Informationally Weak Measurements

• Informationally Weak Measurements are gentle quantum interactions that extract complex amplitude data with minimal disturbance and preserved coherence.
• They enable direct extraction of the real and imaginary parts of weak values, thereby providing insights into the interference among Feynman paths.
• Practical limitations include the inability to recover full probability distributions, necessitating complementary strong measurements for complete state reconstruction.

Informationally weak measurements are a class of quantum measurements in which the interaction between the system and the measurement apparatus (“meter” or “pointer”) is sufficiently gentle that only a vanishingly small amount of information about the observable is extracted per individual run, and correspondingly the disturbance to the system is minimal. These measurements provide direct access to complex amplitudes, rather than probabilities, and are distinguished by their operational, information-theoretic, and foundational characteristics. The regime of informationally weak measurements is central to understanding the modern framework of quantum measurement, quantum information, and foundational quantum mechanics.

1. Quantum Framework and Definitions

In an N-dimensional quantum system preselected in state $|\psi\rangle$ and postselected in $|\phi\rangle$, the transition amplitude over time $T$ is

$$A^{\phi \leftarrow \psi} \equiv \langle \phi | e^{-i\hat{H} T} | \psi \rangle\,.$$

Inserting a resolution of the identity at some intermediate time (e.g., $t = T/2$) in the eigenbasis $\{|i\rangle\}$ of an operator $\hat{S}$ decomposes the amplitude into $N$ virtual (Feynman) paths:

$$A^{\phi \leftarrow \psi} = \sum_{i=1}^N A^{\phi \leftarrow \psi}_i, \ \ \ A^{\phi \leftarrow \psi}_i \equiv \langle \phi | e^{-i\hat{H} T/2} | i \rangle \langle i | e^{-i\hat{H} T/2} | \psi \rangle.$$

A measurement coupling at this intermediate point is realized via the von Neumann interaction, typically modeled as an impulsive coupling Hamiltonian $$H_\text{int} = -i\,\delta(t-T/2)\,\partial_f\,\hat{S}$$, where $f$ is the pointer coordinate. The initial pointer state is a wavepacket $G(f)$ of width $\Delta f$.

In the weak-coupling regime ($\Delta f \to \infty$), the measurement operators act as near-identity transformations, so the coherent superposition of virtual paths remains essentially intact. The key figure of merit is the weak value (Aharonov–Albert–Vaidman value)

$$A_w = \frac{ \langle \phi | \hat{S} | \psi \rangle }{ \langle \phi | \psi \rangle },$$

which, more generally, corresponds to a linear combination of relative path amplitudes

$$\sum_i S_i\,\alpha_i,\quad\text{where}\ \alpha_i = \frac{ A^{\phi \leftarrow \psi}_i }{ \sum_j A^{\phi \leftarrow \psi}_j }\,.$$

The average pointer shift and its conjugate encode $\operatorname{Re}\sum_i \alpha_i S_i$ and $\operatorname{Im}\sum_i \alpha_i S_i$, respectively.

2. Information Content and Operational Limitations

Informationally weak measurements are characterized by the information-disturbance balance and by which statistical properties they reveal:

• Strong/projective measurement ($\Delta f\to0$): Yields classical probabilities $\omega_i = |A_i|^2 / \sum_j |A_j|^2$, from which all statistical moments and the full distribution $P(S_i) = \omega_i$ can be computed.
• Weak measurement ($\Delta f\to\infty$): Yields only linear combinations of the relative amplitudes $\alpha_i$, i.e., $\sum_i S_i\,\operatorname{Re} \alpha_i$ and $\sum_i S_i\,\operatorname{Im} \alpha_i$. No higher-order statistical moments (variances, higher cumulants) or the full distribution of $S_i$ are extractable without further (strong) measurements.

Key consequences:

• The full set $\{\alpha_i\}$ for all $i$ may in principle be reconstructed by simultaneous (or sequential) weak measurements of all projectors $\{|i\rangle\langle i|\}$ at the intermediate point. This provides a tomographic description of the Feynman-path amplitude composition.
• Weak measurements never, by themselves, produce the probability distribution $P(S_i)$. Frequencies only materialize upon projective collapse, which converts amplitudes into probabilities.
• Second and higher nonlinear statistical functions of the classical distribution $\omega_i$ are inaccessible unless one combines weak measurement data with additional strong measurements.

3. Measurement Theory: Formal and Model Aspects

Let $\hat{S}$ have eigenvalues $\{S_i\}$. After the weak measurement interaction and subsequent postselection, the pointer state becomes

$$G'(f) = \sum_{i=1}^N A^{\phi \leftarrow \psi}_i\,G(f - S_i).$$

In the strong limit ($\Delta f\to0$), $G(f-S_i)$ become orthogonal, decohering the paths and yielding projective statistics. In the weak limit ($\Delta f\to\infty$), the states are indistinguishable, and the interference among virtual paths is maintained. A perturbative (first-order) expansion in $1/\Delta f$ yields the weak-value formula for the pointer:

$$\langle f \rangle_\text{weak} = \operatorname{Re} \sum_i \alpha_i S_i, \qquad \langle \lambda \rangle_\text{weak} \propto \operatorname{Im} \sum_i \alpha_i S_i.$$

These results are a direct manifestation of the fact that a weak measurement is a measurement of the complex amplitudes $\alpha_i$ composing the virtual transition, not of the observable's probabilities.

4. Significance and Resolution of Paradoxes

From the amplitude-centric viewpoint, anomalous and counterintuitive weak values, including their “arbitrary” magnitude or values outside the spectrum of $\hat{S}$, are seen to be consequences of tuning the interference pattern between virtual paths rather than expectations from a classical probability ensemble. Three canonical paradoxes are clarified:

• Large weak values (e.g., spin-½ system yielding $\langle f\rangle_\text{weak} = 100$): Such results directly correspond to a pre- and post-selection pair that contrives highly improbable interference, making $\alpha_1 - \alpha_2 = 100$, without contradiction.
• Quantum Cheshire cat: Apparent paradoxes, e.g., finding a particle's presence in one region and its property (such as polarization) in another, reduce to selecting pre- and post-selection states in which only certain amplitude components are present in each measurement context; different measurements define different real path ensembles.
• Three-box and Hardy-type paradoxes: Weak measurements reconstruct only the amplitudes relevant for the specific operator and basis under consideration, avoiding the fallacy of simultaneously “real” alternatives across different measurement partitions.

This perspective abolishes the need for exotic ontologies or the notion of “elements of reality” being unveiled by weak measurements. All observed statistics trace back to standard quantum amplitudes.

5. Fundamental and Practical Limitations

While the weak-measurement paradigm allows the extraction of complex amplitude ratios and their linear combinations, several rigorous signal-processing and information-theoretic boundaries are observed:

• Only position in amplitude space is accessible, not norm (probability) itself, under weak measurement alone.
• Since information is encoded in interference among amplitudes, loss of coherence (e.g., through environmental decoherence or system noise) invalidates the protocol.
• Weak measurements cannot reconstruct the probability distribution over measurement outcomes without further projective measurement or additional informationally complete procedures.
• All paradoxical or “anomalous” weak values are consistently and quantitatively accounted for within the standard quantum formalism without invoking new postulates or interpretations.

6. Illustrative Examples and Extensions

The general decomposition into virtual paths and the pointer-state formalism suffice to derive all standard weak-measurement results, including in realistic experimental settings. For example, in tunneling problems,

$\psi^{T}(x, t) = \int A(x')\,G(x - v t - x')\,dx',$

directly mirrors the structure of the weak-measurement pointer formula, with $x'$ analogous to the discrete $S_i$.

Moreover, by performing simultaneous weak measurements across a set of projectors at the intermediate time, one reconstructs the full set of amplitude components contributing to the transition. Such “tomographic” procedures have been discussed in protocols for remote state determination and weak-value tomography.

7. Summary and Outlook

Informationally weak measurements, as precisely defined by their gentle system-pointer coupling that preserves interference, realize experimental access to the complex probability amplitudes of Feynman paths between pre- and post-selected states. They do not yield hidden variables or classical probabilities in situ, but instead interrogate the “what interferes” of quantum transitions. All classical statistical properties—exclusive alternatives, frequencies, variances—arise only with projective measurement that breaks quantum coherence. The theoretical and experimental paper of weak measurements continues to underpin both conceptual clarifications and the quantification of information extraction versus disturbance in quantum technology, reinforcing that all “unusual” behaviors are direct manifestations of the structure of quantum amplitudes (Sokolovski, 2015).