Trace Preserving Research Articles

Lagrangian representation for fermionic linear optics

Notions of a Gaussian state and a Gaussian linear map are generalized to the case of anticommuting (Grassmann) variables. Conditions under which a Gaussian map is trace preserving and (or) completely positive are formulated. For any Gaussian map an explicit formula relating correlation matrices of input and output states is presented. This formalism allows to develop the Lagrangian representation for fermionic linear optics (FLO). It covers both unitary operations and the single-mode projectors associated with FLO measurements. Using the Lagrangian representation we reduce a classical simulation of FLO to a computation of Gaussian integrals over Grassmann variables. Explicit formulas describing evolution of a quantum state under FLO operations are put forward.

Geometrical conditions for completely positive trace-preserving maps and their application to a quantum repeater and a state-dependent quantum cloning machine

We address the problem of finding optimal CPTP (completely positive, trace preserving) maps between a set of binary pure states and another set of binary generic mixed state in a two dimensional space. The necessary and sufficient conditions for the existence of such CPTP maps can be discussed within a simple geometrical picture. We exploit this analysis to show the existence of an optimal quantum repeater which is superior to the known repeating strategies for a set of coherent states sent through a lossy quantum channel. We also show that the geometrical formulation of the CPTP mapping conditions can be a simpler method to derive a state-dependent quantum (anti) cloning machine than the study so far based on the explicit solution of several constraints imposed by unitarity in an extended Hilbert space.

Trace resetting density matrix purification in O(N) self-consistent-field theory

A new approach to linear scaling construction of the density matrix is proposed, based on trace resetting purification of an effective Hamiltonian. Trace resetting is related to the trace preserving canonical purification scheme of Palser and Manolopoulos [Phys. Rev. B 58, 12704 (1999)] in that they both work with a predefined occupation number and do not require adjustment or prior knowledge of the chemical potential. In the trace resetting approach, trace conservation is not strictly enforced, allowing greater flexibility in the choice of purification polynomial and improved performance for Hamiltonian systems with high or low filling. However, optimal polynomials may in some cases admit unstable solutions, requiring a resetting mechanism to bring the solution back into the domain of convergent purification. A quartic trace resetting method is developed, along with analysis of stability and error accumulation due to incomplete sparse-matrix methods that employ a threshold τ to achieve sparsity. It is argued that threshold metered purification errors in the density matrix are O(τΔg−1) at worst, where Δg is the gap at the chemical potential. In the low filling regime, purification derived total energies are shown to converge smoothly with τ2 for RPBE/STO-6G C60 and a RPBE0/STO-3G Ti substituted zeolite. For the zeolite, the quartic trace resetting method is found to be both faster and over an order of magnitude more accurate than the Palser–Manolopoulos method. In the low filling limit, true linear scaling is demonstrated for RHF/6-31G** water clusters, and the trace resetting method is found to be both faster and an order of magnitude more accurate than the Palser–Manolopoulos scheme. Basis set progression of RPBE chlorophyll reveals the quartic trace resetting to be up to four orders of magnitude more accurate than the Palser–Manolopoulos algorithm in the limit of low filling. Furthermore, the ability of trace resetting and trace preserving algorithms to deal with degeneracy and fractional occupation is discussed.

Approximate quantum cloning and the impossibility of superluminal information transfer

We show that nonlocality of quantum mechanics cannot lead to superluminal transmission of information, even if most general local operations are allowed, as long as they are linear and trace preserving. In particular, any quantum mechanical approximate cloning transformation does not allow signalling. On the other hand, the no-signalling constraint on its own is not sufficient to prevent a transformation from surpassing the known cloning bounds. We illustrate these concepts on the basis of some examples.

On subdiagonal algebras for subfactors

On subdiagonal algebras for subfactors