PHYSICAL REVIEW E 91, 056102 (2015)
Reply to “Comment on ‘Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle’ ” Sumiyoshi Abe Department of Physical Engineering, Mie University, Mie 514-8507, Japan and Institute of Physics, Kazan Federal University, Kazan 420008, Russia (Received 25 April 2015; published 26 May 2015) In their Comment on the paper [Abe and Okuyama, Phys. Rev. E 83, 021121 (2011)], Gonz´alez-D´ıaz and D´ıaz-Sol´orzano discuss that the initial state of the quantum-mechanical analog of the Carnot cycle should be not in a pure state but in a mixed state due to a projective measurement of the system energy. Here, first the Comment is shown to miss the point. Then, second, multiple projective measurements are discussed as a generalization of the Comment, although they are not relevant to the work commented. DOI: 10.1103/PhysRevE.91.056102
PACS number(s): 05.70.−a, 03.65.−w
Roles of quantum mechanics in thermodynamics are important contemporary issues in connection with, e.g., nanoscience, nanotechnology, and quantum information where temperature is low and the number of particles contained in a system is small. It is therefore of interest to consider, in the “thermodynamicslike” manner, an extreme case when a system has vanishing temperature and consists only of a single particle. Fifteen years ago, Bender et al. [1] conducted an intriguing discussion about a quantum-mechanical analog of the Carnot engine. The system employed there consists of a single particle confined in an infinite potential well with a movable wall but does not contact with heat baths, preserving coherence. The engine works reversible because of the unitarity of the underlying quantum dynamics. Recently, the work mentioned above has been revisited in Refs. [2–5]. The discussion in Ref. [1] has been examined in Ref. [2] in view of a formal similarity between quantum mechanics and thermodynamics. The maximum power output condition and the associated universal efficiency in finite time processes have been studied in Ref. [3], and the role of the quantum-mechanical superposition principle for enhancing the efficiency has been explored in Ref. [4]. In addition, the model has been generalized in Ref. [5] to the case of a general confining potential. These concepts and ideas have further been advanced in Ref. [6]. Now, in their paper in Ref. [7], Gonz´alez-D´ıaz and D´ıazSol´orzano make a critical comment on the work in Ref. [2]. They assume that the engine is initially in a state described by superposition of energy eigenstates. Then, they perform a projective measurement of the system energy, which changes the initial pure state to a mixed state. Since the state is mixed, its von Neumann entropy does not vanish, in contrast to the statement made after Eq. (7) in Ref. [2]. This Reply has two parts. In the first part, we explain that the criticism made by the authors of Ref. [7] misses the point. Then, in the second part, we discuss projective measurements on the processes in a manner more general than that in Ref. [7], although irrelevant to the work in Ref. [2], in any way. (i) Gonz´alez-D´ıaz and D´ıaz-Sol´orzano [7] criticize the work in Ref. [2] by assuming that the engine is initially prepared in a state described by superposition of energy eigenstates. On the other hand, as in Ref. [1], the energy ground state is employed as the initial state in Ref. [2]. This point may be one of the origins of a basic misunderstanding in their Comment. 1539-3755/2015/91(5)/056102(2)
It is quite natural to initially start with the ground state if the vanishing temperature limit is taken for the canonical ensemble. In addition, they quote another work of Bender et al. in Ref. [8] as supporting evidence for their criticism. But, unfortunately, there is a confusing point in Ref. [8]. It is as follows. The initial state considered there is a pure state, unlike the claimed setup in Ref. [7]. In fact, the state employed there is |ψ = n an |φn with {|φ n } n being the set of the energy eigenstates, and p m = |am | 2
(1)
is the probability of finding the system in the energy eigenstate |φ m as can be seen above and below Eq. (2.2) in Ref. [8]. Therefore, the von Neumann entropy S vN [ρ (initial) ] = −k Tr{ρ (initial) ln ρ (initial) } identically vanishes for this initial pure state ρ (initial) = |ψψ|. Equation (4.1) in Ref. [8] is, therefore, not the von Neumann entropy but the Shannon entropy S S [p] = −k m pm ln pm in the basis {|φ n } n . (One of the results presented in Ref. [2] is that, if S S is identified with the Clausius entropy, then transmutation apparently occurs from pure-state quantum mechanics into quantum thermodynamics at finite temperatures.) Thus, the criticism of the authors in Ref. [7] does not apply. (ii) Although the comment made in Ref. [7] is, as seen above, irrelevant to the work in Ref. [2], it may still be interesting to examine it in the context of effects of projective measurement on the performance of an engine. Clearly, this is a highly nontrivial issue and needs separate or independent investigation. Therefore, here, we only consider a possible generalization of the discussion in Ref. [7] to the case of two projective measurements in an isoenergetic process. However, once again, we emphasize that this discussion is not relevant to the work in Ref. [2] being commented on by Gonz´alez-D´ıaz and D´ıaz-Sol´orzano. Consider a set of projection operators {Pk }k , which are required here to be Hermitian, and assume that the system Hamiltonian H is decomposed in terms of them as H = E P , where {E k } k is the energy k k k spectrum. The projection operators satisfy Pk Pl = δkl Pl and k Pl = I with I being the identity operator. Through a measurement with this projection operator on the system in a state ρ, the probability of having the value E k of H is given by pk = Tr(ρPk ). Correspondingly, the state changes from ρ to ρ k = pk−1 Pk ρPk , which is the density matrix of the kth pure state. The pure ground state employed
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©2015 American Physical Society
COMMENTS
PHYSICAL REVIEW E 91, 056102 (2015)
as the initial state in Refs. [1,2] may be realized in this way (by making an ensemble of the states with the lowest value of E k ). A more general discussion about state preparation can be found, for example, in Refs. [9,10]. In Eq. (6) in Ref. [7], the sum is taken over all possible outcomes, resulting in the density matrix of a mixed state:ρ = k pk ρk . Now, let us consider an isoenergetic expansion process A → B (see Fig. 1 in Ref. [2]), which is a basic element of the quantum-mechanical analog of the Carnot engine, and study two projective measurements at A and B. Let us consider an initial density matrix ρ (initial) that can be either pure or mixed. The first projective measurement of the energy at A described by the projection operator P m(A) defines Pm(A) ρ (initial) Pm(A) . During the process, the volume of the system (the width of the infinite potential well) grows with keeping the expectation value of the HamiltonianH constant. This process can be described by a unitary transformation:Pm(A) ρ (initial) Pm(A) → † U Pm(A) ρ (initial) Pm(A) U . Since the one and only requirement is the constancy of H along the process, U is not unique, and so the state at B is not in a definite energy eigenstate, in general. The second projective mea† surement at B yields Pn(B) U Pm(A) ρ (initial) Pm(A) U Pn(B) . Define (B) † (A) (initial) (A) pnm = Tr[Pn U Pm ρ Pm U Pn(B) ]. This quantity is
found to satisfy the rule: pnm = pn|m pm , where pn|m = (A) (A) (initial) Tr[Pn(B) U † P m U ] andpm = Tr[Pm ρ ] fulfilling the conditions n pn|m = m pn|m = 1 and m pm = 1, respectively. p m is nothing but the quantity in Eq. (1) above, whereas pn|m physically describes the transition from A to B. pnm generalizes Eq. (1) to the case of two projective measurements. To summarize, we have discussed that the projective measurement in Ref. [7] is not needed for preparation of the initial state in the quantum-mechanical analog of the Carnot engine. If such a measurement is performed on the system, then reversibility, which is a key concept of the engine, will be lost. Also, the difference between the von Neumann entropy and the Shannon entropy should strictly be distinguished: The latter is measurement dependent (i.e., representation dependent), whereas the former is not. It is the von Neumann entropy that is to be employed for examining reversibility of a process.
[1] C. M. Bender, D. C. Brody, and B. K. Meister, J. Phys. A 33, 4427 (2000). [2] S. Abe and S. Okuyama, Phys. Rev. E 83, 021121 (2011). [3] S. Abe, Phys. Rev. E 83, 041117 (2011). [4] S. Abe and S. Okuyama, Phys. Rev. E 85, 011104 (2012). [5] S. Abe, Entropy 15, 1408 (2013). [6] C. Ou, Z. Huang, B. Lin, and J. Chen, Sci. China - Phys. Mech. Astron. 56, 1815 (2013); ,57, 1266 (2014).
[7] L. A. Gonz´alez-D´ıaz and S. D´ıaz-Sol´orzano, Phys. Rev. E 91, 056101 (2015). [8] C. M. Bender, D. C. Brody, and B. K. Meister, Proc. R. Soc. London A 458, 1519 (2002). [9] G. Harel, G. Kurizki, J. K. McIver, and E. Coutsias, Phys. Rev. A 53, 4534 (1996). [10] T. Wellens, A. Buchleitner, B. K¨ummerer, and H. Maassen, Phys. Rev. Lett. 85, 3361 (2000).
This work was supported, in part, by a Grant-in-Aid of Scientific Research from the Japan Society for Promotion of Science and by the Ministry of Education and Science of the Russian Federation (the program of competitive growth of Kazan Federal University).
056102-2
Reply to “Comment on ‘Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle’ ” Sumiyoshi Abe Department of Physical Engineering, Mie University, Mie 514-8507, Japan and Institute of Physics, Kazan Federal University, Kazan 420008, Russia (Received 25 April 2015; published 26 May 2015) In their Comment on the paper [Abe and Okuyama, Phys. Rev. E 83, 021121 (2011)], Gonz´alez-D´ıaz and D´ıaz-Sol´orzano discuss that the initial state of the quantum-mechanical analog of the Carnot cycle should be not in a pure state but in a mixed state due to a projective measurement of the system energy. Here, first the Comment is shown to miss the point. Then, second, multiple projective measurements are discussed as a generalization of the Comment, although they are not relevant to the work commented. DOI: 10.1103/PhysRevE.91.056102
PACS number(s): 05.70.−a, 03.65.−w
Roles of quantum mechanics in thermodynamics are important contemporary issues in connection with, e.g., nanoscience, nanotechnology, and quantum information where temperature is low and the number of particles contained in a system is small. It is therefore of interest to consider, in the “thermodynamicslike” manner, an extreme case when a system has vanishing temperature and consists only of a single particle. Fifteen years ago, Bender et al. [1] conducted an intriguing discussion about a quantum-mechanical analog of the Carnot engine. The system employed there consists of a single particle confined in an infinite potential well with a movable wall but does not contact with heat baths, preserving coherence. The engine works reversible because of the unitarity of the underlying quantum dynamics. Recently, the work mentioned above has been revisited in Refs. [2–5]. The discussion in Ref. [1] has been examined in Ref. [2] in view of a formal similarity between quantum mechanics and thermodynamics. The maximum power output condition and the associated universal efficiency in finite time processes have been studied in Ref. [3], and the role of the quantum-mechanical superposition principle for enhancing the efficiency has been explored in Ref. [4]. In addition, the model has been generalized in Ref. [5] to the case of a general confining potential. These concepts and ideas have further been advanced in Ref. [6]. Now, in their paper in Ref. [7], Gonz´alez-D´ıaz and D´ıazSol´orzano make a critical comment on the work in Ref. [2]. They assume that the engine is initially in a state described by superposition of energy eigenstates. Then, they perform a projective measurement of the system energy, which changes the initial pure state to a mixed state. Since the state is mixed, its von Neumann entropy does not vanish, in contrast to the statement made after Eq. (7) in Ref. [2]. This Reply has two parts. In the first part, we explain that the criticism made by the authors of Ref. [7] misses the point. Then, in the second part, we discuss projective measurements on the processes in a manner more general than that in Ref. [7], although irrelevant to the work in Ref. [2], in any way. (i) Gonz´alez-D´ıaz and D´ıaz-Sol´orzano [7] criticize the work in Ref. [2] by assuming that the engine is initially prepared in a state described by superposition of energy eigenstates. On the other hand, as in Ref. [1], the energy ground state is employed as the initial state in Ref. [2]. This point may be one of the origins of a basic misunderstanding in their Comment. 1539-3755/2015/91(5)/056102(2)
It is quite natural to initially start with the ground state if the vanishing temperature limit is taken for the canonical ensemble. In addition, they quote another work of Bender et al. in Ref. [8] as supporting evidence for their criticism. But, unfortunately, there is a confusing point in Ref. [8]. It is as follows. The initial state considered there is a pure state, unlike the claimed setup in Ref. [7]. In fact, the state employed there is |ψ = n an |φn with {|φ n } n being the set of the energy eigenstates, and p m = |am | 2
(1)
is the probability of finding the system in the energy eigenstate |φ m as can be seen above and below Eq. (2.2) in Ref. [8]. Therefore, the von Neumann entropy S vN [ρ (initial) ] = −k Tr{ρ (initial) ln ρ (initial) } identically vanishes for this initial pure state ρ (initial) = |ψψ|. Equation (4.1) in Ref. [8] is, therefore, not the von Neumann entropy but the Shannon entropy S S [p] = −k m pm ln pm in the basis {|φ n } n . (One of the results presented in Ref. [2] is that, if S S is identified with the Clausius entropy, then transmutation apparently occurs from pure-state quantum mechanics into quantum thermodynamics at finite temperatures.) Thus, the criticism of the authors in Ref. [7] does not apply. (ii) Although the comment made in Ref. [7] is, as seen above, irrelevant to the work in Ref. [2], it may still be interesting to examine it in the context of effects of projective measurement on the performance of an engine. Clearly, this is a highly nontrivial issue and needs separate or independent investigation. Therefore, here, we only consider a possible generalization of the discussion in Ref. [7] to the case of two projective measurements in an isoenergetic process. However, once again, we emphasize that this discussion is not relevant to the work in Ref. [2] being commented on by Gonz´alez-D´ıaz and D´ıaz-Sol´orzano. Consider a set of projection operators {Pk }k , which are required here to be Hermitian, and assume that the system Hamiltonian H is decomposed in terms of them as H = E P , where {E k } k is the energy k k k spectrum. The projection operators satisfy Pk Pl = δkl Pl and k Pl = I with I being the identity operator. Through a measurement with this projection operator on the system in a state ρ, the probability of having the value E k of H is given by pk = Tr(ρPk ). Correspondingly, the state changes from ρ to ρ k = pk−1 Pk ρPk , which is the density matrix of the kth pure state. The pure ground state employed
056102-1
©2015 American Physical Society
COMMENTS
PHYSICAL REVIEW E 91, 056102 (2015)
as the initial state in Refs. [1,2] may be realized in this way (by making an ensemble of the states with the lowest value of E k ). A more general discussion about state preparation can be found, for example, in Refs. [9,10]. In Eq. (6) in Ref. [7], the sum is taken over all possible outcomes, resulting in the density matrix of a mixed state:ρ = k pk ρk . Now, let us consider an isoenergetic expansion process A → B (see Fig. 1 in Ref. [2]), which is a basic element of the quantum-mechanical analog of the Carnot engine, and study two projective measurements at A and B. Let us consider an initial density matrix ρ (initial) that can be either pure or mixed. The first projective measurement of the energy at A described by the projection operator P m(A) defines Pm(A) ρ (initial) Pm(A) . During the process, the volume of the system (the width of the infinite potential well) grows with keeping the expectation value of the HamiltonianH constant. This process can be described by a unitary transformation:Pm(A) ρ (initial) Pm(A) → † U Pm(A) ρ (initial) Pm(A) U . Since the one and only requirement is the constancy of H along the process, U is not unique, and so the state at B is not in a definite energy eigenstate, in general. The second projective mea† surement at B yields Pn(B) U Pm(A) ρ (initial) Pm(A) U Pn(B) . Define (B) † (A) (initial) (A) pnm = Tr[Pn U Pm ρ Pm U Pn(B) ]. This quantity is
found to satisfy the rule: pnm = pn|m pm , where pn|m = (A) (A) (initial) Tr[Pn(B) U † P m U ] andpm = Tr[Pm ρ ] fulfilling the conditions n pn|m = m pn|m = 1 and m pm = 1, respectively. p m is nothing but the quantity in Eq. (1) above, whereas pn|m physically describes the transition from A to B. pnm generalizes Eq. (1) to the case of two projective measurements. To summarize, we have discussed that the projective measurement in Ref. [7] is not needed for preparation of the initial state in the quantum-mechanical analog of the Carnot engine. If such a measurement is performed on the system, then reversibility, which is a key concept of the engine, will be lost. Also, the difference between the von Neumann entropy and the Shannon entropy should strictly be distinguished: The latter is measurement dependent (i.e., representation dependent), whereas the former is not. It is the von Neumann entropy that is to be employed for examining reversibility of a process.
[1] C. M. Bender, D. C. Brody, and B. K. Meister, J. Phys. A 33, 4427 (2000). [2] S. Abe and S. Okuyama, Phys. Rev. E 83, 021121 (2011). [3] S. Abe, Phys. Rev. E 83, 041117 (2011). [4] S. Abe and S. Okuyama, Phys. Rev. E 85, 011104 (2012). [5] S. Abe, Entropy 15, 1408 (2013). [6] C. Ou, Z. Huang, B. Lin, and J. Chen, Sci. China - Phys. Mech. Astron. 56, 1815 (2013); ,57, 1266 (2014).
[7] L. A. Gonz´alez-D´ıaz and S. D´ıaz-Sol´orzano, Phys. Rev. E 91, 056101 (2015). [8] C. M. Bender, D. C. Brody, and B. K. Meister, Proc. R. Soc. London A 458, 1519 (2002). [9] G. Harel, G. Kurizki, J. K. McIver, and E. Coutsias, Phys. Rev. A 53, 4534 (1996). [10] T. Wellens, A. Buchleitner, B. K¨ummerer, and H. Maassen, Phys. Rev. Lett. 85, 3361 (2000).
This work was supported, in part, by a Grant-in-Aid of Scientific Research from the Japan Society for Promotion of Science and by the Ministry of Education and Science of the Russian Federation (the program of competitive growth of Kazan Federal University).
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