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arXiv:2507.09716

When the Weak Becomes Strong: Effective Observables via Time-Symmetric Quantum Selection

Mirco A. Mannucci

Published Jul 13, 2025• 0 citations
arXivHTMLPDF

Abstract

We investigate the sequential composition of weak values in the framework of time-symmetric quantum mechanics. Specifically, we consider a forward'' weak measurement from a preselected state $\ketψ$ to a post-selected state $\ketϕ$, followed by a reverse'' weak measurement. We show that the product of these two weak values corresponds to the normalized expectation value of a strong, state-conditioned observable $B = A P_ψA$, where $P_ψ= \ketψ\braψ$ is the projector onto the preselected state. Analyzing the structure of $B$, we demonstrate how it encodes interference information, particularly when $\ketψ$ is a superposition rather than an eigenstate of $A$. This formulation extends naturally to mixed states by replacing $P_ψ$ with a generic density matrix $ρ$, linking the construction to the formalism of generalized quantum measurements. We illustrate practical applications in quantum information, including state-specific error witnessing in quantum computing, and show how the phase of a weak value can be inferred via strong measurements in the pure-state case.

DOI
10.48550/arxiv.2507.09716

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We investigate the sequential composition of weak values in the framework of time-symmetric quantum mechanics. Specifically, we consider a forward'' weak measurement from a preselected state ∣ψ⟩\ketψ∣ψ⟩ to a post-selected state ∣ϕ⟩\ketϕ∣ϕ⟩, followed by a reverse'' weak measurement. We show that the product of these two weak values corresponds to the normalized expectation value of a strong, state-conditioned observable B=APψAB = A P_ψAB=APψ​A, where Pψ=∣ψ⟩⟨ψ∣P_ψ= \ketψ\braψPψ​=∣ψ⟩⟨ψ∣ is the projector onto the preselected state. Analyzing the structure of BBB, we demonstrate how it encodes interference information, particularly when ∣ψ⟩\ketψ∣ψ⟩ is a superposition rather than an eigenstate of AAA. This formulation extends naturally to mixed states by replacing PψP_ψPψ​ with a generic density matrix ρρρ, linking the construction to the formalism of generalized quantum measurements. We illustrate practical applications in quantum information, including state-specific error witnessing in quantum computing, and show how the phase of a weak value can be inferred via strong measurements in the pure-state case.
In this thesis, we develop a probabilistic description of the time-symmetric formulation of quantum mechanics.We derive a generalized form of the probability distribution of a quantum process which is determined by performing pre-and post-selections on a system.It gives the well-known value for an observable associated to a quantum process, called weak value, as an expectation value.We show that the generalized probability satisfies the law of total probability and admits the transitive form by means of conditional probabilities.This motivates us to step forward to a classical ensemble picture in quantum mechanics.In addition, we present the description of quantum measurement in terms of the generalized probability.It is suggested that the difference between the projective measurement and the weak measurement corresponds to the choice of selections of the forward and backward propagating processes. Consistency of the Probability in the Forward and Backward Processes 6 Conclusion and Discussion Bibliography
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.
Weak values (WVs) arise from weak measurements performed within a time-symmetric formulation of quantum mechanics, where a system is both pre- and post-selected. Anomalous WVs that lie far outside the eigenvalue spectrum of the observable hold both fundamental and practical significance. However, their generation typically relies on near-orthogonal pre- and post-selection, which confines them to a single post-selection outcome with extremely low success probability. This constraint limits experimental accessibility and hinders the full exploitation of time symmetry. To overcome these limitations, we utilize quantum entanglement and post-selection-controlled operations to generate correlated pairwise two-state vectors. By changing the role of post-selection from passive filtering to active engineering, this approach enables the observation of anomalous complementary WVs associated with mutually exclusive post-selection branches. Our results extend the operational accessibility of time-symmetric quantum structures associated with the two-state vector formalism, and open new avenues for exploring the applications of time symmetry in quantum information processing.
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable AAA weakly followed by the strong measurement of another observable BBB (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable BBB weakly followed by the strong measurement of the observable AAA (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.