Weak Values

A weak value is a conditional response coefficient obtained from a weak measurement followed by postselection. For a preselected pure state $\lvert\psi_i\rangle$, a later postselected pure state $\lvert\psi_f\rangle$, and an observable $A$, the standard weak value is

$$A_w = \frac{ \langle\psi_f|A|\psi_i\rangle }{ \langle\psi_f|\psi_i\rangle }, \qquad \langle\psi_f|\psi_i\rangle\ne0.$$

This expression is simple enough to invite overinterpretation. Operationally, $A_w$ is not a single-shot detector reading and not usually an eigenvalue. It is the complex coefficient that controls the first-order response of a weak pointer in the subensemble that passes the final selection.

The useful slogan is:

$$\text{weak value} \quad=\quad \text{transition-amplitude ratio} \quad=\quad \text{postselected weak-response coefficient}.$$

The page Weak Measurements develops the weak-coupling regime itself. This page is the canonical home for the weak-value quantity and its measurement-theoretic interpretation.

Preselection and Postselection

A weak-value experiment has three ingredients:

1. prepare an initial ensemble in $\lvert\psi_i\rangle$;
2. weakly couple the system to a meter so that little information is gained in each retained trial;
3. perform a later measurement and keep only runs with outcome $\lvert\psi_f\rangle$.

The retained data set is not the original ensemble. It is a postselected subensemble with zeroth-order success probability

$$p_f^{(0)} = \lvert\langle\psi_f|\psi_i\rangle\rvert^2.$$

When this probability is small, the denominator of $A_w$ is small and the weak value can become large. The same small denominator also means that the successful sample is rare and sensitive to higher-order effects, technical background, and statistical uncertainty.

Pointer Model

Use the standard von Neumann meter. The system observable $A$ couples impulsively to a pointer coordinate $Q$ through its conjugate momentum $P$:

$$U_g = \exp \!\left( -\frac{i}{\hbar}g A\otimes P \right), \qquad [Q,P]=i\hbar.$$

Let the initial pointer state be $\lvert\phi\rangle$, centered so that $\langle Q\rangle_\phi=\langle P\rangle_\phi=0$. After the coupling and successful postselection, the unnormalized pointer state is

$$\lvert\Phi_f\rangle = \langle\psi_f|U_g|\psi_i\rangle \lvert\phi\rangle.$$

For weak coupling,

$$\langle\psi_f|U_g|\psi_i\rangle = \langle\psi_f|\psi_i\rangle \left[ 1 - \frac{i g}{\hbar} A_w P + O(g^2) \right].$$

More explicitly, the second-order term contains the weak value of $A^2$:

$$\langle\psi_f|U_g|\psi_i\rangle = \langle\psi_f|\psi_i\rangle \left[ 1 - \frac{i g}{\hbar}A_w P - \frac{g^2}{2\hbar^2}(A^2)_w P^2 + O(g^3) \right],$$

where

$$(A^2)_w = \frac{ \langle\psi_f|A^2|\psi_i\rangle }{ \langle\psi_f|\psi_i\rangle }.$$

The first-order approximation is therefore not just the statement that $g$ is small. It also requires the postselected expansion to be controlled after division by $\langle\psi_f|\psi_i\rangle$.

Pointer Shifts

For a real centered Gaussian pointer with no initial $Q$-$P$ correlation, the first-order shifts are

$$\Delta\langle Q\rangle_f = g\,\operatorname{Re} A_w,$$

and

$$\Delta\langle P\rangle_f = \frac{2g(\Delta P)^2}{\hbar} \operatorname{Im} A_w.$$

The real part shifts the pointer position. The imaginary part changes the conjugate momentum distribution and is also related to the first-order change in the postselection probability. Different pointer preparations can add correlation-dependent terms, so these clean formulas should be read with their assumptions attached.

The normalized postselected pointer mean is a conditional statistic:

$$\langle Q\rangle_f = \frac{ \langle\Phi_f|Q|\Phi_f\rangle }{ \langle\Phi_f|\Phi_f\rangle }.$$

It is measured by averaging many successful trials. A single retained event does not display the weak value.

Spectral Form

If

$$A=\sum_a a\,\Pi_a,$$

then

$$A_w = \sum_a a\,(\Pi_a)_w, \qquad (\Pi_a)_w = \frac{ \langle\psi_f|\Pi_a|\psi_i\rangle }{ \langle\psi_f|\psi_i\rangle }.$$

The quantities $(\Pi_a)_w$ sum to one:

$$\sum_a(\Pi_a)_w=1.$$

They are not ordinary probabilities. They can be negative or complex, and they need not lie between $0$ and $1$. This is one way to see how $A_w$ can lie outside the eigenvalue range of $A$: the weak value is a weighted transition-amplitude ratio, not a convex average with positive weights.

Complex Weak Values

Complex weak values are not a mathematical accident. The weak interaction probes an amplitude, and amplitudes have phase. In the Gaussian pointer convention above,

$$\operatorname{Re}A_w \quad \text{controls the coordinate shift,}$$

while

$$\operatorname{Im}A_w \quad \text{controls the conjugate-momentum shift.}$$

The imaginary part can also be understood from the change in postselection probability under a small unitary disturbance generated by $A$. If the coupling slightly rotates the system before the final filter, the rate at which the success probability changes contains the imaginary part of the same transition-amplitude ratio.

This is why it is misleading to ask whether a complex weak value is “the value of $A$.” A Hermitian observable has real projective eigenvalues, but its weak value is a complex conditional response.

Ordinary Limits

Weak values reduce to familiar quantities in special cases.

If the preselected and postselected states are the same,

$$\lvert\psi_f\rangle=\lvert\psi_i\rangle,$$

then

$$A_w = \langle\psi_i|A|\psi_i\rangle.$$

If the postselected state is an eigenstate of $A$,

$$A\lvert a\rangle=a\lvert a\rangle, \qquad \langle a|\psi_i\rangle\ne0,$$

and $\lvert\psi_f\rangle=\lvert a\rangle$, then

$$A_w=a.$$

Anomalous weak values therefore require a nontrivial relation among preparation, observable, and postselection. They are not produced merely by making a meter noisy.

A Qubit Example

Let

$$\lvert\psi_i\rangle = \lvert +x\rangle = \frac{ \lvert +z\rangle+\lvert -z\rangle }{\sqrt 2},$$

and choose the postselected state

$$\lvert\psi_f\rangle = \frac{ \lvert +z\rangle-\alpha\lvert -z\rangle }{ \sqrt{1+\alpha^2} }, \qquad 0\le\alpha\lt1.$$

For $A=\sigma_z$,

$$\langle\psi_f|\psi_i\rangle = \frac{1-\alpha} {\sqrt{2(1+\alpha^2)}},$$

while

$$\langle\psi_f|\sigma_z|\psi_i\rangle = \frac{1+\alpha} {\sqrt{2(1+\alpha^2)}}.$$

Thus

$$(\sigma_z)_w = \frac{1+\alpha}{1-\alpha}.$$

For $\alpha$ close to $1$, this weak value is much larger than the eigenvalue range $[-1,1]$. The same limit makes the postselection probability small:

$$p_f^{(0)} = \frac{(1-\alpha)^2} {2(1+\alpha^2)}.$$

The anomalous value and the rare successful postselection are two sides of the same amplitude ratio.

Time-Dependent Form

If the weak coupling occurs at an intermediate time $t$ between preparation at $t_i$ and postselection at $t_f$, the natural Heisenberg-picture-looking expression is

$$A_w(t) = \frac{ \langle\psi_f| U(t_f,t)\,A\,U(t,t_i) |\psi_i\rangle }{ \langle\psi_f| U(t_f,t_i) |\psi_i\rangle }.$$

This formula is often useful, but the notation should not be read as assigning an ordinary possessed value to the system at time $t$. It is a conditional amplitude ratio for a particular preparation, dynamics, weak coupling, and final filter.

Mixed States and Effects

For a mixed initial state $\rho_i$ and a final postselection represented by an effect $E_f$, the common trace form is

$$A_w = \frac{ \operatorname{Tr}(E_f A\rho_i) }{ \operatorname{Tr}(E_f\rho_i) }, \qquad \operatorname{Tr}(E_f\rho_i)\ne0.$$

This expression keeps the time ordering appropriate to a weak measurement of $A$ before the final effect $E_f$. If the intermediate operation is not a simple weak von Neumann coupling, or if the final selection is part of a more complicated instrument, the instrument must be specified. The numerator is not just a classical conditional expectation.

Statistical Estimation

In an experiment, one estimates weak values from many trials. Let $N$ be the number of prepared systems and let $N_f$ be the number that pass postselection. Roughly,

$$N_f\approx N p_f,$$

where $p_f$ includes the effects of the weak measurement and technical imperfections. The uncertainty of the pointer mean scales with the number of successful runs, not with the number initially prepared:

$$\delta\langle Q\rangle_f \sim \frac{\Delta Q}{\sqrt{N_f}}.$$

A large $\operatorname{Re}A_w$ can amplify a pointer displacement in the retained ensemble, but it can simultaneously reduce $N_f$. Whether this improves a real measurement depends on the noise model: technical offsets, detector saturation, background counts, and imperfect postselection matter. Weak-value amplification is therefore a metrological technique with conditions, not a general way to beat quantum statistics.

Relation to Measurement Backaction

Weak values are sometimes described as revealing something “without disturbing the system.” That is too strong.

The weak interaction entangles the system and pointer. If the pointer is ignored, the system undergoes a small nonselective channel. If the pointer is read, the system is conditionally updated. If postselection is applied afterward, the retained ensemble is biased by the final filter. The disturbance can be perturbatively small per trial, but it is not zero.

The clean interpretation is:

$$A_w \quad \text{is a first-order coefficient in a controlled weak-interaction expansion.}$$

This connects weak values directly to Kraus Operators, Quantum Instruments, and Measurement Backaction.

Interpretive Caution

Weak values have been important in discussions of measurement, contextuality, quantum paradoxes, and pre/postselected ensembles. Those discussions can be illuminating, but they should be separated from the basic measurement theory.

The settled technical statement is that weak values are experimentally accessible response coefficients under specified weak-coupling and postselection conditions. Stronger claims about what a quantum system “really had” between two measurements require an interpretation and should not be presented as automatic consequences of the formula.

For broader foundations context, see What These Experiments Do and Do Not Prove.

Common Mistakes

Treating a weak value as an eigenvalue

A weak value may equal an eigenvalue in special cases, but it can also be complex or outside the spectrum. It is not the outcome of a projective measurement of $A$.

Forgetting the successful sample size

Nearly orthogonal preselection and postselection can make $A_w$ large. They also make successful postselection rare, so statistical and technical errors become more important.

Calling every noisy measurement weak

Weakness is about small system-meter coupling and small per-trial backaction in a controlled model. A sharp measurement followed by noisy classical readout is not automatically a weak measurement.

Ignoring higher-order terms

The expansion is organized after postselection. A small denominator can make second-order terms important unless the coupling is reduced accordingly.

Dropping the instrument

The same POVM effects can arise from different instruments. Weak-value predictions depend on the weak interaction and on the final selection procedure, not only on a list of probabilities.

Exercises

Same preselection and postselection

Show that if $\lvert\psi_f\rangle=\lvert\psi_i\rangle$, the weak value of $A$ is the ordinary expectation value of $A$ in the state $\lvert\psi_i\rangle$.

Solution

Substitute $\lvert\psi_f\rangle=\lvert\psi_i\rangle$:

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

For a normalized state, $\langle\psi_i|\psi_i\rangle=1$, so

$$A_w=\langle\psi_i|A|\psi_i\rangle.$$

Eigenstate postselection

Let $A\lvert a\rangle=a\lvert a\rangle$ and let $\lvert\psi_f\rangle=\lvert a\rangle$. Assume $\langle a|\psi_i\rangle\ne0$. Show that $A_w=a$.

Solution

Using $\langle a|A=a\langle a|$ for Hermitian $A$,

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

Anomalous qubit weak value

For the qubit states in the example above, compute $(\sigma_z)_w$ and identify why it can exceed $1$.

Solution

The overlaps are

$$\langle\psi_f|\psi_i\rangle = \frac{1-\alpha} {\sqrt{2(1+\alpha^2)}}, \qquad \langle\psi_f|\sigma_z|\psi_i\rangle = \frac{1+\alpha} {\sqrt{2(1+\alpha^2)}}.$$

Therefore

$$(\sigma_z)_w = \frac{1+\alpha}{1-\alpha}.$$

As $\alpha$ approaches $1$, the denominator approaches zero. The postselection becomes nearly orthogonal to the preselection, so the weak value grows while the successful sample becomes rare.

Pointer shift from the first-order state

Assume a centered real Gaussian pointer with no initial $Q$-$P$ correlation. Starting from

$$\lvert\Phi_f\rangle \propto \left( 1 - \frac{i g}{\hbar}A_w P \right) \lvert\phi\rangle,$$

show that the first-order position shift is $g\,\operatorname{Re}A_w$.

Solution

Write $A_w=a+ib$. To first order in $g$,

$$\langle Q\rangle_f = \langle Q\rangle_\phi + \frac{i g}{\hbar} \left( A_w^*\langle P Q\rangle_\phi - A_w\langle QP\rangle_\phi \right).$$

For a centered real Gaussian with no $Q$-$P$ correlation,

$$\langle Q\rangle_\phi=0, \qquad \langle QP\rangle_\phi=\frac{i\hbar}{2}, \qquad \langle P Q\rangle_\phi=-\frac{i\hbar}{2}.$$

Substitution gives

$$\langle Q\rangle_f = g a = g\,\operatorname{Re}A_w.$$

Cross-Links

• Weak Measurements
• Protective Measurements
• Unsharp Measurements
• POVMs
• Kraus Operators
• Quantum Instruments
• Measurement Backaction
• Von Neumann Measurement Model
• Selective and Nonselective Measurements
• Compatible, Incompatible, and Sequential Measurements
• Stochastic Master Equations
• What These Experiments Do and Do Not Prove
• Glossary

References

• Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Physical Review Letters 60, 1351–1354 (1988).
• I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Physical Review D 40, 2112–2117 (1989).
• Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Physical Review A 41, 11–20 (1990).
• R. Jozsa, “Complex weak values in quantum measurement,” Physical Review A 76, 044103 (2007).
• J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Reviews of Modern Physics 86, 307–316 (2014).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).