Quantum Markov chain
Encyclopedia
In mathematics
, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain
, replacing the classical definitions of probability with quantum probability
. Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automata, 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
.
More precisely, a quantum Markov chain is a pair
with 
a density matrix
and
a quantum channel
such that
is a completely positive trace-preserving map, and
a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that
for all
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
, replacing the classical definitions of probability with quantum probability
Quantum probability
Quantum probability was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes. One of its aims is to clarify the mathematical foundations of quantum theory and its statistical interpretation....
. Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automata, with some important substitutions: the initial state is to be replaced by a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
, and the projection operators are to be replaced by positive operator valued measures
POVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...
.
More precisely, a quantum Markov chain is a pair
with a density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
and
a quantum channelQuantum 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...
such that
is a completely positive trace-preserving map, and
a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, thatfor all
.