mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the quantum Markov chain is a reformulation of the ideas of a classical

Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

, replacing the classical definitions of probability with quantum probability.

Introduction

Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automaton, with some important substitutions: the initial state is to be replaced by a density matrix, and the projection operators are to be replaced by positive operator valued measures.

Formal statement

More precisely, a quantum Markov chain is a pair

(E,\rho)

with

\rho

a density matrix and

E

quantum channel In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information i ...

such that :

E:\mathcal\otimes\mathcal\to\mathcal

is a completely positive trace-preserving map, and

\mathcal

a C^*-algebra of bounded operators. The pair must obey the quantum Markov condition, that :

\operatorname \rho (b_1\otimes b_2) = \operatorname \rho E(b_1, b_2)

for all

b_1,b_2\in \mathcal

References

*Gudder, Stanley.
Quantum Markov chains
" Journal of Mathematical Physics 49.7 (2008): 072105. {{DEFAULTSORT:Quantum Markov Chain Exotic probabilities Quantum information science Markov models

Introduction

Formal statement

See also

References