LETTER TO THE EDITOR Entropy production rates of bistochastic strictly contractive quantum channels on a matrix algebra

We derive, for a bistochastic strictly contractive quantum channel on a matrix algebra, a relation between the contraction rate and the rate of entropy production. We also sketch some applications of our result to the statistical physics of irreversible processes and to quantum information processing. PACS numbers: 03.65.Yz, 03.67.-a, 05.30.-d Let A be the algebra of observables (say, a C*-algebra with identity), associated with a quantum-mechanical system Σ. A general evolution of Σ is described, in the Heisenberg picture, by a map T : A → A which is (i) unital: T (11) = 11, and (ii) completely positive: for any nonnegative integer n, the map T⊗id : A⊗Mn → A⊗Mn, where Mn is the algebra of n×n complex matrices, sends positive operators to positive operators [1]. We shall henceforth refer to such maps as (quantum) channels [2]. If Σ is an N -level system, then its algebra of observables is isomorphic to MN , which is exclusively the case we shall consider in this letter. A celebrated result of Kraus [1] then says that, for any channel T , there exists a collecton of at most N operators Vi ∈ MN , which we shall call the Kraus operators associated with T , such that (i) T (A) = ∑ i V ∗ i AVi, and (ii) ∑ i V ∗ i Vi = 11. It is now easy to see that T (A ∗) = T (A)∗ for all A ∈MN , i.e., any channel maps Hermitian operators to Hermitian operators. Given a channel T , the corresponding Schrödinger-picture channel T̂ is defined via the duality tr [T̂ (A)B] = tr [AT (B)], whence it follows that T̂ is a completely positive map which preserves the trace, i.e., tr T̂ (A) = tr A for all A ∈ MN . In other words, T̂ maps the set DN of N × N density matrices into itself. Furthermore, in terms of the Kraus operators Vi, we have T̂ (A) = ∑ i ViAV ∗ i , so that T̂ (A ∗) = T̂ (A)∗ as well. The set {T }n∈N is a discrete-time quantum dynamical semigroup generated by T , i.e., T T = T n+m and we take T 0 ≡ id (the identity channel). It is easy to show that, § E-mail address: [email protected] Letter to the Editor 2 for any channel T , there exists at least one density operator ρ such that T̂ (ρ) = ρ [3]. A question of clear physical importance is to determine whether the dynamics generated by T is relaxing [3], i.e., whether there exists a density operator ρ such that, for any density operator σ, the orbit {T̂ n(σ)} converges to ρ in the trace norm ‖A‖1:= tr (A∗A)1/2. One way to show that a dynamics is relaxing relies on the so-called Liapunov’s direct method [4]. Let X be a compact separable space, and let φ : X → X be a continuous map, such that (i) φ has a unique fixed point x0 ∈ X , and (ii) there exists a Liapunov function for φ, i.e., a continuous functional S on X such that, for all x ∈ X , S[φ(x)] ≥ S(x), where equality holds if and only if x ≡ x0. Then, for any x ∈ X , the sequence {φn(x)} converges to x0. Let T be a bistochastic channel, i.e., one for which T (11) = T̂ (11) = 11. If we treat MN as a Hilbert space with the Hilbert-Schmidt inner product, 〈A,B〉:= tr (A∗B), then an easy calculation shows that the Schrödinger-picture channel T̂ is precisely the adjoint of T with respect to 〈·, ·〉, i.e., 〈A, T (B)〉 = 〈T̂ (A), B〉 for all A,B ∈Mn. The composite map T ◦ T̂ (which we shall write henceforth as T T̂ ) is also a bistochastic channel, which is, furthermore, a Hermitian operator with respect to 〈·, ·〉. In [5], Streater proved the following result. Theorem 1 Let T : MN →MN be a bistochastic channel. Suppose that T̂ is ergodic with a spectral gap γ ∈ [0, 1), i.e., (i) up to a scalar multiple, the identity matrix 11 is the only fixed point of T̂ in all of MN , and (ii) the spectrum of T T̂ is contained in the set [0, 1− γ] ∪ {1}. Then, for any σ ∈ DN , we have S[T̂ (σ)]− S(σ) ≥ γ 2 ‖σ −N11‖2, (1) where S(σ):=− tr (σ ln σ) is the von Neumann entropy of σ and ‖A‖2:=[ tr (A∗A)]1/2 is the Hilbert-Schmidt norm of A. In other words, if a bistochastic channel T̂ is ergodic, then the dynamics generated by T is relaxing by Liapunov’s theorem†. Furthermore, the relaxation process is accompanied by entropy production at a rate controlled by the spectral gap. Now we have an interesting “inverse” problem. Consider a bistochastic channel T on MN with T̂ strictly contractive [6]. That is, T̂ is uniformly continuous on DN (in the trace-norm topology) with Lipschitz constant C ∈ [0, 1): for any pair σ, σ′ ∈ DN , we have ‖T̂ (σ)− T̂ (σ′)‖1 ≤ C‖σ − σ′‖1. Then by the contraction mapping principle [7], N−111 is the only density matrix left invariant by T̂ , and furthermore ‖T̂ (σ) −N−111‖1 → 0 as n → ∞ for any σ ∈ DN , i.e., the dynamics generated by T is relaxing. The question is, does the entropy-gain estimate (1) hold, and, if so, how does the spectral gap γ depend on the contraction rate C? † Endowing the set DN with the trace-norm topology takes care of all the continuity requirements imposed by Liapunov’s theorem. Letter to the Editor 3 This problem was motivated in the first place by the following observation. In the case of M2, the action of a bistochastic strictly contractive channel T̂ can be given a direct geometric interpretation. Recall that the density matrices in M2 are in a oneto-one correspondence with the points of the closed unit ball in R. Then the image of D2 under a strictly contractive channel T̂ with contraction rate C will be contained inside the closed ball of radius C, centered at the origin [6], i.e., the image of D2 under T̂ will consist only of mixed states. This geometric illustration suggests that the rate of entropy increase under T̂ must be related to the contraction rate. Now even though in the case of MN with N ≥ 3 we no longer have such a convenient geometric illustration, nevertheless it seems plausible that the rate of entropy production under a bistochastic strictly contractive channel would still be controlled by the contraction rate. Indeed it turns out that the contraction rate is related to the rate of entropy production, as stated in the following theorem. Theorem 2 Let T be a bistochastic channel on MN , such that T̂ is strictly contractive with rate C. Then T̂ is ergodic with spectral gap γ ≥ 1− C, so that, for any σ ∈ DN , we have S[T̂ (σ)]− S(σ) ≥ 1− C 2 ‖σ −N11‖2. (2) Proof. We first prove that T̂ is ergodic. As we noted before, T and T̂ are adjoints of each other with respect to the Hilbert-Schmidt inner product. Using the KadisonSchwarz inequality [8] Φ(A∗A) ≥ Φ(A)∗Φ(A) for any channel Φ on a C*-algebra A, as well as the fact that tr T (A) = tr [T̂ (11)A] = tr A for any A ∈MN , we find that ‖T (A)‖2 = tr [T (A)∗T (A)] ≤ tr [T (A∗A)] = tr (A∗A) = ‖A‖2, and the same goes for T̂ . That is, both T and T̂ are contractions on MN (in the Hilbert-Schmidt norm), hence their fixed-point sets coincide [9]. By hypothesis, T̂ leaves invariant the density matrix N−111, which is invertible. In this case a theorem of Fannes, Nachtergaele, and Werner [10, 11] says that T (X) = X if and only if ViX = XVi for all Vi, where Vi are the Kraus operators associated with T . It was shown in [6] that if T̂ is strictly contractive, then the set of all X such that ViX = XVi for all Vi consists precisely of multiples of the identity matrix. We see, therefore, that T (X) = X if and only if X = χ11 for some χ ∈ C, whence it follows that T̂ (X) = X if and only if X is a multiple of 11. This proves ergodicity of T̂ . Our next task is to establish the spectral gap estimate γ ≥ 1 − C. Let X be a Hermitian operator with tr X = 0. In that case we can find a density operator ρ and Letter to the Editor 4 a sufficiently small number 2 > 0 such that σ:=ρ + 2X is still a density operator [3]‡. Then ‖T̂ (X)‖1 = (1/2)‖T̂ (σ)− T̂ (ρ)‖1 ≤ (C/2)‖σ − ρ‖1 = C‖X‖1. (3) Because one is a simple eigenvalue of both T and T̂ , it is also a simple eigenvalue of T T̂ . Hence 1−γ (which we may as well assume to belong to the spectrum of T T̂ ) is the largest eigenvalue of the restriction of T T̂ to traceless matrices. Let Y be the corresponding eigenvector. Without loss of generality we may choose Y to be Hermitian§. Then, using Eq. (3) and the fact that ‖Φ(A)‖1 ≤ ‖A‖1 for any trace-preserving completely positive map Φ [13], we may write (1− γ)‖Y ‖1 = ‖T T̂ (Y )‖1 ≤ ‖T̂ (Y )‖1 ≤ C‖Y ‖1, which yields the desired spectral gap estimate. The entropy gain bound (2) now follows from Theorem 1. ¥ Remark. Eq. (3) can also be proved using the following finite-dimensional specialization of a general result due to Ruskai [13]. If T : MN → MN is a channel, then sup A=A∗; tr A=0 ‖T̂ (A)‖1 ‖A‖1 = 1 2 sup ψ,φ∈CN ;〈ψ|φ〉=0 ‖T̂ (|ψ〉〈ψ| − |φ〉〈φ|)‖1. (4) Because T̂ is strictly contractive, the right-hand side of (4) is bounded from above by C, and (3) follows. The supremum on the left-hand side of (4) is the “Dobrushin contraction coefficient,” studied extensively by Lesniewski and Ruskai [14] in connection with the contraction of monotone Riemannian metrics on quantum state spaces under (duals of) quantum channels. ¤ Note that in some cases the sharper estimate S[T̂ (σ)]− S(σ) ≥ 1− C 2 2 ‖σ −N11‖2 (5) may be shown to hold. Consider, for instance, the case T = T̂ , so that the eigenvalues of T̂ are all real. Let λ1, . . . , λL, L = N 2 − 1, be the eigenvalues of T̂ that are distinct from unity. Then we claim that maxj |λj| ≤ C, which can be proved via reductio ad absurdum. Suppose that there exists some X (which we may take to be Hermitian) with tr X = 0 such that T̂ (X) = λX with |λ| > C. We may use the same trick as in the proof above to show that there exist two density operators, σ and ρ, such that ‖T̂ (σ)− T̂ (ρ)‖1 > C‖σ− ρ‖1, which would contradict the strict contractivity of T̂ . ‡ This may be seen as a simple consequence of the following fact [12]. The set Dinv N of all invertible N×N density matrices is a smooth manifold, where the tangent space at any ρ ∈ Dinv N can be naturally identified with the set of N ×N traceless Hermitian matrices. § Recall that 1 − γ is real, and T T̂ (A∗) = [T T̂ (A)]∗ for all A, which implies that Y + Y ∗ is also an eigenvector of T T̂ with the same eigenvalue. Letter to the Editor 5 Because T T̂ = T̂ , we have 1−γ = (maxj |λj|) ≤ C, which confirms (5). Furthermore, using a theorem of King and Ruskai [15], the bound (5) can be established for all bistochastic strictly contractive channels on M2, as well as for tensor products of such channels. The proof of this last assertion goes as follows. Let T be a channel on M2 such that T̂ is strictly contractive. Then T̂ is ergodic, the proof of which can be taken verbatim from the proof of Theorem 2. It is left to show that 1− γ ≤ C. Any density operator in M2 can be written as