Multiple-time states and multiple-time measurements in quantum mechanics

We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to an alternative approach to quantum mechanics. In particular, we introduce the idea of multitime quantum states which are the appropriate tools for describing these experimental situations. We also describe multitime measurements and discuss their relation to multitime states. A consequence of our formalism is to put states and operators on an equal footing. Finally we discuss the implications of our approach to quantum mechanics for the problem of the flow of time.

Figure 1

An experiment consisting of three “preparation” stages and two “measurement” stages.

Figure 2

(a) At time $t_1$ the system is prepared in state $|\Psi ⟩$ and at time $t_2$ an operator $B$ is measured. The outcome of $B$ happens to be $b$, the (nondegenerate) eigenvalue corresponding to the eigenstate $|\Phi ⟩$. (b) This situation we describe by the two-time state ${}_{t_2}⟨\Phi |{|\Psi ⟩}_{t_1}$

Figure 3

(a) The system $S$ interacts with the ancilla $A$. The arrows represent states propagating “forward” and “backward” in time, i.e., ket and bra vectors. The wiggled line connecting the forward in time propagating states (i.e., ket vectors) describes (arbitrary) entanglement. The continuous line connecting the backward in time propagating states (i.e., bra vectors) illustrates maximal entanglement. More precisely, since it refers to the bra vectors, it denotes post-selecting the maximally entangled state for the system and the ancilla. The dotted line illustrates how entanglement is transferred from the ancilla onto the system. The diagram (b) illustrates the same situation as (a) but from the point of view of the system alone.

Figure 4

Different multi-time situations. In (a) both the past and the future are well defined, i.e., they are part of “preparation” stages. In (b) the future is uncertain, in (c) the past is uncertain while in (d) both the future and the past are uncertain.

Figure 5

(a) The second and the fourth “measurement” stages, i.e., from $t_3$ to $t_4$ and from $t_7$ to $t_8$ are correlated with each other but not with the other two “measurement” stages. If we are interested in measurements that occurred during the first and the third stages, the second and the fourth stages are irrelevant. (b) An effective state describing the periods $t_1$ to $t_2$ and $t_5$ to $t_6$ can be obtained simply by ignoring the other stages.

Figure 6

(a) The system $S$ and the three ancillas, $A_1$, $A_2$ and $A_3$ start in an entangled state that gets transferred onto the system via the interactions of the system and the ancillas and via appropriate post-selections. The continuous line describes maximal entanglement while the wiggled line describes arbitrary entanglement and the arrows represent states propagating “forward” and “backward” in time, i.e., ket and bra vectors. Note the SWAP interaction between the system $S$ and the ancilla $A_2$; it is nothing other that maximal entanglement between bra states of the system and ket states of the ancilla and vice versa. The dotted line illustrates how entanglement is transferred from the ancillas onto the system. The diagram (b) illustrates the same situation as (a) but from the point of view of the system alone.

Figure 7

A simple one-time state with uncertain past.

Figure 8

Using measurements for preparation. To the original state $\Psi$ we add the information that the result of a POVM performed between $t_5$ to $t_6$ and $t_7$ to $t_8$ yielded the outcome $k$ corresponding to the Krauss operator $A_k$, The operator $A_k$ can also be viewed as a multi-time state ${\Phi }_k$. The new state that takes into account all the information is $\Psi \otimes {\Phi }_k$ which is simply the composition of $\Psi$ and ${\Phi }_k$, as illustrated in (b).