermodynamics
• Extension of the analysis to classical invasive theories
measurements influence the statistics
Signaling in timeAcknowledgments
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