Entropy exchange for infinite-dimensional systems

In this paper the entropy exchange for channels and states in infinite-dimensional systems are defined and studied. It is shown that, this entropy exchange depends only on the given channel and the state. An explicit expression of the entropy exchange in terms of the state and the channel is proposed. The generalized Klein’s inequality, the subadditivity and the triangle inequality about the entropy including infinite entropy for the infinite-dimensional systems are established, and then, applied to compare the entropy exchange with the entropy change.

In quantum mechanics a quantum system is associated with a separable complex Hilbert space H. A quantum state ρ is a density operator, that is, the set of all states in the quantum system associated with H. A state ρ is called a pure state if ρ 2 = ρ; otherwise, ρ is called a mixed state.
Consider two quantum systems associated with Hilbert spaces H and K respectively. Recall that a quantum channel between these two systems is a trace-preserving completely positive linear map from H ( )  into K ( )  . It is known 1-4 that every channel Φ → H K : ( ) ( )   has an operator-sum representation max{ , } such that = ∑ = F u E l k N kl k 1 , = … l M 1, 2, , . This fact is so-called the unitary freedom in the operator-sum representation for quantum channels. However, unitary freedom is no longer valid for infinite-dimensional systems 5 . In fact, what we have is so-called the bi-contractive freedom, which asserts that, if a channel   Φ → H K : ( ) ( ) has two operator-sum representations Φ = ∑ =∑ † † , then there exist contractive matrices Ω = (ω ij ) and Γ = (γ ji ) such that ω = ∑ A B i j ij j for each i and γ = ∑ B A j i ji i for each j. The converse is also true. Particularly, if Ω = (ω ij ) is an isometry so that j holds for any X. Let R and Q be two quantum systems described by Hilbert spaces H R and H Q , respectively. Suppose that the joint system RQ is prepared in a pure entangled state Ψ RQ and the initial state of system Q is ρ = Ψ Ψ Tr Q R RQ RQ . The system R is dynamically isolated and has a zero internal Hamiltonian, while the system Q undergoes some evolution that possibly involves interaction with the environment E. The final state of RQ is possibly mixed and is described by the density operator ρ RQ′ . Thus, if the dynamical evolution that Q is subjected to is described by Φ Q , then the final state is ρ = ⊗ Φ |Ψ 〉〈Ψ | For finite-dimensional systems there is another quantity concerning channels and states that is intrinsic to subsystem Q. This quantity is called the entropy exchange. For a given state ρ Q and a given channel Φ Q in a finite-dimensional system Q, recall that the entropy exchange S e is defined by refs 1, 6 and 12-14 RQ and Ψ RQ is a purification of ρ Q . It was shown 1,6 that the entropy exchange S e is independent of the choice of purification Ψ RQ of the state ρ Q . It was also shown 1 that the entropy exchange S e has another explicit formulation Q the sequence of the Kraus operators of an operator-sum representation of Φ Q and the minimum is taken over all operator-sum representations of Φ Q .
It is clear that Eq. (3) can be naturally generalized to infinite-dimensional case to give a definition of the entropy exchange for channels and states in infinite-dimensional systems. In continuous variable systems, Chen and Qiu 15 studied the coherent information I e = S(ρ Q ) − S e of the thermal radiation signal ρ Q transmitted over the thermal radiation noise channel, one of the most essential quantum Gaussian channels, and derived an analytical expression for computation of the value of it. However, as the von Neumann entropy S(ρ) of a non-Gaussian state in an infinite-dimensional system may be + ∞ 16 , we may have S e = + ∞ . In this paper we consider general states and channels and show that the definition Eq. (3) does not depend on the choice of the purification of the state either, and Eq. (4) is still true for infinite-dimensional systems.
For finite-dimensional systems, it is known 6 that the entropy exchange is larger than or equal to the change of the entropy, that is, The second purpose of the present paper is to compare the entropy exchange with the change of the entropy and to check whether or not the inequality (5) is still valid in infinite-dimensional systems. We show that, for infinite-dimensional case, what we can have are the following three inequalities: To prove the above inequalities, we need the subadditivity and the triangle inequality of von Neumann entropies for infinite-dimensional quantum systems. These two inequalities were established in a more general frame of von Neumann algebras for normal states with finite entropy 17 . However, for the convenience of readers, we present some elementary proofs including the case of infinite von Neumann entropy here by establishing the generalized Klein's inequality for infinite-dimensional case. We also give some examples which illustrates that the entropy exchange is different from the change of entropy.

Entropy exchange for infinite-dimensional systems
In this section, we mainly give some properties of the entropy exchange for infinite-dimensional systems. In fact, the results in this section hold for both finite-and infinite-dimensional cases.
Recall that a linear operator U from a Hilbert space into another is called an isometry if = † Proof. By the assumption, |φ〉 and |ψ〉 have respectively the Schmidt decompositions φ λ to an orthonormal basis {|i′ R 〉 , |j′ R 〉 } of the system R. In the same way, extend {|i Q 〉 } to an orthonormal basis {|i Q 〉 , |l Q 〉 }, and {|i′ Q 〉 } to an orthonormal basis {|i′ Q 〉 , |l′ Q 〉 } of the system Q. Denote the cardinal number of a set  by . Let unitary operators U on system R and V on system Q be defined respectively by Case 2. d 1 = d 2 and d 3 < d 4 . Let U be defined as in Case 1 and V be defined by V|i Q 〉 = |i′ Q 〉 for 1 ≤ i ≤ N and Then U is a unitary operator on system R and V is an isometry V on system Q sat- Case 3. d 1 = d 2 and d 3 > d 4 . Define U on system R as in Case 1 and define V on system Q by In a similar way, it is obvious to see that There is an isometry U on system R and a unitary V on system Q such that There is an isometry U on system R and a coisometry V on system Q such that There is a coisometry U on system R and a unitary V on system Q such that There is a coisometry U on system R and an isometry V on system Q such that are purifications of a state ρ Q to a composite system RQ, then there exists an isometry V R on system R such that either have identical Schmidt coefficients. Making use of lemma 1, there is an isometry or a coisometry U R on system R such that are two purifications of a state ρ Q to a composite system RQ, and each is subjected to the same evolution superoperator ⊗ Φ I R Q with the resulting states respectively ρ ′ RQ 1 Proof. By lemma 2, there exists an isometry transformation V R acting on system R such that either . Wit hout l o ss of ge ne r a l it y, assu me t hat holds, then we have

Lemma 4. If A is a bounded self-adjoint operator on a complex Hilbert space and f is a continuous function on σ(A), the spectrum of A, then, for any isometric operator V, we have
is a bounded closed set. Because f is a continuous function on σ(A), we can apply the Weierstrass theorem to find a sequence of polynomials {P n } such that The following result reveals that, for infinite-dimensional systems, similar to the entanglement fidelity 5 , the value of entropy exchange is also independent of the choice of purifications of the initial state.
By lemma 3, there is an isometry V R so that the resulting states ρ ′ RQ 1 . Then, by lemma 4, as desired. ◽ In the sequel, analogue to Eq. (4) for finite-dimensional systems, we derive an explicit expression for S e in terms of ρ Q and Φ Q for infinite-dimensional systems.
To do this, we need some more lemmas.
Proof. Fix an orthonormal basis {|i〉 } of H B . Then B can be written in a matrix B = (b ij ), and ⊗ A B and ρ can be w r itten in op erator mat r ices , resp e c t ively. Thus we have Proof. By lemma 6 and with the same symbols as in the proof of lemma 6, we have is an operator-sum representation for the channel Φ Q . If ρ Q is a state of system Q and Ψ RQ is a purification of ρ Q into composite system RQ, then, for any μ, let |Φ 〉 = ⊗ |Ψ 〉 Q is a sequence of Kraus operators of an operator-sum representation of Φ Q , that is, Q , and the minimum is taken over all operator-sum representations of Φ Q .
Proof. For given state ρ Q and quantum channel Φ Q , if {A μ } is the sequence of Kraus operators of an operator-sum representation of Φ Q , then by lemma 8 and the discussion previous theorem 9, , that is, W is the matrix of ρ E in an appropriate basis; then S e = S(W). Since W is a matrix representation of the environmental density operator, it may be diagonalized by a unitary matrix U = (u μν ), i.e.,

Comparison with entropy change
The entropy exchange S e simply characterizes the information exchange between the system Q and the external world during the evolution given by Φ Q . It is interesting to explore the relationship between the entropy exchange and the entropy change during the same evolution. Such a question was studied for finite-dimensional systems and the inequality (5) was established 6 . However, the inequality (5) does not always valid in infinite-dimensional case. To solve the question for infinite-dimensional systems, we need the subadditivity and the triangle inequality of von Neumann entropies for infinite-dimensional systems which was established in the textbook 17 for normal states with finite entropy in a more general frame of von Neumann algebras. However, we have to deal with the states with infinite entropy. Here we present somewhat elementary proofs for these two inequalities by generalizing the generalized Klein's inequality from finite-dimensional systems to the infinite-dimensional systems and clarify when the inequalities are still valid for states with infinite entropy.
The following lemma 10 and 11 are obvious 18 .
Lemma 10. If f is a convex (concave) function, then f is continuous.

Lemma 12. Suppose f is a convex (concave) function and A is a bounded self-adjoint operator on a Hilbert space H
Proof. By lemma 10, f is continuous. Let be the spectral decomposition of the self-adjoint operator A. Assume that f is convex. For any unit vector φ ∈ H, denote by μ the probability measure defined by Similarly, if f is concave, then one gets ◽ In finite-dimensional case, the following result is valid and is called the generalized Klein's inequality. We generalize it to infinite-dimensional case.

Lemma 14. (Generalized Klein's inequality) Let A, B be two positive operators of trace-class on a Hilbert space H. If
, then Proof. Take f so that f(x) = − x log x for x > 0 and f(0) = 0. Then f(x) is a concave function with Making use of this result, we see that the relative entropy ρ σ ρ ρ σ = − S ( , ) Tr( (log log )) is also non-negative for the infinite-dimensional quantum systems whenever S(σ) < ∞ .

Corollary 15. For any two density operators ρ,
Proof. Since ρ, σ are two density operators, Tr ρ = Tr σ = 1. Substituting these in the inequality (24), we have ρ ρ ρ σ − ≤ − Tr( log ) Tr( log ). ◽ Next, we apply the corollary 15 to prove the subadditivity inequality (27) and the triangle inequalities (29) and (30) for Von Neumann entropy. If S(σ) < ∞ , corollary 15 and the above equations imply ρ ρ σ B holds for any bipartite states and is called the triangle inequality. In infinite-dimensional case, this inequality may be not valid except the case when both S(ρ A ), S(ρ B ) are finite. What we can have is the triangle inequalities of the following kind. Proof. To prove the inequality (29), we introduce a system C which purifies the system AB. Let Ψ ABC be a purification of ρ AB ; then Since Ψ ABC is a pure state, S(ρ AB ) = S(ρ C ) and S(ρ AC ) = S(ρ B ). Hence the previous inequality is the same as B . By symmetry between the systems A and B one sees that ρ ρ ρ A is also true. ◽ Now, we relate the entropy exchange to change in the entropy of the system Q for infinite-dimensional quantum systems.
Theorem 18. For any evolution Φ Q and initial state ρ Q in an infinite-dimensional system Q, with ρ Q′ = Φ Q (ρ Q ), the following inequalities are true. Q , which means that the entropy exchange is not less than the change in entropy of the system Q. In general, the entropy exchange is different from the change in entropy of the system Q, that is, Q holds for some channels and states.
Examples. The following is an example for finite-dimensional case.
with dim H Q = 2. The bit flip channel Φ Q flips the state of a qubit from |0〉 to |1〉 with probability 1 − p. It has operation elements On the other hand, note that Ψ = + ( 01 10 ) D a a ( ) exp( ) is the displacement operator, and N n is the average photon number of the output state if the input is the vacuum. If the input state ρ Q is a thermal noise signal with its average photon number N s , then the output state ρ Q′ will be a thermal noise signal with its average photon number N s + N n 19 . We know that the entropy of any Gaussian state ρ is finite and is formulated by S(ρ) = g(N), where g(x) = (x + 1) ln(x + 1) − x ln x is a monotonically increasing convex function and N is the average number of photons of the Gaussian state ρ. Thus, we can get s . Now, we introduce a reference system R, initially, the joint system RQ is prepared in a pure entangled states Ψ RQ with ρ Ψ Ψ = Tr R RQ RQ Q , i.e., the pure state Ψ RQ is a purification of the state ρ Q . The system R is dynamically isolated and has a zero internal Hamiltonian, while the system Q undergoes an internal with above thermal noise channel Φ Q . The final state of RQ is described by the state ρ RQ′ . Then the entropy exchange S e = S(ρ RQ′ ) = g(N 1 ) + g(N 2 ), where

Discussion
The notion of entropy exchange can be introduced in infinite-dimensional quantum systems with the same form as that in finite-dimensional systems if we allow it may take infinity value. Thus, for a state ρ Q and a channel Φ Q in an infinite-dimensional system Q, the entropy exchange S e is defined as S e = S(ρ RQ′ ), where ρ = ⊗ Φ Ψ Ψ ′ I ( )

RQ R Q RQ
RQ and Ψ RQ is a purification of ρ Q in a larger system RQ. This quantity does not depend on the choice of purifications of the state ρ Q and characterizes the information exchange between the system Q and the external world during the evolution given by Φ Q . An explicit expression for S e in terms of ρ Q and Φ Q is established, which asserts that = −∑ µ µ Q the sequence of Kraus operators in an operator-sum representation of Φ Q , and the minimum is taken over all operator-sum representations of Φ Q . In general, the entropy exchange is not equal to the change in entropy We also give some examples which illustrates that the entropy exchange is different from the change of entropy. In general the entropy exchange is larger than the change of entropy.