Weak Measurements and Weak Values

Weak values describe a conditional response of a weakly coupled meter for a preselected and postselected ensemble. Their foundational interest is that this response can lie outside an observable’s eigenvalue range or be complex. The technical definition and pointer derivation belong to Weak Values; this page examines what an anomalous response licenses us to infer.

An anomalous qubit weak value

Prepare $\lvert i\rangle=\lvert+x\rangle$ and postselect

$$\lvert f\rangle=\cos\epsilon\,\lvert-x\rangle+ \sin\epsilon\,\lvert+x\rangle, \qquad 0<\epsilon<\pi/2.$$

For $A=Z$, the transition-amplitude ratio is

$$A_w=\frac{\langle f|Z|i\rangle}{\langle f|i\rangle} =\cot\epsilon, \qquad p_f^{(0)}=\sin^2\epsilon.$$

At $\tan\epsilon=1/10$, $A_w=10$ even though the projective outcomes of $Z$ are $+1$ and $-1$. The zeroth-order postselection probability is $1/101$. This is a property of an interference-sensitive conditional ensemble, not a tenth eigenvalue or a detector reporting $10$ on every retained trial.

A von Neumann interaction $U=\exp(-igZ\otimes P/\hbar)$ shifts a sufficiently broad real meter wave packet by approximately $g\operatorname{Re}A_w$ after successful postselection. The imaginary part generally changes conjugate-meter statistics and depends on the initial meter correlations. The pointer-response derivation states the required assumptions about the initial meter.

A large number does not remove the weak-coupling condition

The expansion is controlled by the actual conditioned meter response. For typical meter momenta, one requires the first-order term to remain small, in particular $|g A_w|\Delta P/\hbar\ll1$ in the simple model. Taking $\epsilon\to0$ at fixed $g$ invalidates this approximation before an infinite physical shift appears. Exactly orthogonal pre- and postselection makes the usual ratio undefined.

The tradeoff between amplification, success probability, detector noise, and resource accounting prevents a universal sensitivity claim. Discarded trials still consume prepared systems. Whether amplification helps a practical experiment depends on technical limitations and the estimator, as reviewed by Dressel and collaborators.

What the anomaly says about properties

A classical conditional mean of a variable constrained to $[-1,1]$ remains in that interval if the conditioning merely selects members of a fixed distribution. A weak measurement is a physical intervention, and quantum transition amplitudes need not form that classical conditional distribution. This identifies the assumption to investigate.

Under additional experimentally checkable noncontextuality and disturbance conditions, anomalous weak values can support contextuality witnesses. An anomaly alone, without those conditions, is not a complete contextuality proof. Nor does a weak value generally describe a trajectory or a value possessed independently of the postselection. Interpretive readings must specify their additional ontology.

Exercise

In the example, compute $p_f^{(0)}|A_w|^2$. Explain its behavior as $\epsilon\to0$.

Solution

It equals $\cos^2\epsilon$, bounded by one. The divergence of the conditional coefficient is accompanied by a vanishing successful fraction. This elementary identity is not a full Fisher-information calculation, but it demonstrates why amplification alone is insufficient to establish an unlimited resource advantage.

References

• Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N., and Boyd, R. W., “Colloquium: Understanding quantum weak values: Basics and applications,” Reviews of Modern Physics 86, 307–316 (2014), doi:10.1103/RevModPhys.86.307 — operational response and experimental uses.
• Pusey, M. F., “Anomalous weak values are proofs of contextuality,” Physical Review Letters 113, 200401 (2014), doi:10.1103/PhysRevLett.113.200401 — the additional hypotheses behind the contextuality inference.