The trace operator Tr extracts a complex number from an operator.
We now show that although its definition (6.67) is in terms of a partic-ular basis {|mi}, its value is independent of the basis used. Let {|qji}
be any other basis. Then we insert identity operators I =P
j|qjihqj|
Another useful result is that for any two operators A and B, Tr(AB) = Tr(BA): By making the substitutions B → C and A → AB in this result we infer that
Tr(ABC) = Tr(CAB). (3)
This equation shows that whereas ρ is represented by a diagonal matrix in the {|ni} basis, in the {|qji} basis ρ is represented by a non-diagonal matrix. This contrast arises because in writing equation (6.64) we as-sumed that our system was in one of the states of the set {|ni}, although we were unsure which one. In general if the system is in one of these states, it will definitely not be in any of the states {|qji} because each
|ni will be a non-trivial linear combination of kets |qji. Thus when ρ is expanded in this basis, the expansion does not simply specify a proba-bility to be in each state. Instead it includes complex off-diagonal terms pjk=P
npnhqj|nihn|qki that have no classical interpretation. When we have incomplete knowledge of the state of our system, we will generally not know that the system is in some state of a given complete set, so we should not assume that the off-diagonal elements of ρ vanish. Never the less, we may safely use equation (6.64) because whatever matrix repre-sents ρ in a given basis, ρ is a Hermitian operator and will have a complete set of eigenkets. Equation (6.64) gives the expansion of ρ in terms of its eigenkets. In practical applications we may not know what the eigenkets
|ni are, but this need not prevent us using them in calculations.
The importance of ρ is that through equation (6.67) we can obtain from it the expectation value of any observable. As the system evolves, these observables will evolve because ρ evolves. To find its equation of
motion, we differentiate equation (6.64) with respect to time and use the tdse. The differentiation is straightforward because pn is time-independent: if the system was in the state |ni at time t, at any later time it will certainly be in whatever state |ni evolves into. Hence we have
dρ
This equation of motion can be written more simply
i¯hdρ
dt = [H, ρ]. (6.70)
To obtain the equation of motion of an arbitrary expectation value Q = Tr(ρQ), we expand the trace in terms of a time-independent basis {|ai} and use equation (6.70):
i¯hdQ
where the last equality uses equation (3) of Box 6.3. Ehrenfest’s theorem (2.34) states that the rate of change of the expectation value Q for a given quantum state is the expectation value of [Q, H] divided by i¯h, so equation (6.71) states that when the quantum state is uncertain, the expected rate of change of Q is the appropriately weighted average of the rates of change of Q for each of the possible states of the system.
Notice that the density operator and the operators for the Hamilto-nian and other observables encapsulate a complete, self-contained theory of dynamics. If we have incomplete knowledge of our system’s initial state, use of this theory is mandatory. If we do know the initial state, we can still use this apparatus by assigning our system the density operator
ρ = |ψihψ| (6.72)
rather than using the tdse and extracting amplitudes for possible out-comes of measurements. However, when ρ takes the special form (6.72), the use of the density operator becomes optional (Problem 6.8).
6.3.1 Reduced density operators
We have seen that any physical interaction between two quantum systems is likely to entangle them. No man is an island and no system is truly isolated (except perhaps the entire universe!). Consequently, a real system is constantly entangling itself with its environment. We now show that even if our system starts in a pure state, once it has entangled itself with its environment, it will be in an impure state.
We consider a system that is comprised of two subsystems: A, which will represent our system, and B, which will represent the environment – the environment consists of anything that is dynamically coupled to our system but not observed in sufficient detail for its dynamics to be followed. Let the density operator of the entire system be
ρAB=X
ijkl
|A; ii|B; jiρijklhA; k|hB; l|. (6.73)
Let Q be an observable property of subsystem A. The expectation value of Q is where the second equality exploits the fact that Q operates only on the states of subsystem A, and also uses the orthonormality of the states of each subsystem: hA; k|A; mi = δkm, etc. We now define the reduced density operator of subsystem A to be
ρA≡X
where the second equality uses equation (6.73). In terms of the reduced density operator, equation (6.74) can be written
Q =X
m
hA; m|QρA|A; mi = Tr QρA. (6.76)
Thus the reduced density operator enables us to obtain expectation val-ues of subsystem A’s observables without bothering about the states of subsystem B. It is formed from the density operator of the entire system by taking the partial trace over the states of subsystem B (eq. 6.75).
Suppose both subsystems start in well-defined states. Then under the tdse the composite system will evolve through a series of pure states
|ψ, ti, and at time t the density operator of the composite system will be (cf. 6.72)
ρAB= |ψ, tihψ, t|. (6.77)
If the two subsystems have not become entangled, so |ψ, ti = |A, ti|B, ti, then the reduced density operator for A is
ρA= |A, tihA, t|X
i
hB; i|B, tihB, t|B; ii = |A, tihA, t|, (6.78)
where we have used the fact that the set {|B; ii} is a complete set of states for subsystem B. Equation (6.78) shows that so long as the subsystems remain unentangled, the reduced density operator for A has the form expected for a system that is in a pure state. To show that entanglement will generally lead subsystem A into an impure state, we consider the simplest non-trivial example: that in which both subsystems are qubits.
Suppose they have evolved into the entangled state
|ψ, ti = 1
√2(|A; 0i|B; 0i + |A; 1i|B; 1i) . (6.79)
Then evaluating the trace over the two states of B we find
ρA= 12hB; 0| (|A; 0i|B; 0i + |A; 1i|B; 1i) (hA; 0|hB; 0| + hA; 1|hB; 1|) |B; 0i +12hB; 1| (|A; 0i|B; 0i + |A; 1i|B; 1i) (hA; 0|hB; 0| + hA; 1|hB; 1|) |B; 1i
= 12(|A; 0ihA; 0| + |A; 1ihA; 1|) ,
(6.80) which is the density operator of a very impure state. Physically this result makes perfect sense: in equation (6.80) ρA states that subsystem A has equal probability of being in either |0i or |1i, which is consistent with the state (6.79) of the entire system. In that state these two possibilities were associated with distinct predictions about the state of subsystem B, but in passing from ρAB to ρA we have lost track of these correlations: if we choose to consider system A in isolation, we lose the information carried by these correlations, with the result that we have incomplete information about system A. In this case system A is in an impure state. So long as we recognise that A is part of the larger system AB and we retain the ability to measure both parts of AB, we have complete information, so AB is in a pure state.
In this example system A represents the system under study and sys-tem B represents the environment of A, which we defined to be whatever is dynamically coupled to A but incompletely instrumented. If, for exam-ple, A is a hydrogen atom, then the electromagnetic field inside the vessel containing the atom would form part of B because a hydrogen atom, be-ing comprised of two movbe-ing charged particles, is inevitably coupled to the electromagnetic field. If we start with the atom in its first excited state and the electromagnetic field in its ground state, then atom, field and atom-plus-field are initially all in pure states. After some time the atom-plus-field will evolve into the state
|ψ, ti = a0(t)|A; 0i|F; 1i + a1(t)|A; 1i|F; 0i, (6.81) where |A; ni is the nth excited state of the atom, while |F; ni is the state of the electromagnetic field when it contains n photons of the frequency associated with transitions between the atom’s ground and first-excited states. In equation (6.81), a0(t) is the amplitude that the atom has de-cayed to its ground state while a1(t) is the amplitude that it is still in its excited state. When neither amplitude vanishes, the atom is entangled
with the electromagnetic field. If we fail to monitor the electromagnetic field, we have to describe the atom by its reduced density operator
ρA= |a0|2|A; 0ihA; 0| + |a1|2|A; 1ihA; 1|. (6.82) This density operator indicates that the atom is now in an impure state.
In practice a system under study will sooner or later become entan-gled with its environment, and once it has, we will be obliged to treat the system as one for which we lack complete information. That is, we will have to predict the results of measurements with a non-trivial den-sity operator. The transition of systems in this way from pure states to impure ones is called quantum decoherence. Experimental work di-rected at realising the possibilities offered by quantum computing is very much concerned with arresting the decoherence process by weakening all couplings to the environment.
6.3.2 Shannon entropy
Once we recognise that systems are typically in impure states, it’s natural to want to quantify the impurity of a state: for example, if in the definition (6.64) of the density operator, p3= 0.99999999, then the system is almost certain to be found in the state |3i and predictions made by assuming that the system is in the pure state |3i will not be much in error, while if the largest probability occurring in the sum is 10−20, the effects of impurity will be enormous.
A probability distribution {pi} provides a certain amount of informa-tion about the outcome of some investigainforma-tion. If one probability is close to unity, the information it provides is nearly complete. Conversely, if all the probabilities are small, no outcome is particularly likely and the missing information is large. The question we now address is “what is the appropriate measure of the missing information that remains after a probability distribution {pi} has been specified?”
Logic dictates that the required measure s(p1, . . . , pn) of missing in-formation must have the following properties:
• s must be a continuous, symmetric function of the pi;
• s should be largest when every outcome is equally likely, i.e., when pi = 1/n for all i. We define
s(n1, . . . ,n1) = sn (6.83) and require that sn+1> sn (more possibilities implies more missing information).
• s shall be consistent in the sense that it yields the same missing information when there are different ways of enumerating the possible outcomes of the event.
To grasp the essence of the last requirement, consider an experiment with three possible outcomes x1, x2 and x3 to which we assign probabilities p1, p2 and p3, yielding missing information s(p1, p2, p3). We could group the last two outcomes together into the outcome x23, by which we mean