Partial Trace As A Quantum Operation
The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product of Hilbert spaces. A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product The partial trace of ρ with respect to the system B, denoted by, is called the reduced state of ρ on system A. In symbols,
To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is . However one chooses to define a reduced state, there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in and that of when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:
We see that this is satisfied if is as defined above via the partial trace. Furthermore it is the unique such operation.
Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map
is completely positive and trace-preserving.
The partial trace map as given above induces a dual map between the C*-algebras of bounded operators on and given by
maps observables to observables and is the Heisenberg picture representation of .
Read more about this topic: Partial Trace
Famous quotes containing the words partial, trace, quantum and/or operation:
“The one-eyed man will be King in the country of the blind only if he arrives there in full possession of his partial faculties—that is, providing he is perfectly aware of the precise nature of sight and does not confuse it with second sight ... nor with madness.”
—Angela Carter (1940–1992)
“Superstition saw
Something it had never seen before:
Brown eyes that loved without a trace of fear,
Beauty so sudden for that time of year.”
—Jean Toomer (1894–1967)
“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)
“Human knowledge and human power meet in one; for where the cause is not known the effect cannot be produced. Nature to be commanded must be obeyed; and that which in contemplation is as the cause is in operation as the rule.”
—Francis Bacon (1560–1626)