Nonlocal Weak Values in Quantum Systems

• Nonlocal weak values are defined as conditioned averages obtained from weak measurements on spatially separated or entangled subsystems using pre- and post-selected states.
• They encapsulate interference effects via quasi-probability distributions, often producing anomalous values that lie outside the expected eigenvalue range.
• These values serve as practical indicators of entanglement and contextuality in experimental setups like Mach–Zehnder interferometers, Hardy’s, and EPR–Bohm experiments.

Searching arXiv for relevant papers on nonlocal weak values, weak values as interference, and related experimental/theoretical work. Nonlocal weak values are weak values whose operational meaning is tied to spatially separated subsystems, to a single-particle wavefunction spread over distinct locations, or to globally defined pre- and post-selected interference patterns. In the pre-/post-selected framework, a weak value is not merely a detector readout; it is a conditioned quantity determined by both boundary conditions, and it can govern measurement statistics, reduced dynamics, and multi-particle correlations. In this setting, “nonlocal weak value” can mean either the weak value of a nonlocal observable, or a weak value of a local observable whose value depends on nonlocal interference structure, such as entanglement, modular variables, or a remote interferometric phase (Dressel, 2014).

1. Formal definitions and operational setting

For a pre-selected state $|\psi_i\rangle$, a post-selected state $|\psi_f\rangle$, and an observable $A$, the standard Aharonov–Albert–Vaidman weak value is

$$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$

In Dressel’s time-dependent form, if the system is prepared in $|i\rangle$ at time $0$, evolves with $U_t=e^{-iHt/\hbar}$, and is post-selected in $|f\rangle$ at time $T$, then the intermediate weak value is

$$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$

For mixed states and generalized measurements this becomes

$|\psi_f\rangle$0

with $|\psi_f\rangle$1 the predictive state and $|\psi_f\rangle$2 the retrodictive effect. This is the form used in past-quantum-state and quantum-smoothing analyses (Dressel, 2014).

Operationally, weak values emerge from a weak von Neumann coupling

$|\psi_f\rangle$3

followed by a strong post-selection. In the weak-coupling regime, the conditioned average of a calibrated detector observable converges to the weak value, so weak values arise as conditioned averages of weak measurements that minimally disturb the pre-selected state. Dressel’s review also stresses that the same quantity appears as the optimal estimate of the average value of $|\psi_f\rangle$4 between two measurements, given only the initial state, final state, and Hamiltonian (Dressel, 2014).

Nothing in the definition restricts $|\psi_f\rangle$5 to be local. For a bipartite system,

$|\psi_f\rangle$6

with $|\psi_f\rangle$7 any joint observable such as $|\psi_f\rangle$8. The same formalism also extends to “history weak values,” where the relevant object is a projector onto an intermediate path or history. In the square nested Mach–Zehnder analysis, the weak value of the history “source $|\psi_f\rangle$9 arm $A$0 $A$1 detector $A$2” is written as

$A$3

so the weak value is the amplitude for one history divided by the total Feynman sum for the transition (Georgiev et al., 2018).

2. Interference structure and anomalous values

A central result of the interference-based formulation is that weak values are best understood as conditioned interference amplitudes. If $A$4, then

$A$5

with

$A$6

The corresponding joint quasi-probability

$A$7

is the Terletsky–Margenau–Hill distribution, namely the real part of the Kirkwood–Dirac quasi-probability. In this representation, the numerator and denominator of the weak value are built from amplitudes associated with alternative intermediate paths, and post-selection reweights those paths (Dressel, 2014).

This formulation makes anomalous weak values precise. A weak value is anomalous when it lies outside the spectral range, has very large magnitude, or has a significant imaginary part. If $A$8 lies outside the eigenvalue range of $A$9, then at least one $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$0 must be negative. The anomaly is therefore equivalent to negativity in the underlying quasi-probability, and Dressel explicitly links this negativity to nonclassicality and contextuality. Pusey’s theorem, cited there, is used to frame anomalous weak values as proofs of contextuality (Dressel, 2014).

The same interference logic underlies nonlocal weak values. When the intermediate alternatives are joint multi-particle paths, or spatially separated arms of a delocalized wavefunction, the weak value still takes the form of an eigenvalue average over a quasi-distribution, but that quasi-distribution now lives on a multi-particle or spatially extended configuration space. A plausible implication is that “nonlocal weak value” does not denote a separate formalism; it denotes the nonlocal regime of the same interference-based construction.

3. Forms of nonlocality in pre-/post-selected ensembles

Nonlocal weak values appear in several distinct but mathematically continuous forms. One form is the weak value of an explicitly nonlocal observable, such as $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$1 or a Bell operator. Another is the weak value of a local observable evaluated in a nonlocal pre-/post-selected context, so that the value depends on the full multiparticle wavefunction or on remote interferometric settings.

Within the Two-State-Vector Formalism, the system between preparation and post-selection is described by the two-state vector $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$2, and weak values encode properties of this two-state description. This is particularly clear in Hardy’s experiment, the Cheshire Cat scenario, the EPR–Bohm experiment, and the quantum pigeonhole setting. In Hardy’s setup, the weak values satisfy

$$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$3

while the joint projector has

$$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$4

The local occupations are therefore classical-looking, but the nonlocal joint occupation is anomalous. In the Cheshire Cat setting,

$$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$5

so the joint weak value cannot be reconstructed from the local ones. In the EPR–Bohm example, the joint weak value $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$6 plays the same role. The product rule can fail,

$$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$7

even when $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$8 and $$A_w=\frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}.$$9 act on different subsystems (Aharonov et al., 2015).

The same paper emphasizes a more unusual case: even product pre-selected and product post-selected states can generate strong nonlocal intermediate correlations. For two spins pre-selected in $|i\rangle$0 and post-selected in $|i\rangle$1, the local weak value is

$|i\rangle$2

while

$|i\rangle$3

Because the joint observable is dichotomic, this yields perfect anti-correlation in the intermediate time, which the paper interprets as an “emerging correlation” and relates to the quantum pigeonhole effect (Aharonov et al., 2015).

A related but more kinematic form of nonlocality arises in bipartite momentum weak values. For a two-particle wavefunction

$|i\rangle$4

the momentum weak value of particle $|i\rangle$5 at post-selected position $|i\rangle$6 is

$|i\rangle$7

with

$|i\rangle$8

For entangled states, both $|i\rangle$9 and $0$0 depend on both coordinates, so the local weak value of particle $0$1 depends on the distant coordinate $0$2. The weak correlation

$0$3

then provides a strong entanglement criterion: in one dimension,

$0$4

Here “A-entanglement” means entanglement encoded in the amplitude $0$5, and “P-entanglement” means entanglement encoded in the phase $0$6 (Valdés-Hernández et al., 2018).

4. Single-particle nonlocality, modular variables, and dynamical weak values

Single-particle nonlocal weak values arise when the particle’s wavefunction occupies spatially separated paths and the weak value depends on the total interference context. A paradigmatic example is the modular-momentum experiment in a twin Mach–Zehnder interferometer. The measured observable is the local projector

$0$7

but the weak value of this local projector depends on whether the dark path $0$8 in the second interferometer is open or blocked. With both effective “slits” open,

$0$9

When the dark path $U_t=e^{-iHt/\hbar}$0 is blocked,

$U_t=e^{-iHt/\hbar}$1

In the control configuration,

$U_t=e^{-iHt/\hbar}$2

The crucial point is that the only difference between the first two cases is a blocking action in a dark arm of the distant interferometer, yet the weak value of the upstream local projector changes from $U_t=e^{-iHt/\hbar}$3 to $U_t=e^{-iHt/\hbar}$4. This is interpreted as an effect induced by a nonlocal exchange of modular momentum, governed by the Heisenberg equation

$U_t=e^{-iHt/\hbar}$5

whose right-hand side depends on potentials at two separated locations (Spence et al., 2010).

A second single-particle realization uses weak values of histories in a square nested Mach–Zehnder interferometer with Alice’s weak meter on one arm and Bob’s phase shifter on a spacelike separated arm. For post-selection at detector $U_t=e^{-iHt/\hbar}$6,

$U_t=e^{-iHt/\hbar}$7

and for post-selection at $U_t=e^{-iHt/\hbar}$8,

$U_t=e^{-iHt/\hbar}$9

while $|f\rangle$0 for $|f\rangle$1. Thus the weak value on Alice’s arm depends explicitly on Bob’s remote phase $|f\rangle$2, because the denominator is the total Feynman sum over all coherent histories. The pointer distributions in both position and momentum bases inherit this dependence, and the corresponding post-selected Wigner functions can develop negative regions even when the pre-$|f\rangle$3 Wigner function is positive (Georgiev et al., 2018).

Nonlocal weak values also arise as dynamical variables. In the dispersive cQED example with Hamiltonian

$|f\rangle$4

the qubit coherence obeys

$|f\rangle$5

where

$|f\rangle$6

The real part gives the ac Stark shift, and the imaginary part gives the ensemble-average dephasing rate. Here a weak value of the cavity photon number, defined between cavity states correlated with different qubit states, controls the local reduced dynamics of the qubit. Dressel presents this as an instance of effective nonlocal behavior in Hilbert space, and in realistic architectures also in space (Dressel, 2014).

5. Bell operators, entanglement witnesses, and post-selected correlation bounds

The explicitly nonlocal weak value of a Bell operator has become a central object in recent work on post-selected nonlocal correlations. For the CHSH operator

$|f\rangle$7

with local dichotomic observables, the nonlocal weak value is

$|f\rangle$8

At fixed overlap

$|f\rangle$9

the relevant quantities are the two-state norms $T$0, $T$1, and $T$2, corresponding respectively to unrestricted boundary states, one separable boundary state, and both boundary states separable. They satisfy

$T$3

In this framework, post-selection and entanglement are distinct operational resources: post-selection alone can enhance correlations, but entanglement is necessary to exceed the separable PPS bound, and their combination yields the strongest attainable correlations. The same work shows that nonlocal weak values provide post-selected entanglement witnesses, and it gives a constructive protocol that detects every pure two-qubit source state with nonzero concurrence in the ideal state-adapted setting, including regimes where the corresponding standard CHSH entanglement test is inconclusive (Cohen et al., 1 Jul 2026).

The momentum-based criterion supplies a different route to the same theme. Since

$T$4

the real part corresponds to the Bohmian momentum field, while the imaginary part encodes the diffusive or osmotic contribution. In the bipartite case, both become nonlocal fields on configuration space, and their weak correlations detect both the presence and the location of entanglement in the wavefunction—whether in amplitude or in phase (Valdés-Hernández et al., 2018).

The earlier TSVF literature presents these developments in concrete paradoxical examples rather than as optimized witnesses. In the EPR–Bohm case, Hardy’s paradox, and the Cheshire Cat scenario, nonlocal weak values were used to exhibit joint properties that are invisible to local marginals and incompatible with naive product rules. In the emerging-correlation and quantum-pigeonhole examples, the same formalism shows that product pre- and post-selected ensembles can support intermediate joint correlations that mimic maximal entanglement at the level of weak values (Aharonov et al., 2015).

6. Interpretation, classical models, and limits

A recurring controversy is whether anomalous weak values are merely disturbance artifacts. Dressel’s analysis rejects that reduction. In classical noisy conditioned averages, anomalies outside the range of the measured variable generally require disturbance of the variable itself prior to conditioning. By contrast, the most anomalous quantum weak values arise in the weak limit, when the intermediate measurement minimally disturbs the state, and they also appear in contexts where no intermediate measurement is performed at all, as in the cQED Stark-shift example. The interference/quasi-probability account therefore treats anomalies as consequences of negativity and interference, not of classical disturbance (Dressel, 2014).

The distinction becomes sharper in the sequential setting. For a single weak measurement without post-selection,

$T$5

so no anomaly occurs. But for sequential weak measurements of $T$6 followed by $T$7, the no-post-selection sequential weak value

$T$8

can be anomalous, because it is not, in general, the expectation value of a Hermitian observable. In the qubit example with two projectors, the weak-limit pointer correlation becomes

$T$9

which lies outside the classical range for products of $$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$0-valued observables. The same paper emphasizes, however, that for a genuinely bipartite nonlocal observable $$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$1 without post-selection one only obtains

$$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$2

so anomalous nonlocal weak values in the spatial sense still require nontrivial post-selection. The anomaly mechanism without post-selection is therefore temporal and sequential, not a replacement for the spatially nonlocal PPS framework (Abbott et al., 2018).

A separate line of work introduces null weak values. In that protocol, the first measurement is strong when it occurs but occurs with small probability, and the system is then post-selected. The null weak value is

$$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$3

so its numerator is an ordinary expectation value rather than the transition matrix element $$A_w(t)=\mathrm{Re}\,\frac{\langle f|U_{T-t}AU_t|i\rangle}{\langle f|U_T|i\rangle}.$$4. The paper presenting null weak values does not treat nonlocal weak values explicitly, but it states that the formalism is general enough to extend to nonlocal observables and multipartite Hilbert spaces (Zilberberg et al., 2013).

Single-particle nonlocality analyses also place stringent demands on local hidden-variable models. In the square nested Mach–Zehnder case, reproducing the dependence of post-selected pointer statistics on Bob’s spacelike separated phase requires hidden signaling and a list of contextual instructions, and the authors show that local hidden-variable models could rely neither on only two hidden variables for position and momentum, nor on simultaneous factorizability of both the hidden probability densities and weights of splitting, if they are to reproduce the correct quantum distributions. Their conclusion is not a blanket impossibility proof for all nonfactorizable local hidden-variable models, but a demonstration that weak values of quantum histories necessitate contextual splitting of prior commitments to measurement outcomes because of their functional dependence on the total Feynman sum (Georgiev et al., 2018).

Taken together, these developments support a unified interpretation. Nonlocal weak values are conditioned quantities associated with pre- and post-selected quantum processes whose defining structure is interference of amplitudes rather than averaging over positive probabilities. They may attach to explicitly nonlocal observables, to local observables in entangled or delocalized contexts, to quantum histories, or to effective dynamical fields. Their anomalies track quasi-probability negativity, contextuality, and entanglement; their measurement requires weak coupling and conditioning; and their most distinctive feature is that they encode nonlocal quantum structure in a form that appears directly in conditioned averages, pointer shifts, and reduced dynamics.