Postselection conditions a quantum measurement on a specified outcome. If its branch acts by a linear operator on a normalized input , its probability is , and its normalized output is . This conditional state is defined only when the probability is nonzero. Physically unsuccessful trials are discarded; postselection is not a deterministic implementation of the nonlinear normalization map.
For an ideal projective measurement with the Lüders rule, condition on both an initial quantum state preparation and a final successful postselection. Its outcome probabilities are proportional to , where is the forward-evolved initial state and the backward-evolved final state. The normalization must be nonzero. The expression is symmetric in these two boundary states, although its derivation uses the ordinary Born rule and conditional state update.
In the ABL rule, probabilities depend on the full intermediate quantum measurement instrument. A binary Lüders rule quantum measurement preserves coherence within its unresolved complement, whereas a fully resolved projective measurement removes it. Summing fine-grained probabilities after quantum measurement does not generally reproduce the coarse-grained experiment. Different postselection success rates are part of the distinction.
For a uniform prestate in dimensions and poststate proportional to , each binary question versus has conditional probability one for . A fully resolved quantum measurement instead assigns to each of the first outcomes and to the last. The context dependence of pre- and post-selected measurements explains why these alternative certainties do not imply simultaneous properties.
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Postselection is a concept primarily used in quantum mechanics and quantum information theory. It refers to the process of selecting certain outcomes from a quantum experiment after measurement has taken place, effectively discarding other outcomes that do not meet specific criteria. In quantum systems, measurements can yield a range of possible results due to the probabilistic nature of quantum mechanics. Postselection involves analyzing the outcomes and only retaining those results that align with a predetermined condition.