Time-Symmetric Quantum Mechanics

Time-symmetric descriptions use both preparation and later selection to characterize an intermediate quantum experiment. In their operational form they reorganize ordinary conditional probabilities; they need not introduce a new force from the future. A stronger interpretation of final boundary conditions is an additional proposal.

Conditioning on preparation and postselection

Prepare $\lvert i\rangle$, perform an intermediate ideal projective measurement with projectors $P_k$, and later retain runs passing $\lvert f\rangle$. Absorb intervening unitary evolution into the states or projectors. The joint probability for intermediate result $k$ and final success is

$$p(k,f\mid i)=|\langle f|P_k|i\rangle|^2.$$

Dividing by the sum over the actually performed intermediate measurement gives the Aharonov–Bergmann–Lebowitz rule,

$$p(k\mid i,f,\{P_j\})= \frac{|\langle f|P_k|i\rangle|^2} {\sum_j|\langle f|P_j|i\rangle|^2}.$$

The denominator must be nonzero. The full instrument matters for degenerate measurements: the displayed expression assumes Lüders projection, not an arbitrary refinement followed by forgetting its result. The measurement context is written explicitly because different interventions can change the postselection probability.

A qubit with certain alternative outcomes

Let $\lvert i\rangle=\lvert+z\rangle$ and $\lvert f\rangle=\lvert+x\rangle$. If the intermediate measurement is $Z$, the ABL rule gives $+z$ with certainty. If it is instead $X$, it gives $+x$ with certainty.

These statements concern different experiments. They do not assign simultaneously sharp $X$ and $Z$ values to one undisturbed qubit. Performing both measurements in sequence changes the intervention and requires another calculation. A claim about “what would have happened” must preserve which preparation, apparatus, and postselection procedure is held fixed.

Time reversal is a different question

Two-boundary conditioning is not identical to invariance under physical time reversal. For a time-reversal invariant model, an antiunitary operator transforms states, reverses momenta and spins as appropriate, and relates the forward Hamiltonian to the reversed experiment. Fields and boundary conditions may also need reversal.

The ABL conditional-probability expression can be formed even for a driven protocol whose physical reversal is not the same laboratory procedure. Likewise, the thermodynamic asymmetry between preparing and postselecting ensembles is not removed by a symmetric-looking formula. A preparation can be imposed; a desired postselection result generally cannot be forced.

Why there is no message from the future

Before final outcomes are communicated, the relevant intermediate probability is obtained by summing over all final outcomes. Completeness restores the ordinary Born probability for the actual intermediate measurement. Conditioning on the successful subset is available only after the success labels are known.

The Two-State-Vector Formalism packages the same boundary information into forward and backward mathematical objects. Retrocausality discusses the further claim that future settings help determine earlier ontic variables.

Exercise

For the pre- and postselection above, perform an intermediate measurement of $Y$. Calculate the probabilities of its two outcomes in the retained ensemble.

Solution

Both $|\langle+x|\pm y\rangle|^2$ and $|\langle\pm y|+z\rangle|^2$ equal $1/2$. Each joint weight is therefore $1/4$, and normalization gives $1/2,1/2$. Certainty in the alternative $X$ and $Z$ protocols does not transfer to a third context.

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 — two-boundary probability rule.
• Vaidman, L., “The Two-State Vector Formalism” (2007), arXiv:0706.1347 — formalism and its interpretive reading.