Two-State-Vector Formalism

The two-state-vector formalism describes a preselected and postselected ensemble by a forward-evolving ket and a backward-propagated bra at an intermediate time. It is a useful organization of boundary information. Treating both objects as independently real physical states is a further interpretive commitment, not required for the conditional predictions.

Two boundary objects at one time

For preparation $\lvert i\rangle$ at $t_i$ and final selected result $\lvert f\rangle$ at $t_f$, define

$$\lvert\psi(t)\rangle=U(t,t_i)\lvert i\rangle,\qquad \langle\phi(t)\rvert=\langle f|U(t_f,t).$$

The bra carries information about a selected future result backward through the model. It is not an operation that sends a signal to the earlier apparatus. With no intervening measurement, $\langle\phi(t)|\psi(t)\rangle=\langle f|U(t_f,t_i)|i\rangle$ is independent of the chosen intermediate time.

An intermediate instrument with one Kraus operator $M_k$ per outcome gives joint weight $|\langle\phi|M_k|\psi\rangle|^2$. For multiple Kraus operators, sum the squared amplitudes over the unresolved instrument label. Merely giving POVM effects does not in general determine the postselected statistics because different instruments can produce different disturbances.

Weak response and generalized conditioning

For a sufficiently weak coupling to $A$, the conditional coefficient is

$$A_w(t)= \frac{\langle\phi(t)|A|\psi(t)\rangle} {\langle\phi(t)|\psi(t)\rangle}.$$

The denominator must be nonzero. Technical response and validity conditions belong to Weak Values. The two boundary objects do not define an ordinary density operator by simply multiplying them: $\lvert\psi\rangle\langle\phi\rvert$ need not be Hermitian or positive.

A mixed preparation $\rho$ and positive final effect $E$ give a more general probability rule for an intermediate instrument,

$$p(k\mid E)= \frac{\operatorname{tr}[E\,\mathcal I_k(\rho)]} {\sum_j\operatorname{tr}[E\,\mathcal I_j(\rho)]}.$$

The future effect is propagated by the adjoint dynamics. This operational formulation makes the conditioning and instrument dependence explicit without requiring a new ontology.

A three-box calculation

Prepare $\lvert i\rangle=(\lvert1\rangle+\lvert2\rangle+\lvert3\rangle)/\sqrt3$ and postselect $\lvert f\rangle=(\lvert1\rangle+\lvert2\rangle-\lvert3\rangle)/\sqrt3$. The overlap is $1/3$, and the projector weak values are

$$(P_1)_w=1,\qquad(P_2)_w=1,\qquad(P_3)_w=-1.$$

Their sum is one, as required by $P_1+P_2+P_3=I$. These are not three ordinary occupancy probabilities. A strong measurement resolving all three boxes has equal conditional probabilities $1/3$, because it uses three separate squared amplitudes.

A binary Lüders test “box 1 or not box 1” instead gives box 1 with conditional certainty: the two amplitudes in the complement cancel. The corresponding box 2 test also gives certainty in a different context. Interference in a coarse projector and incoherent summation over resolved outcomes are different operations.

Exercise

For the three-box states, calculate the postselection success probability with no intermediate measurement and with a fully resolving three-outcome measurement.

Solution

Without an intermediate measurement it is $|\langle f|i\rangle|^2=1/9$. With full resolution, the three joint weights are each $1/9$, so the success probability is $1/3$. This difference is why one cannot transfer conditional claims between measurement contexts while holding the selected ensemble fixed by name alone.

References

• Aharonov, Y., Bergmann, P. G., and Lebowitz, J. L., “Time symmetry in the quantum process of measurement,” Physical Review 134, B1410–B1416 (1964), doi:10.1103/PhysRev.134.B1410 — conditional probability foundation.
• Vaidman, L., “The Two-State Vector Formalism” (2007), arXiv:0706.1347 — two-state descriptions and examples.