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 .
Famous quotes containing the words quantum and/or chain:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)