Some Properties of Generalized Quantum Operations
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Some Properties of Generalized Quantum Operations Yuan Li · Xiu-Hong Sun · Yan-Ni Dou
Received: 27 December 2012 / Accepted: 3 January 2013 / Published online: 12 January 2013 © Springer Science+Business Media New York 2013
Abstract Let B(H) be the set of all bounded linear operators on the separable Hilbert space H. A (generalized) quantum operation is a bounded linear operator defined on B(H), ∗ which has the form ΦA (X) = ∞ i=1 Ai XAi , where Ai ∈ B (H) (i = 1, 2, . . .) satisfy ∞ ∗ i=1 Ai Ai ≤ I in the strong operator topology. In this paper, we establish the relation† with respect to the ship between the (generalized) quantum operation ΦA and its dual ΦA set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel. Keywords Quantum operation · Fixed point · Noiseless subspace
1 Introduction It is well known that a quantum mechanical system is represented by a complex Hilbert space H, and a bounded self-adjoint operator A on H is called a bounded quantum observable. Let B(H) be the set of all bounded linear operators on H and T (H) be the set of all trace class operators on H. For an operator A ∈ B(H), we denote by N (A), R(A) and σ (A) the null space, the range and the spectrum of A, respectively. A partial isometry is a linear transformation that is isometric on the orthogonal complement of its kernel. Let PM represent the orthogonal projection onto the subspace M ⊆ H.
Y. Li () · Y.-N. Dou College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China e-mail: [email protected] X.-H. Sun School of Science, Xi’an University of Science and Technology, Xi’an 710054, People’s Republic of China
2084
Int J Theor Phys (2013) 52:2083–2091
A (generalized) quantum operation is a bounded linear operator defined on B(H) which has the form ΦA (B) =
∞
Ai BA∗i ,
(1.1)
i=1
∗ where Ai ∈ B(H) (i = 1, 2, . . .) satisfies ∞ i=1 Ai Ai ≤ I in the strong operator topology and ∗ A = {Ai , Ai : Ai ∈ B(H), i = 1, 2, . . .} is called a family of Kraus operators for ΦA . In this case, the dual of ΦA is defined by † (X) = ΦA
∞
A∗i XAi
for X ∈ T (H).
i=1 † It is clear that | Tr[ΦA (X)Y ]| = | Tr[XΦA (Y )]| ≤ ΦA Y Tr(|X|), for X ∈ T (H) and † † on T (H). In general, ΦA can’t be extended Y ∈ B(H), so ΦA (X) ∈ T (H) is well ∞defined ∗ from T (H) into B(H). However, if i=1 Ai Ai is convergent in the strong operator topology, † is well defined on B(H) and is normal. A normal completely positive map Φ, which then ΦA is trace preserving (Φ † (I ) = I ) is called a quantum channel. We say that B ∈ B(H) is a fixed point ΦA , if ΦA (B) = B. We also denote A = {S : SB = BS for B ∈ A}. In more recent papers [1–6, 11–19, 21], the fixed points of quantum operations attracts the attention of a number of mathematicians. In particular, Arias, Gheondea, and Gudder in [1] extensively investigated
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