Abstract We study completely positive and trace-preserving equivariant maps between operators on irreducible representations of $${{\,\textrm{SU}\,}}(2)$$ SU ( 2 ) . We find asymptotic approximations of channels in the limit of large output representation and we compute traces of functions of channel outputs. Our main tool is quantization using coherent states. We provide quantitative error bounds for various semiclassical formulas satisfied by quantizations of functions on the sphere.
Abstract We study completely positive and trace-preserving equivariant maps between operators on irreducible representations of SU(2) SU ( 2 ) . We find asymptotic approximations of channels in the limit of large output representation and we compute traces of functions of channel outputs. Our main tool is quantization using coherent states. We provide quantitative error bounds for various semiclassical formulas satisfied by quantizations of functions on the sphere.
In this thesis, we introduce EPOSIC channels, a class of SU(2) -covariant quantum channels. For each of them, we give a Stinespring representation, a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that these channels are the extreme points of all SU(2) -irreducibly covariant channels. As an application of these channels to the theory of quantum information, we study the minimal output entropy of EPOSIC channels, and show that a large class of these channels is a potential example of violating the well-known problem, the additivity problem. We determine the cases where their minimal output entropy is not zero, and obtain some partial results on the fulfillment of their entanglement breaking property. We find a bound of the minimal output entropy of the tensor product of two SU(2) -irreducibly covariant channels. We also get an example of a positive map that is not completely positive.
In matrix algebras, we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the Weyl operators. In particular, we analyze the irreducibly covariant quantum channels, that is, the completely positive and trace-preserving linear maps. Interestingly, imposing additional symmetries leads to the so-called generalized Pauli channels, which were recently considered in the context of the non-Markovian quantum evolution. Finally, we provide examples of irreducibly covariant positive but not necessarily completely positive maps.
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.
We develop an approximation approach to infinite-dimensional quantum channels based on a detailed investigation of continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair “channel, input state.” Obtained results are then applied to the problems of continuity of the χ -capacity as a function of a channel, strong additivity of the χ -capacity for infinite-dimensional channels, and approximating representation for the convex closure of the output entropy of an arbitrary quantum channel.
We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II error exponent equals the regularized Umegaki channel relative entropy minimized over the alternatives, and both defining limits exist. Inputs may be entangled across channel uses and with a reference system. The main technical result constructs exactly trace-preserving approximations with exponentially small diamond error, dominated in completely positive order by free channels at every rate above the common exponent. The proof combines uniform auxiliary map approximation, iterative reduction of the domination rate, and tensor amplification. It yields asymptotic equipartition with trace-preserving smoothing at every subexponentially vanishing error, an exponential strong converse, and stability under vanishing diamond-norm perturbations. Under additional permutation invariance and closure under insertion of a fixed faithful input state and discarding of its output, we also prove an explicit finite-block completion bound. This quantitative construction uses weighted discarding, local operator corrections, and comparison with auxiliary extensions. Applications include state-preserving, coherence-restricted, covariant, entanglement-breaking, and positive-partial-transpose channels. For isometric targets against entanglement-breaking or positive-partial-transpose alternatives, we obtain exact finite-block testing and smoothing formulas.
From our recently developed representation of the path integral in terms of coherent states of the dynamical group SU(1,1), we obtain the semiclassical limit as a large-N limit. Specifically, the parameter k=21(l+2N) which labels the relevant SU(1,1) representations plays the role of ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. A phase-integral quantization rule is given and used to calculate the energy eigenvalues for various isotropic anharmonic oscillators. We obtain results comparable to the first-order JWKB calculations even at N=1.