Search Results for « Unitary transformations »

In quantum mechanics and quantum information science, state evolution occurs via two fundamental mechanisms: deterministic unitary gate transformations and projective (von Neumann) measurements. This paper formalizes the three mathematical and physical pillars that differentiate projective measurements from unitary operations: stochastic behavior governed by the Born rule, non-linearity introduced via wave function collapse normalization, and operational irreversibility. It provides rigorous mathematical derivations for state update rules, explores the role of eigenvalues and observables, and demonstrates the non-linearity of projective measurements through concrete mathematical examples.

The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system A to output system B , system A cannot influence system B . Conversely, given a unitary U with a no-influence relation from input A to output B , it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of U with no path from A to B . However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident simultaneously . To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A causally faithful extended circuit decomposition, representing a unitary U , is then one for which there is a path from an input A to an output B if and only if there actually is influence from A to B in U . We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

The article is devoted to algebraic modeling of universal rules of stochastic organization of genomic DNA of higher and lower organisms, previously published by the author. The proposed algebraic apparatus, which uses formalisms of quantum mechanics and quantum informatics and which is based on the so-called tensor-unitary transformations of vectors that generate families of interrelated stochastic-deterministic vectors of increased dimensions. The features of the vectors' interconnections in these families model the stochastic-deterministic properties of the named phenomenological rules. There are new approaches to modeling of developing multi-parameter biosystems, whose number of parameters increases in the course of step-by-step development. In the light of the presented materials, the issues of fractal-like organization in genetically inherited biosystems are considered. The development of the theory of stochastic determinism as an antipode of deterministic chaos is discussed.

Abstract Unitary transformations of operators and variables are considered for those cases when a Hermitian operator in the expression U = exp ( i S ) explicitly depends on the projection operators. Some simple examples of such transformations are given and the equations obtained which define projectors (among special sets of them), thus bringing forth better estimations for upper and lower bounds. The unitary transformations are also applied to the reduced resolvent operator and equations are presented for the operators U that minimize the energy functional.

We study the unitary time evolution of a simple quantum Hamiltonian describing two harmonic oscillators coupled via a three-level system. The latter acts as an engine transferring energy from one oscillator to the other and is driven in a cyclic manner by time-dependent external fields. The S-matrix (scattering matrix) of the cycle is obtained in analytic form. The total number of quanta contained in the system is a conserved quantity. As a consequence, the spectrum of the S-matrix is purely discrete, and the evolution of the system is quasi-periodic. The explicit knowledge of the S-matrix makes it possible to do accurate numerical evaluations of the time-dependent wave function. They confirm the quasi-periodic behavior. In particular, the energy flows back and forth between the two oscillators in a quasi-periodic manner.