Measurement-based formulation of quantum heat engine
andoptimal performance of quantum heat engines
arXiv:1405 .6457 (2014)arXiv:1504 .06150 (2015)

Riken CEMS Nakamura Laboratory
Hiroyasu Tajima

Collaborated with Prof. Masahito Hayashi at Nagoya University.
YKIS2016, 2016/6/14

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Research Theme:

Information theoretical analysis for heat engines;
Data compression：・・・

Cryptography:・・・

Heat Engine：Apply!

What is the merit ?
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Merits of information theoreticalapproach

Merit 1: A new sight helps us to find hidden
problems in previous formulations.

Result 1: Measurement based
formulation for quantum heat engine

Merit 2: A new mathematical tool helps us to try
unsolved problems.

Result 2: Asymptotic analysis for the optimal
performance of quantum heat engine
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Two formulations for work extraction
from quantum system

Operational scenario: Internal unitary model

I:Baths and working bodyU I of work,
When we request the ditectability
the time evolution of I cannot be unitary….

arXiv:1504 .06150 (2015)

J. Kurchan, arXiv:cond-mat/0007360(2000).

H. Tasaki, arXiv:cond-mat/0009244 (2000).
T. Sagawa and M. Ueda, Phys. Rev. Lett, 100 080403 (2008).

Autonomous scenario:
Total unitary model

Ex: work storage

Vor deterministic work extraction

P. Skrzypczyk, et. al Nat. Comm. 5, 4185, (2014)
F. G. S. L. Brandao, et. al., arXiv:1305.5278, (2013).
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H. Tasaki, arXiv:1511.01999 (2015).
M. Horodecki and J. Oppenheim, Nat. Commun. 4, 2059 (2013).

“Detectability” of work

The work extraction translates the energy …
To detactable form
From undetactable form

We measure only
here!

As the minimal request for the “useful form,” wedemand the following;
We can detect the amount of the extracted
energy only by measuring the external system.

Can Internal-unitary model satisfy this request?

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A hint from quantum communication

A well-known fact in quantum communication field;“When a system evolutes approximately unitarily, it
is difficult to obtain information from the system.”
The time evolution is the more close to unitary,V

In fact, we can derive a trade-off relation for the internal unitary6/32
model.

This state is the more independent ofTrade-off relation

Whenever the setthen,satisfies

for any,Vforditectability
any
andof work,.
When we request thethe time evolution of I cannot be unitary….

arXiv:1504 .06150 (2015)• The Brues distance expresses the limit of theSoof
wethe
need
anotherdistinguishability
stateswithoperational
the POVMformulation!
measurements.

Whenever, the measurements on EX can not distinguish
the final states of EX whose initial state of I are different.

Because the energy gain changes from its minimum to its
maximum during the initial state of I changes, we can not know
the energy gain of EX by the measurements on EX, after all.

Reconsider about “definition of work”

In macroscopic heat engine…

Macro system

Macro system
In quantum heat engine…
quantum system

Macro systemSo, work extraction from quantum system=Discernable change of macro system caused by quantum
system =measurement process!

Measurement-based formulation

In general, an arbitrary measurement process is described as aset of completely positive (CP) maps;
CP map on system A; For arbitrary positive,
is positive.

So, we formulate the work extraction as a set of CP maps;
withis completely positeve (CP) map, and.

is a function of j.

energy conservation conditionarXiv:1504 .06150 (2015)

Relation to the total unitary models

arXiv:1504 .06150 (2015)

Our formulation is consistent with total unitary model,
because of direct-indirect measurement relation.
CP-work extraction！

VSo, now we have an operational scenario which isconsistent with the autonomous scenario.

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Merits of information theoretical
approach

Merit 1: A new sight helps us to find hidden
problems in previous formulations.

Result 1: Measurement based
formulation for quantum heat engine

Merit 2: A new mathematical tool helps us to try
unsolved problems.

Result 2: Asymptotic analysis for the optimal
performance of quantum heat engine
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Researches for the optimal
performance of heat engines

In Thermodynamics;

The optimal performances for macroscopic heatengines are given.

In Statistical mechanics(and quantum information);
Par cle number: ∞Still
unsolved
・・・

In i.i.d. case, i.e.,,the optimal performances are given.

H. Tasaki, arXiv:cond-mat/0009244 (2000).
Kurchan, arXiv:cond-mat/0007360(2000).
We clarify J.here
in i.i.d case,
P. Skrzypczyk,et.al., Nat. Com., 4185, (2014).

by asymptotic theory
in data compression.
In generalcase, the optimalarXiv:1405.6457 (2014)
Particle number:1〜20
performances are given.

M. Horodecki and J. Oppenheim, Nat. Comm. 4, 2059 (2013).
F. G. S. L. Brandao, et. al., PNAS, 112,3215(2015).

Heat engine with finite-size heat baths
Operational form:

CP-work
extraction

Autonomous form:WW

We measure here.

We can add arbitrary
catalytic system.

Asymptotic evaluation ofoptimal efficiency arXiv:1405 .6457 (2014)

Let us define the optimal efficiency as follows;

When the particles of each heat bath do not interact each others, i.e.,
The optimal efficiency satisfies

We can easily compute the coefficients;
Our results give computable approximation of optimal efficiency
even whenEvaluation of the quality of energy gain

We can give a concrete dynamics satisfying the asymptotic
expansion of the efficiency, and the followings;

arXiv:1405 .6457 (2014)

When,

holds.
Optimal process is an example of translation from “heat” to “work”.

Summary 1/2:

arXiv:1504 .06150 (2015)

As operational formulation of quantum heat engines,“internal unitary model” has been widely used.
UWe find that this model is not consistent with theautonomous formulation by considering the detectability ofwork.

As another operational formulation, we propose ameasurement-based formulation, which isconsistent with the autonomous formulation.
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Summary 2/2:

arXiv:1405 .6457 (2014)

We give an asymptotic expansion of optimal efficiency.

We give the optimal process as a concrete dynamics.
It is a good example of translation from “heat” to “work.”