Quantum Transition Matrix

Jean Salmon

Stanley Gudder

A new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum operation. A quantum Markov chain is defined to be a pair (G,E) where G is a directed graph and E=[Eij] is a TOM whose entry Eij labels the edge from vertex j to vertex i. We think of the vertices of G as sites that a quantum system can occupy and Eij is the transition operation from site j to site i in one time step. The discrete dynamics of the system is obtained by iterating the TOM E. We next consider a special type of TOM called a transition effect matrix. In this case, there are two types of dynamics, a state dynamics and an operator dynamics. Although these two types are not identical, they are statistically equivalent. We next give examples that illustrate various properties of quantum Markov chains. We conclude by showing that our formalism generalizes the usual framework for quantum random walks.

Jincheng Zhang

This paper explores a novel quantum state transition mechanism based on non-linear matrix dynamics, aiming to overcome the limitations of traditional quantum entanglement. We propose a framework where the matrix's self-adaptability is leveraged to facilitate faster and more flexible quantum state transitions. This approach offers the potential to significantly enhance quantum information processing and quantum computation by enabling dynamic and controlled quantum evolutions. The core mechanism involves employing a matrix representation to dynamically adjust quantum properties, allowing for rapid and efficient transitions between states. This research investigates the theoretical underpinnings of this approach, outlines the design of the matrix structure and its associated operations, and discusses the potential benefits and challenges associated with its implementation. We demonstrate through simulations that the proposed framework exhibits superior transition speeds compared to conventional methods, significantly advancing the capabilities of quantum systems.

D. I. Blokhint︠s︡ev

Bo-Bo Wei

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derive a set of universal functional relations between the matrix elements of a two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first-order phase transitions are signaled by the matrix elements of the reduced density matrix while the second-order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near the second-order phase-transition point, we show that the first derivative of the reduced-density-matrix elements present universal scaling behaviors. Finally, we establish a theorem which connects the phase transitions and bipartite entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.

Stanley Gudder