H-theorem for Systems with an Interaction Invariant Distribution Function
aa r X i v : . [ qu a n t - ph ] O c t H-theorem for Systems with an Interaction Invariant DistributionFunction
A. V. Lebedev and G. B. Lesovik
1, *
Submitted by S. A. Grigoryan Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, 141700, Moscow District,Russia
Received May 8, 2019; revised May 13, 2019; accepted May 18, 2019
Abstract—H-theorem gives necessary conditions for a system to evolve in time with a non-diminishing entropy. In a quantum case the role of H-theorem plays the unitality criteria of aquantum channel transformation describing the evolution of the system’s density matrix underthe presence of the interaction with an environment. Here, we show that if diagonal elements ofthe system’s density matrix are robust to the presence of interaction the corresponding quantumchannel is unital.2010 Mathematical Subject Classification:
Keywords and phrases: Quantum channels, quantum information, unitality
1. INTRODUCTIONThe second law of thermodynamics poses a constrain on the time direction of natural ther-modynamic processes. Its classical formulation states that the entropy of an isolated system cannot decrease over time. The celebrated H-theorem of Ludwig Boltzmann [1, 2] guarantees the non-diminishing entropy increase of a classical system those dynamics is governed by the kinetic equationof a specific form. Boltzmann’s kinetic equations relies on the molecular chaos hypothesis whichassumes the chaotic uncorrelated initial state of colliding particles [3]. Therefore, the Boltzmann’sH-theorem can be viewed as a sufficient constraint on dynamics of classical particles to develop witha non-diminishing entropy.In a quantum realm the entropy of an isolated quantum system stays constant and the classicalformulation of the second law is trivial. Therefore, to make the meaningful formulation of the secondlaw in quantum case one needs to consider the entropy dynamics of an open quantum system. Here,the quantum master equation [4] describing the evolution of the system’s density matrix ˆ ρ is ananalog of the Boltzmann’s kinetic equation for a classical case. For a quantum system interactingwith a memoryless environment (or reservoir) the evolution of the system’s density matrix can bedescribed by Marcovian (or equivalently Lindblad) master equation [5]. In this situation one canprove the non-negativity of the entropy production rate σ (ˆ ρ ) that is the amount of entropy producedper unit time by the system [6]. In non-equilibrium thermodynamics σ (ˆ ρ ) obeys a balance equation σ = ddt S (cid:0) ˆ ρ t (cid:1) + J , where S (ˆ ρ ) = − k B tr { ˆ ρ ln ˆ ρ } is the von Neumann entropy and J is the entropy fluxthat is an amount of entropy which is exchanged between the open system and reservoir per unittime. For a thermal equilibrium reservoir at a temperature T the entropy flux J = − T dQdt , where Q id the heat dissipated by the system into the environment. For an energy isolated open system J = 0 and the inequality σ (ˆ ρ ( t )) ≥ expresses the second law of thermodynamics.The quantum information theory [7, 8] (QIT) suggests a more general approach to the dynamicsof open quantum systems which goes far beyond the Marcovian approximation. Within the QITframework the evolution of the system’s density matrix is described via the so called quantumchannel: ρ → Φ( ρ ) ≡ ˆ ρ t that is a completely positive, trace-preserving linear map of a density * E-mail: [email protected]
A. V. LEBEDEV AND G. B. LESOVIK matrix ˆ ρ . Given the uncorrelated initial product state ˆ ρ ⊗ ˆ π of the system’s density matrix ˆ ρ and its environment ˆ π , the explicit form of the quantum channel can be found by tracing out theenvironmental degrees of freedom in a time-evolved state of a grand system (a given quantum systemplus its environment), Φ( ρ ) = Tr env { ˆ U t (ˆ ρ ⊗ ˆ π ) ˆ U † t (cid:9) , where a joint unitary evolution operator ˆ U t = exp( − it ˆ H/ ~ ) is generated by the Hamiltonian ˆ H = ˆ H S + ˆ H R + ˆ H int comprised of the freeHamiltonians of the quantum system ˆ H S and the reservoir ˆ H R and the interaction term ˆ H int . Inthis context the Lindbald dynamics can be described within the so called collision model [9], wherea quantum system interacts locally in time and only once with different uncorrelated environmentaldegrees of freedom or sub-environments. The resulting quantum channel then possesses a divisibilityproperty: Φ = Φ N ◦ · · · ◦ Φ , where Φ i is a quantum channel corresponding to the interaction withan i th sub-environment followed by a free evolution of a quantum system.However, in the presence of a correlated environment [10] and/or finite time reservoir memoryeffects [11] the quantum channel is not divisible and hence cannot be described by the Lindbladmaster equation. This may result in a non-monotonic entropy dynamics and the negative entropygain in general. One of the remarkable results of the QIT is the lower bound for the entropy gainof an arbitrary quantum channel [12], S (cid:0) Φ(ˆ ρ ) (cid:1) − S (ˆ ρ ) = − k B Tr { Φ(ˆ ρ ) ln Φ(ˆ1) } , (1)where ˆ1 is the identity operator. In [13] this result was further generalized for a tensor product ofa dephasing channel and an arbitrary quantum channel. It follows from the right hand side of (1)that for a wide class the so called unital quantum channels defined by the relation Φ(ˆ1) = ˆ1 theentropy gain is non-negative.In [14] the unitality constraint has been considered as a quantum analog of the Boltzmann’s H-theorem and the unitality criteria was formulated in terms of the joint system-reservoir unitaryevolution operators. Consider a decomposition of the unitary evolution operator ˆ U t of thegrand system in interaction representation ˆ U t = ˆ U S ˆ U R ˆ U int , where ˆ U S = exp( − it ˆ H S / ~ ) and ˆ U R =exp( − it ˆ H R / ~ ) are the free evolution operators of the two subparts of the grand system and ˆ U int = T exp( − i R t dt ′ ˆ H int ( t ′ ) / ~ ) is the joint evolution operator. Then the operator ˆ U t can berepresented in the factorised form, ˆ U t = P ki | ˜ ψ k ih ψ i | ˆ U R ˆ V ki , where ˆ V ki ≡ h ψ k | ˆ U int | ψ i i (2)is a set of operators acting in the reservoir Hilbert space, | ψ i i is a complete orthonormal set of statesin the system’s Hilbert space and | ˜ ψ k i ≡ ˆ U S | ψ k i . Then the unitality condition can be expressed as (cid:2) Φ(ˆ1) (cid:3) kk ′ − δ kk ′ = X i D(cid:2) ˆ V † k ′ i , ˆ V ki (cid:3)E , (3)where (cid:2) Φ(ˆ1) (cid:3) kk ′ = h ˜ ψ k | Φ(ˆ1) | ˜ ψ k ′ i . The necessary and sufficient requirement for the channel Φ to beunital then can be found in the vanishing of the right hand side of (3). This unitality criteria wasfound to be an effective tool in describing the entropy production in exemplary physical phenomenalike electron inelastic scattering, electron-phonon interaction and so on, see [14]. Moreover, for anenergetically isolated quantum system this criteria can identify a non-unital channels preservingthe energy of the system and which can lower its entropy without heat exchange with the reservoir.Such a situation corresponds to an action of a quantum Maxwell demon [15] and was demonstratedin the system of interacting qubits [16]. In this article we prove a new unitality criteria which isexpressed in terms of invariance of the diagonal density matrix elements i.e. classical distributionfunction of the system to the presence of the interaction with an environment. Namely, we provethe followingTheorem. Consider a quantum system endowed with a finite dimensional Hilbert space initiallyuncorrelated and disentangled from a reservoir. If there is a basis where for all initial states of thesystem the diagonal elements of its density matrix with and without interaction with the reservoircoincide at the end of the evolution then the system evolves under a unital quantum channel.First, we prove the following LOBACHEVSKII JOURNAL OF MATHEMATICS -THEOREM FOR SYSTEMS 3
Lemma. Let all the requirements of the Theorem hold true and | ˜ ψ k i is the specific basis, where ∀ ˆ ρ, k h ˜ ψ k | Φ(ˆ ρ ) | ˜ ψ k i = h ˜ ψ k | Φ (ˆ ρ ) | ˜ ψ k i , (4) Φ(ˆ ρ ) and Φ (ˆ ρ ) are the quantum channels describing the evolution of the quantum system with andwithout interaction with the reservoir initially prepared in a state ˆ π . Let ˆ U S and ˆ U R are the freeevolution operators for the system and the reservoir respectively, and ˆ U int is the interacting partof the global evolution operator ˆ U = ˆ U S ˆ U R ˆ U int . Then for any eigenstate | n i of the initial densitymatrix of the reservoir with h n | ˆ π | n i > U int (cid:0) | ψ k i ⊗ | n i (cid:1) = | ψ k i ⊗ | ϕ n,k i ∀ k, where | ψ k i = ˆ U † S | ˜ ψ k i and | ϕ n,k i is a final normalized state of the reservoir.Proof. Let the initial state of the grand system comprising the system and the reservoir has theform ˆ R = P ii ′ ρ ii ′ | ψ i ih ψ i ′ | ⊗ ˆ π , where ρ ii ′ = h ψ i | ˆ ρ | ψ i ′ i are the system’s density matrix elements.Then under the unitary evolution ˆ U the resulting state of the grand system is ˆ R = X ii ′ ρ ii ′ X kk ′ | ˜ ψ k i ˆ U R ˆ V ki ˆ π ˆ V † k ′ i ′ ˆ U † R h ˜ ψ k ′ | , where a set of reservoir’s operators ˆ V ki is defined in (2). Tracing out the reservoir degrees of freedomone gets the quantum channel Φ(ˆ ρ ) describing the evolution in the interacting case Φ(ˆ ρ ) = X kk ′ | ˜ ψ k ih ˜ ψ k ′ | X ii ′ ρ ii ′ Tr { ˆ π ˆ V † k ′ i ′ ˆ V ki } , while in the non-interacting situation one has Φ (ˆ ρ ) = P kk ′ | ˜ ψ k ih ˜ ψ k ′ | ρ kk ′ . Then the requirementgiven by (4) can be expressed in the form ∀ k, ˆ ρ : P ij ρ ij Tr { ˆ π ˆ V † kj ˆ V ki } = ρ kk . Let us introduce aset of matrixes H k : H k,ij ≡ Tr { ˆ π ˆ V † kj ˆ V ki } and rewrite (4) as ∀ k, ˆ ρ : Tr { ρ H k } = ρ kk , (5)where ρ is the system’s density matrix in the specific representation | ψ k i . The matrixes H k are clearly Hermitian and hence for each k can be diagonalized in a specific basis H k,ij = P α h ( k ) α ξ ( k ) α,i ξ ( k ) ∗ α,j , where all h ( k ) α are real and ~ξ ( k ) α form a complete and orthonormal set ofeigenvectors. According to the requirements of the Theorem the Eq. (5) is satisfied for all ρ and hence for the density matrixes diagonal in ~ξ ( k ) α representation ρ ij = P α p α ξ ( k ) α,i ξ ( k ) ∗ α,j . Then (5)can be rewritten as X α h ( k ) α p α = X α p α (cid:12)(cid:12) ξ ( k ) α,k (cid:12)(cid:12) , ∀ p α ≥ , X α p α = 1 . Therefore, h ( k ) α = | ξ ( k ) α,k | are non-negative numbers and as follows from completeness of the set ~ξ ( k ) α the matrix H k has a unit trace: P α h ( k ) α = P α | ξ ( k ) α,k | = 1 . Choosing a pure initial state with ρ ij = 0 for all i, j = k and ρ kk = 1 one gets from the (5) H k,kk = 1 , H k,ii = 0 ∀ i = k. (6)Consider a basis in the reservoir Hilbert space diagonalizing the initial density matrix ˆ π : ˆ π = P n π n | n ih n | . The reservoir operators ˆ V ki can be represented as ˆ V ki = P nm | n ih m | v ( ki ) nm and thenthe diagonal elements H k,ii = P n π n P m | v ( ki ) nm | ≥ . Hence, from the Eq. (6) it follows v ( ki ) nm = 0 or equivalently ˆ V ki | n i = 0 for all i = k . The latter proves the Lemma: ∀ k : ˆ U int | ψ k i ⊗ | n i = X i | ψ i i ⊗ ˆ V ik | n i = | ψ k i ⊗ | ϕ n,k i , LOBACHEVSKII JOURNAL OF MATHEMATICS
A. V. LEBEDEV AND G. B. LESOVIK where the final state of the reservoir | ϕ n,k i ≡ ˆ V kk | n i is normalized due to H k,kk = 1 .Proof of the Theorem. Making use the results and definitions of the above Lemma the densitymatrix of the grand system after interaction with the environment has the form ˆ R = X n π n X kk ′ ρ kk ′ ˆ U S ˆ U R ˆ U int | ψ k i| n ih n |h ψ k ′ | ˆ U † int ˆ U † R ˆ U † S = X n π n X kk ′ ρ kk ′ | ˜ ψ k ih ˜ ψ k ′ | ⊗ ˆ U R | ϕ n,k ih ϕ n,k ′ | ˆ U † R . Therefore, the induced quantum channel in the specific basis set | ˜ ψ k i has the form Φ(ˆ ρ ) = X n π n X kk ′ ρ kk ′ | ˜ ψ k ih ˜ ψ k ′ | h ϕ n,k ′ | ϕ n,k i . Introducing a set of Gramian matrixes [ γ n ] kk ′ ≡ h ϕ n,k ′ | ϕ n,k i , the quantum channel Φ can berewritten as a convex combination Φ(ˆ ρ ) = ˆ U S (cid:16)P n π n Φ n (ˆ ρ ) (cid:17) ˆ U † S , where each channel Φ n correspondsto the dephasing channel in the | ψ k i = ˆ U † S | ˜ ψ k i representation: Φ n (ˆ ρ ) = P kk ′ | ψ k ih ψ k ′ | ρ kk ′ (cid:2) γ n (cid:3) kk ′ . The dephasing quantum channel is known to be unital [7, 8] and hence the convex linear combinationof the dephasing channels is unital as well. This proves the statement of the Theorem.We thank G. M. Graf from ETH Zurich for comments and discussions. The research wassupported by the Government of the Russian Federation (Agreement 05.Y09.21.0018), by the RFBRGrants No. 17- 02-00396A and 18-02-00642A, Foundation for the Advancement of TheoreticalPhysics “BASIS”, the Ministry of Education and Science of the Russian Federation 16.7162.2017/8.9(A.V.L.). REFERENCES
1. L. Boltzmann, “Weitere Studien ¨uber das W¨armegleichgewicht unter Gasmolek¨ulen, ”Wiener Berichte75, 62 (1872).2. L. Boltzmann, “Entgegnung auf die w¨arme-theoretischen Betrachtungen des Hrn. E. Zermelo,”Annalender Physik (Leipzig) 57, 773 (1896) [Translated and reprinted in S. G. Brush, Kinetic Theory 2, PergamonElmsford, New York (1966)].3. J. L. Lebowitz, “Statistical mechanics: A selective review of two central issues,”Rev. Mod. Phys. 71,S346 (1999).4. H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford Univ. Press, 2002).5. G. Lindblad, “On the generators of quantum dynamical semigroups,”Comm. Math. Phys. 48, 119 (1976).6. G. Lindblad, “Completely positive maps and entropy inequalities,”Comm. Math. Phys. 40, 147 (1975).7. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge Univ.Press, 2011).8. A. S. Holevo, Quantum systems, channels, information. A mathematical introduction (De Gryter, 2012).9. M. Ziman et al., “Diluting quantum information: An analysis of information transfer in system-reservoirinteractions,”Phys. Rev. A 65, 042105 (2002).10. T. Ryb´ar, S.N. Filippov, M. Ziman, and V. Buzek, “Simulation of indivisible qubit channels in collisionmodels,”J. Phys. B 45, 154006 (2012).11. F. Ciccarello, G. M. Palma, and V. Giovannetti, “Collision-model-based approach to non-Markovianquantum dynamics,”Phys. Rev. A 87, 040103 (2013).12. A. S. Holevo, “The entropy gain of infinite-dimensional quantum channels,”Doklady Math. 82, 730 (2010).13. G. G. Amosov, “Estimating the output entropy of a tensor product of two quantum channels,”Theoret.and Math. Phys. 182, 397 (2015).14. G. B. Lesovik, A. V. Lebedev, I. A. Sadovskyy, M. V. Suslov, and V. M. Vinokur, “H-theorem in quantumphysics,”Scientific Reports 6, 32815 (2016).15. S. Lloyd, “Quantum-mechanical Maxwell’s demon,”Phys. Rev. A 56, 3374 (1997).16. N. S. Kirsanov, A. V. Lebedev, M. V. Suslov, V. M. Vinokur, G. Blatter, and G. B. Lesovik, “Entropydynamics in a system of interacting qubits,”J. Russ. Laser Res. 39, 120 (2018); N. S. Kirsanov,A. V. Lebedev, I. A. Sadovskyy, M. V. Suslov, V. M. Vinokur, G. Blatter, and G. B. Lesovik, “H-theorem and Maxwell Demon in Quantum Physics,”AIP Conf. Proc. 1936, 020026 (2018).1. L. Boltzmann, “Weitere Studien ¨uber das W¨armegleichgewicht unter Gasmolek¨ulen, ”Wiener Berichte75, 62 (1872).2. L. Boltzmann, “Entgegnung auf die w¨arme-theoretischen Betrachtungen des Hrn. E. Zermelo,”Annalender Physik (Leipzig) 57, 773 (1896) [Translated and reprinted in S. G. Brush, Kinetic Theory 2, PergamonElmsford, New York (1966)].3. J. L. Lebowitz, “Statistical mechanics: A selective review of two central issues,”Rev. Mod. Phys. 71,S346 (1999).4. H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford Univ. Press, 2002).5. G. Lindblad, “On the generators of quantum dynamical semigroups,”Comm. Math. Phys. 48, 119 (1976).6. G. Lindblad, “Completely positive maps and entropy inequalities,”Comm. Math. Phys. 40, 147 (1975).7. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge Univ.Press, 2011).8. A. S. Holevo, Quantum systems, channels, information. A mathematical introduction (De Gryter, 2012).9. M. Ziman et al., “Diluting quantum information: An analysis of information transfer in system-reservoirinteractions,”Phys. Rev. A 65, 042105 (2002).10. T. Ryb´ar, S.N. Filippov, M. Ziman, and V. Buzek, “Simulation of indivisible qubit channels in collisionmodels,”J. Phys. B 45, 154006 (2012).11. F. Ciccarello, G. M. Palma, and V. Giovannetti, “Collision-model-based approach to non-Markovianquantum dynamics,”Phys. Rev. A 87, 040103 (2013).12. A. S. Holevo, “The entropy gain of infinite-dimensional quantum channels,”Doklady Math. 82, 730 (2010).13. G. G. Amosov, “Estimating the output entropy of a tensor product of two quantum channels,”Theoret.and Math. Phys. 182, 397 (2015).14. G. B. Lesovik, A. V. Lebedev, I. A. Sadovskyy, M. V. Suslov, and V. M. Vinokur, “H-theorem in quantumphysics,”Scientific Reports 6, 32815 (2016).15. S. Lloyd, “Quantum-mechanical Maxwell’s demon,”Phys. Rev. A 56, 3374 (1997).16. N. S. Kirsanov, A. V. Lebedev, M. V. Suslov, V. M. Vinokur, G. Blatter, and G. B. Lesovik, “Entropydynamics in a system of interacting qubits,”J. Russ. Laser Res. 39, 120 (2018); N. S. Kirsanov,A. V. Lebedev, I. A. Sadovskyy, M. V. Suslov, V. M. Vinokur, G. Blatter, and G. B. Lesovik, “H-theorem and Maxwell Demon in Quantum Physics,”AIP Conf. Proc. 1936, 020026 (2018).