Tensor Products of Quantum Mappings
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TENSOR PRODUCTS OF QUANTUM MAPPINGS S. N. Filippov
UDC 519.72, 530.145
Abstract. In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2 . For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive and Φt . A criterion of annihilation of entanglement by an divisibility of dynamical mappings Φ⊗2 t arbitrary qubit mapping Φ⊗2 is found. Keywords and phrases: quantum channel, complete positivity, positive mapping, divisibility, tensor product. AMS Subject Classification: 15A69, 46L06
1. Introduction. Linear mappings naturally arise in the problems of quantum evolution of density operators (see [6, 27, 39]) since the fundamental quantum equations of motion are linear differential equations of the first order in time. In this paper, we consider finite-dimensional quantum systems whose states are described by linear density operators (t) ∈ B(H), where H is a finite-dimensional Hilbert (unitary) space, dim H = d < ∞, and B(H) is the set of operators acting on H. The density operator (t) is Hermitian († (t) = (t)) and nonnegative definite (ψ|† (t)|ψ ≥ 0 for all |ψ ∈ H; here and below we use the standard Dirac notation), which has a unit trace (tr[(t)] = 1). In the case of the absence of initial correlations between a quantum system and its environment, the evolution is described by a dynamic mapping (t) = Φt [(0)], where Φt is a completely positive, tracepreserving mapping called a quantum channel (see, e.g., [27]). The physical requirement of complete positivity instead of simple positivity of the mapping is explained by the fact that the initial state of the system considered can be entangled with another (auxiliary) system (see [31, 48, 49]). The auxiliary system can have an arbitrary dimension k; its trivial evolution is governed by the identity mapping Idk . The density matrix of the initial entangled state must turn into a certain density matrix; therefore, the mapping Φt ⊗ Idk must be positive for any k: this is the definition of complete positivity of the mapping Φt . It is remarkable that for completely positive mappings, there exists a convenient criterion based on the Choi–Jamiolkowski isomorphism (see [7, 9, 32]), while for positive mappings such a criterion has not been found in the general case. The existence of entangled states implies the structure of the tensor product of the space H. An entangled state of a two-part system is a density operator AB ∈ B(HA ) ⊗ B(HB ), which cannot be represented as the closure of the convex sum of tensor products of local density operators (see [53]), i.e., (k) (k) p k A ⊗ B . (1) AB = k
States described by the right-hand side of Eq. (1) are said to be separable (or disentangled); they can be prepared by using local operations and classical communication in remote laboratories A and B. Assume that two physical carriers of information A and B evolve independently. In particular, this occurs
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