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OPEN
Entropy exchange for infinitedimensional systems Zhoubo Duan* & Jinchuan Hou*
received: 26 July 2016 accepted: 29 December 2016 Published: 06 February 2017
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, ρ ∈ T (H ) ⊆ B (H ) which is positive and has trace 1, where (H ) and (H ) denote the von Neumann algebras of all bounded linear operators and the space of all trace-class operators with T 1 = Tr(T †T ) < ∞, respectively. Let us denote by (H ) the set of all states in the quantum system associated with H. A state ρ is called a pure state if ρ2 = 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 known1–4 that every channel Φ : (H ) → (K ) has an operator-sum representation Φ (ρ ) =
N
∑ E kρEk†,
(1)
k= 1
{E k}kN=1
where 1 ≤ N ≤ ∞ and ⊆ (H , K ) is a sequence of bounded linear operators from H into K with † ∑ kN=1 Ek E kI . Eks are called the operation elements or Kraus operators of the quantum channel Φ. The representation of Φin Eq. (1) is not unique. If both H and K are finite-dimensional, it is well known that N ≤ dim H dim K 0 with ∑ iN=1 λ i2 = 1. Extend {|i R〉 } to an orthonormal basis {|j R 〉 }{|i R〉 }, {|j R 〉 } and {|j′ R 〉 } to an orthonormal basis {|i′R〉, |j′R〉} of the system R. In the same way, extend {|iQ〉} to an orthonormal basis {|iQ〉, |lQ〉}, and {|i′Q〉} to an orthonormal basis {|i′Q〉, |l′Q〉} of the system Q. Denote the cardinal number of a set by . Let {|j R¯ 〉 } = d1, {|j′¯R〉} = d 2, {|l Q¯ 〉 } = d 3 and {|l ′¯Q〉} = d 4. Clearly, we have 9 possible cases. Case 1. d1 = d2 and d3 = d4. Let unitary operators U on system R and V on system Q be defined respectively by U|iR〉 = |i′R〉 for 1 ≤ i ≤ N and U|jR〉 = |j′R〉 for 1 ≤ j ≤ d1 = d2; V|iQ〉 = |i′Q〉 for 1 ≤ i ≤ N and V|lQ〉 = |l′Q〉 for 1 ≤ l ≤ d3 = d4. Then φ = (U ⊗ V ) ψ . Case 2. d1 = d2 and d3 0. By lemma 13, we have Tr[ − B log B + A log A] ≤ Tr[(B − A)( − log A − 1)] = Tr[ − B log A + A log A − B + A] .
(25)
Since TrA log A
OPEN
Entropy exchange for infinitedimensional systems Zhoubo Duan* & Jinchuan Hou*
received: 26 July 2016 accepted: 29 December 2016 Published: 06 February 2017
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, ρ ∈ T (H ) ⊆ B (H ) which is positive and has trace 1, where (H ) and (H ) denote the von Neumann algebras of all bounded linear operators and the space of all trace-class operators with T 1 = Tr(T †T ) < ∞, respectively. Let us denote by (H ) the set of all states in the quantum system associated with H. A state ρ is called a pure state if ρ2 = 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 known1–4 that every channel Φ : (H ) → (K ) has an operator-sum representation Φ (ρ ) =
N
∑ E kρEk†,
(1)
k= 1
{E k}kN=1
where 1 ≤ N ≤ ∞ and ⊆ (H , K ) is a sequence of bounded linear operators from H into K with † ∑ kN=1 Ek E kI . Eks are called the operation elements or Kraus operators of the quantum channel Φ. The representation of Φin Eq. (1) is not unique. If both H and K are finite-dimensional, it is well known that N ≤ dim H dim K 0 with ∑ iN=1 λ i2 = 1. Extend {|i R〉 } to an orthonormal basis {|j R 〉 }{|i R〉 }, {|j R 〉 } and {|j′ R 〉 } to an orthonormal basis {|i′R〉, |j′R〉} of the system R. In the same way, extend {|iQ〉} to an orthonormal basis {|iQ〉, |lQ〉}, and {|i′Q〉} to an orthonormal basis {|i′Q〉, |l′Q〉} of the system Q. Denote the cardinal number of a set by . Let {|j R¯ 〉 } = d1, {|j′¯R〉} = d 2, {|l Q¯ 〉 } = d 3 and {|l ′¯Q〉} = d 4. Clearly, we have 9 possible cases. Case 1. d1 = d2 and d3 = d4. Let unitary operators U on system R and V on system Q be defined respectively by U|iR〉 = |i′R〉 for 1 ≤ i ≤ N and U|jR〉 = |j′R〉 for 1 ≤ j ≤ d1 = d2; V|iQ〉 = |i′Q〉 for 1 ≤ i ≤ N and V|lQ〉 = |l′Q〉 for 1 ≤ l ≤ d3 = d4. Then φ = (U ⊗ V ) ψ . Case 2. d1 = d2 and d3 0. By lemma 13, we have Tr[ − B log B + A log A] ≤ Tr[(B − A)( − log A − 1)] = Tr[ − B log A + A log A − B + A] .
(25)
Since TrA log A
