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arXiv:2401.14344 · Infinite Dimensional Analysis Quantum Probability and Related Topics

From the Choi formalism in infinite dimensions to unique decompositions of generators of completely positive dynamical semigroups

Frederik vom Ende

Published Jul 19, 2024• 0 citations
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Abstract

Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.

DOI
10.1142/s0219025724500115

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Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.
We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product.
In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.
Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.
Let H\mathcal{H}H be a complex finite-dimensional or infinite-dimensional separable Hilbert space, B(H)\mathcal{B(H)}B(H) and T(H)\mathcal{T(H)}T(H) be the Banach spaces of all bounded linear operators and of all trace class operators on H,\mathcal{H},H, respectively. In this paper, we give a concrete description of the linear maps Φ:T(H)→B(H⊗H)Φ:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}Φ:T(H)→B(H⊗H) that are continuous relative to the norm topology and covariance under unitary evolution (i.e., Φ(UXU∗)=(U⊗U)Φ(X)(U∗⊗U∗)Φ(UXU^*)=(U\otimes U)Φ(X)(U^*\otimes U^*)Φ(UXU∗)=(U⊗U)Φ(X)(U∗⊗U∗) for all X∈T(H)X\in\mathcal{T(H)}X∈T(H) and unitary operators U∈B(H)).U\in\mathcal{B(H)}).U∈B(H)). Using this, we obtain the equivalent conditions for this class of maps to be self-adjoint or positive. As a corollary, we get that the virtual broadcasting map Bvb:T(H)→B(H⊗H)\mathcal{B}_{vb}:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}Bvb​:T(H)→B(H⊗H) with the form Bvb(X)=12[S(I⊗X)+S(X⊗I)]\mathcal{B}_{vb}(X)=\frac{ 1}{2}[S(I\otimes X)+S(X\otimes I)]Bvb​(X)=21​[S(I⊗X)+S(X⊗I)] is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where S∈B(H⊗H)S\in\mathcal{B(H\otimes H)}S∈B(H⊗H) is the swap operator. Moreover, the linear maps Ψ:B(H)→B(H⊗H)Ψ:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}Ψ:B(H)→B(H⊗H) that are continuous relative to the W∗W^*W∗-topology and covariance under unitary evolution are also characterized.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
Abstract The Lindblad equation embodies a fundamental paradigm of the quantum theory of open systems, and the Gorini–Kossakowski–Sudarshan–Lindblad generation theorem says precisely which superoperators can appear on its right-hand side. These are the generators of completely positive (CP) trace-preserving (or nonincreasing) semigroups. We prove a generalization, with time-dependent generator, as an application of an investigation of the geometry of the class of CP maps. The treatment of the finite-dimensional setting is based on a basis-free Choi–Jamiołkowski type isomorphism. The infinite-dimensional case is bootstrapped from the finite-dimensional theory via a sequence of finite-dimensional approximations. Kraus decomposition is established along the way, in the guise of an extremal decomposition of the closed convex cone of CP maps. No appeal is made to results from the representation theory of operator algebras.