© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics Communications

Impossible weak values

Yakov Bloch

Published Jun 1, 2026• 0 citations

Abstract

Abstract Weak values characterize quantum systems that are both pre- and post-selected. Such systems are described by two wavefunctions, the overlap of which defines the success probability of the experiment. In weak value theory, it is usually assumed that the weak value does not exist for zero-overlapping pre- and post-selected wavefunctions. In this paper, we show that weak values can be defined even when the success probability of the experiment tends to zero. We call such weak values impossible. Standard quantum expectation values can be constructed from weak values. Remarkably, impossible weak values vanish in the expectation value, but may appear in its dynamics. We show that impossible weak values are not unique for a given pre- and post-selected state pair. They depend on the path in state space along which the pair is approached.

DOI: 10.1088/2399-6528/ae779c

Related papers

On the Two-State Vector Reformulation of Quantum Mechanics

Yakir Aharonov, Lev Vaidman A

A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.

Physica Scripta•Jan 1, 1998•19 citations

Anomalous Weak Values Without Post-Selection

Alastair A. Abbott, Ralph Silva, Julian Wechs +2 more

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.

arXiv:1805.09364•Quantum•Oct 14, 2019•13 citations

Weak Measurements

Lev Vaidman A

Weak measurement is a standard measuring procedure with two changes: it is performed on pre- and post-selected quantum systems and the coupling to the measuring device is weakened. The outcomes of weak measurements, ``weak values'' are very different for the eigenvalues of the measured operators. The weak values yield novel rich structure of the quantum world. Weak values help explaining peculiar quantum phenomena and finding new effects which might have practical applications.

arXiv:hep-th/9408154•Aug 28, 1994•9 citations

Strange weak values

Akio Hosoya, Yutaka Shikano

We develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter-factual processes. We present the condition for the choice of the post-selected state to give a negative weak value of a given projection operator and strange values of an observable in general. The general framework is applied to Hardy's paradox and the spin-1/2 system to explicitly address the issues of counter-factuality and strange weak values. The counter-factual arguments which characterize the paradox specify the pre-selected state, and a complete set of the post-selected states clarifies how the strange weak values emerge.

•Jan 1, 2010•36 citations

Understanding Weak Values Without New Probability Theory

Riuji Mochizuki

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable $\hat {A}$ with post-selection 〈F| may be out of the eigenvalue range of $\hat {A}$ . This is because the weak value of $\hat {A}$ with the post-selection is, in general, not the expectation value of $\hat {A}$ , but the expectation value of $\hat {A}| F\rangle \langle F|$ boosted by the post-selection.

arXiv:1604.02197•International Journal of Theoretical Physics•Jan 16, 2018•1 citation

Conditions for Anomalous Weak Value

Arun Kumar Pati, Junde Wu

We show that the weak value of any observable in pre- and post-selected states can be expressed as the sum of the average of the observable in the pre-selected state and an anomalous part. We argue that at a fundamental level the anomalous nature of the weak values arises due to the interference between the post-selected state and another quantum state which is orthogonal to the pre-selected state. This provides a necessary and sufficient condition for the anomalous nature of the weak value of a quantum observable. Furthermore, we prove that for two non-commuting observables the product of their anomalous parts cannot be arbitrarily large.

arXiv:1410.5221•Oct 20, 2014•5 citations

Abstract Weak values characterize quantum systems that are both pre- and post-selected. Such systems are described by two wavefunctions, the overlap of which defines the success probability of the experiment. In weak value theory, it is usually assumed that the weak value does not exist for zero-overlapping pre- and post-selected wavefunctions. In this paper, we show that weak values can be defined even when the success probability of the experiment tends to zero. We call such weak values impossible. Standard quantum expectation values can be constructed from weak values. Remarkably, impossible weak values vanish in the expectation value, but may appear in its dynamics. We show that impossible weak values are not unique for a given pre- and post-selected state pair. They depend on the path in state space along which the pair is approached.

A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.

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.

We develop a formal theory of the weak values with emphasis on the consistency conditions and a probabilistic interpretation in the counter-factual processes. We present the condition for the choice of the post-selected state to give a negative weak value of a given projection operator and strange values of an observable in general. The general framework is applied to Hardy's paradox and the spin-1/2 system to explicitly address the issues of counter-factuality and strange weak values. The counter-factual arguments which characterize the paradox specify the pre-selected state, and a complete set of the post-selected states clarifies how the strange weak values emerge.

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable

\hat {A}

with post-selection 〈F| may be out of the eigenvalue range of

\hat {A}

. This is because the weak value of

\hat {A}

with the post-selection is, in general, not the expectation value of

\hat {A}

, but the expectation value of

\hat {A}| F\rangle \langle F|

boosted by the post-selection.

We show that the weak value of any observable in pre- and post-selected states can be expressed as the sum of the average of the observable in the pre-selected state and an anomalous part. We argue that at a fundamental level the anomalous nature of the weak values arises due to the interference between the post-selected state and another quantum state which is orthogonal to the pre-selected state. This provides a necessary and sufficient condition for the anomalous nature of the weak value of a quantum observable. Furthermore, we prove that for two non-commuting observables the product of their anomalous parts cannot be arbitrarily large.