The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity of two quantum operations and determine what kind of measurement statistics are preserved by a unital quantum operation. Our work extends the results of Lee et al (2013 J. Phys. A: Math. Theor. 46 205305) to the infinite dimensional case and also corrects part (a) of theorem 6 in Lee et al (2013 J. Phys. A: Math. Theor. 46 205305).
Quantum operations fixing a convex cone of density operators on T(H)
The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity of two quantum operations and determine what kind of measurement statistics are preserved by a unital quantum operation. Our work extends the results of Lee et al (2013 J. Phys. A: Math. Theor. 46 205305) to the infinite dimensional case and also corrects part (a) of theorem 6 in Lee et al (2013 J. Phys. A: Math. Theor. 46 205305).
We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation ϕ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with ϕ, we can completely characterize the kind of quantum states that are fixed by ϕ. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of St{\o}rmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.
We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation ϕ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with ϕ, we can completely characterize the kind of quantum states that are fixed by ϕ. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states --- a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of Størmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on finite-dimensional quantum state.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten p -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗ -algebra.
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.
Let {o^,..., a) n ] and {<TI,..., a n ] be finite sets of normal states over the bounded linear operators acting over some infinite-dimensional separable Hilbert-space.In the paper mutually equivalent sets of necessary and sufficient conditions are derived that there exists a completely positive, unity preserving linear map Ttransforming 0-1,..., a n simultaneously into o)i..., o) n : a) k = a k ° T,k = l,... 9 n.Of particular interest is this 6 '/z-tuple-problem" in case of pairs of density operators (normal states), i.e. for n=2.In this situation possible connections with the notion of "generalized transition probability" are analyzed, and at least in case of normal states over the bounded operators a characterization of functionals of this type is proposed and applied. § lo Main Results, ExamplesLet jtf be a unital C*-algebra ? with unit 7, topological dual jtf*, and group of unitary elements <%(j&).To each Fe ^(j/) let us associate a bounded linear operator T v over the dual by T ¥ (a))(X) = co(VXV*) = co v (X), for all XEJ^ and any COG jtf*.On the set of bounded linear operators ^(j/*) over j/* we may introduce the weak operator topology (w-topology) which is characterized by the system of semi-norms {q^xi coe.s/*,XEJ/} given by q (0 ,x(T) = \T(a>)(X)\, with Te^(X*).Then, by % u (<xf} the weak closure of the convex hull of the operators T v will be denoted: ^M(j^) = conv{T F : VE<%(JZ?)} W .In this paper as an essential result the following theorem is proved (proofs of most of the assertions of this part are given In § §3-5): LI Ttoeoremo Let 3? be an infinite-dimensional separable complex