General open-system quantum evolution in terms of affine maps of the polarization vector

Abstract

The operator-sum decomposition (OS) of a mapping from one density matrix to another has many applications in quantum information science. To this mapping there corresponds an affine map which provides a geometric description of the density matrix in terms of the polarization vector representation. This has been thoroughly explored for qubits since the components of the polarization vector are measurable quantities (corresponding to expectation values of Hermitian operators) and also because it enables the description of map domains geometrically. Here we extend the OS-affine map correspondence to qudits, briefly discuss general properties of the map, the form for particular important cases, and provide several explicit results for qutrit maps. We use the affine map and a singular-value-like decomposition, to find positivity constraints that provide a symmetry for small polarization vector magnitudes (states which are closer to the maximally mixed state) which is broken as the polarization vector increases in magnitude (a state becomes more pure). The dependence of this symmetry on the magnitude of the polarization vector implies the polar decomposition of the map can not be used as it can for the qubit case. However, it still leads us to a connection between positivity and purity for general d-state systems. Contents

Qudit versions of the qubitπ/8gate

“…The orders of the individual group elements are ord(U υ (0, 0, 0)) = 1 (33) ord(U υ (z , 0, )) = 3 (excluding case (33)) (34) ord(U υ (z , γ , )) = 9 (excluding cases (33) and (34)) In fact, for p = 3 we have…”

“…Phase damping is a physically well-motivated noise process (often interpreted as resulting from so-called phase kicks), whose overall effect on a state ρ is to uniformly decrease the amplitude of all off-diagonal elements in ρ (see e.g., [33]). The implementation of a diagonal gate U θ , while suffering phase damping noise (with noise rate ε P D ), results in an overall operation…”

Effects of noise, correlations, and errors in the preparation of initial states in quantum simulations

“…(For higher-dimensional systems, this is referred to as the polarization vector, coherence vector, or generalized block vector. See [102][103][104][105][106][107][108] and references therein.) The magnitude of the Bloch vector is constrained by the condition a 2 x + a 2 y + a 2 z 1, and | a| = 1 represents a pure state.…”

Characterization of qutrit channels in terms of their covariance and symmetry properties

“…,ρ d 2 −1 ) is an affine map, i.e., ρ −→ ρ + T. An analysis of the completely positive maps based on these affine maps, with examples of qutrit channels has been carried out in Ref. [14]. As pointed out in the Introduction, we want to study qutrit channels from the perspective of their covariance and symmetry properties.…”

“…In Ref. [14], general properties of the affine map on polarization vectors, obtained from completely positive maps on d-level states (qudits), has been obtained. In particular special classes of qudit channels with special class of Kraus operators, i.e., unitary, Hermitian, and orthogonal, have been investigated.…”