360 Transfer matrix and diagonalisation of spin chains
Now we work outEq. (11.49)using the basis set|σ〉. We obtain
〈QS〉=∑
σ,,σ′,′Cσ∗Cσ′′〈σ|QS|σ′′〉. (11.52)We note thatQSonly affects the states of S, so the states of E can immediately be
contracted. Using orthonormality of the|〉we obtain
〈QS〉=∑
σ,,σ′C∗σCσ′〈σ|QS|σ′〉. (11.53)Noting that
ρS≡TrEρ=∑
〈|ρ|〉=∑
σ,σ′CσC∗σ′|σ〉〈σ′|, (11.54)we conclude that
〈QS〉=TrS(ρSQS). (11.55)
What have we just shown? Starting from apurestate of the entire (U) system,
we have generated a so-calledreduced density matrixρSfor S, which in general
describes a mixed state, and which is the most complete description of that system.
In other words, when a system S is coupled to an environment E, its state must in
general be described as amixedstate, although system plus environment together
(that is, U) are in apurestate.
A system whose state is mixed due to coupling with an environment is said to
beentangledwith that environment. The simplest example of entanglement is a
system consisting of two degrees of freedom which can both be in two states,| 0 〉
and| 1 〉. For a state
|ψ〉=| 00 〉, (11.56)
where the first 0 refers to S and the second one to E, the reduced density matrix is
ρS=| 0 〉〈 0 |which describes a pure state, reflecting the fact that we know that the
system S is in state| 0 〉. However, the state
|ψ〉=1
√
2
(| 00 〉+| 11 〉) (11.57)
leads to the reduced density matrix
ρS=1
2
(| 0 〉〈 0 |+| 1 〉〈 1 |) (11.58)
which describes a mixed state. The state|ψ〉is therefore an example of an entangled
state – this particular state is one of the fourBell states[26].
Now let us return to the renormalisation procedure. Knowing that U is in the
ground state, we should describe S by a mixed state. But which mixed state? After
all, we do not know the ground state of U, and therefore we cannot take the trace