Counterexamples to the Theory of Thermodynamics With Feedback Proposed By Sagawa and Ueda

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by  Robert R. Tucci

Counterexamples to the Theory
of Thermodynamics With Feedback
Proposed By Sagawa and Ueda
3 Robert R. Tucci
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P.O. Box 226
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2 Bedford, MA 01730

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[email protected]
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9 January 11, 2013

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Abstract

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We give some counterexamples to the theory of thermodynamics with feedback that
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their theory, in its present form, is seriously flawed.
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In a string of papers (Refs.[1] to [7]) dating from 1997 to 2012, Sagawa, Ueda and
coworkers (S-U) have proposed a theory for thermodynamics with feedback. Their
work has generated one published comment in Ref.[8] to which they have replied in
Ref.[9]. This note objects to their theory on different grounds than Ref.[8].
Numerous times throughout Refs.[1] to [7], S-U define what they call the
“classical-quantum information” as
I = I(ρ : X) = S(ρ )− p S(ρ(k)) , (1)
QC 1 1 k 2
Xk
and then they claim that
0 ≤ I ≤ H({p }) . (2)
QC k
For instance, they claim to prove Eq.(2) in Ref.[1], Eqs.(17) and (18). Unfortunately,
their proof of Eq.(2) is wrong; both of these bounds can be violated. Below we will
give an example for which I < 0 and an example for which I > H({p }).
QC QC k
Henceforth, we will denote classical random variables by underlined letters.
In our two examples, all density matrices are simultaneously diagonal and thus cor-
respond to classical probability distributions. Let ρ correspond to a probability
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distribution P , ρ to P , and ρ(k) to P (·|k) for all k. In this case,1
x2 1 x1 2 x2|k
I = H(x )−H(x |k) . (3)
QC 1 2
In both of our examples, we will assume that the probability distribution
P can be described by the Markov chain x ← k ← x where x ,k,x ∈
x2,k,x1 2 1 2 1
{0,1}. Let h(α) = −αlnα−(1−α)ln(1−α) for α ∈ [0,1].
• Example for which I < 0.
QC
Suppose
1 1 1
P = 2 2 , P = . (4)
x2|k (cid:20) 1 1 (cid:21) x1 (cid:20) 0 (cid:21)
2 2
Then H(x ) = 0 and H(x |k) = h(1) = ln2 so I = −ln2.
1 2 2 QC
• Example for which I > H(k).
QC
Suppose
1 0 0 0 1
P = , P = , P = 2 . (5)
x2|k (cid:20) 0 1 (cid:21) k|x1 (cid:20) 1 1 (cid:21) x1 (cid:20) 1 (cid:21)
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1Note that I 6= H(x )−H(x |k) = H(x : k). The mutual information H(x : k) does
QC 1 1 1 1
satisfy 0≤H(x1 : k)≤H(k).
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Then H(x ) = h(1) = ln2 and H(x |k) = 0 so I = ln2. But P =
1 2 2 QC k
0
P P = so H(k) = 0.
k|x1 x1 (cid:20) 1 (cid:21)
In Ref.[5], Eqs.(30) and (31), S-U define quantities I and I . Then they prove
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the following generalization of the Jarzynski equality:
e−σ−Ic = 1 . (6)
(cid:10) (cid:11)
(see Ref.[5], Eq.(66)). They conjecture (but do not prove) that hIi ≈ hI i ≈ I
c QC
in some sense. But is this really a good approximation? Note that hIi and hI i are
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always non-negative whereas we’ve proven that I can be negative. Note also that
QC
Eq.(6) implies that hσ +I i ≥ 0. If we replace hI i by I in this inequality, we get
c c QC
hσi+I ≥ 0 which is violated if the entropy production hσi equals zero and I < 0.
QC QC
S-U may have proven hσ +I i ≥ 0, but they never give a proof or even a good reason
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why we should expect that hσi+I ≥ 0. And a violation of hσi+I ≥ 0 means
QC QC
there can be no generalization of the Jarzynski equality similar to Eq.(6) but for I
QC
instead of I .
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Ref.[7] generalizes the theory of Ref.[1] from a uni-partite to a bi-partite sys-
tem. Ref.[7] reduces to the uni-partite case if one of the two particles is constrained
to occupy a single pure state. Thus, all of the above objections also apply to Ref.[7].
In Ref.[10], I describe my own theory for generalizing thermodynamics so that
it applies to processes with feedback. I have profitted immensely from reading the
wonderful work of S-U. My theory agrees very much in spirit with the S-U theory, but
differs from it in some important details. Of course, whether the objections to the
S-U theory raised in this note are valid or not is totally independent of the contents
of Ref.[10].
References
[1] T. Sagawa , M. Ueda, “Second Law of Thermodynamics with Discrete Quantum
Feedback Control”, arXiv:0710.0956
[2] T. Sagawa , M. Ueda, “Minimal Energy Cost for Thermodynamic Information
Processing: Measurement and Information Erasure”, arXiv:0809.4098
[3] T. Sagawa , M. Ueda, “Generalized Jarzynski Equality under Nonequilibrium
Feedback Control”, arXiv:0907.4914
[4] S. Toyabe, T. Sagawa , M. Ueda , E. Muneyuki, M. Sano, “Experimental demon-
stration of information-to-energy conversion and validation of the generalized
Jarzynski equality”, arXiv:1009.5287
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[5] T. Sagawa , M. Ueda, “Nonequilibrium thermodynamics of feedback control”,
arXiv:1105.3262
[6] T. Sagawa, “Second Law-Like Inequalities with Quantum Relative Entropy: An
Introduction”, arXiv:1202.0983
[7] K. Funo, Y. Watanabe, M. Ueda, “Thermodynamic Work Gain from Entangle-
ment”, arXiv:1207.6872
[8] R. Dillenschneider, E. Lutz, Comment on “Minimal Energy Cost for Thermody-
namic Information Processing: Measurement and Information Erasure”, Phys.
Rev. Lett. 104, 198903 (2010).
[9] T.SagawaandM.Ueda, “SagawaandUedaReply”, Phys. Rev.Lett.104,198904
(2010).
[10] R.R. Tucci, “Maxwell Demon from a Quantum Bayesian Networks Perspective”,
arXiv:1301.1284
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