Weak Measurements

A weak measurement is a measurement that extracts only a small amount of information in a single trial and produces correspondingly small conditional backaction in that trial. Repeating the measurement many times, or taking a continuous-time limit, can still produce strong information and substantial cumulative disturbance.

The safe definition is operational:

$$\text{weak} \quad \text{means low signal-to-noise per shot,} \quad \text{not disturbance-free.}$$

Weak measurements are best specified by an instrument or by an explicit system-pointer model. A POVM may be unsharp while the underlying instrument is still highly disturbing, so the phrase “weak” should not be inferred from effects alone.

Relation to Unsharp Measurements

An unsharp measurement has nonprojective effects. A weak measurement is a regime of small information gain and small state change per measurement step.

For the qubit unsharp $\sigma_z$ effects

$$E_\pm = \frac{1}{2} \left( I\pm\epsilon\sigma_z \right), \qquad 0\le\epsilon\ll1,$$

a gentle square-root instrument has measurement operators

$$M_\pm = \sqrt{\frac{1}{2}} \left( \sqrt{1\pm\epsilon}\,P_+ + \sqrt{1\mp\epsilon}\,P_- \right),$$

where $P_\pm=(I\pm\sigma_z)/2$. Each result only slightly biases the state toward one $\sigma_z$ eigenspace.

The associated nonselective channel multiplies the off-diagonal element in the $\sigma_z$ basis by

$$\sqrt{1-\epsilon^2} \approx 1-\frac{\epsilon^2}{2}.$$

Thus the unread disturbance is second order in the small measurement strength for this instrument. That gentle scaling is an instrument property, not a consequence of the POVM alone. See Unsharp Measurements.

Weak Pointer Coupling

The standard von Neumann pointer model makes the weak regime geometric. Let the measured observable be

$$A=\sum_a aP_a,$$

and let a pointer have coordinate $Q$ and conjugate momentum $P_Q$. During the measurement interaction,

$$H_{\mathrm{int}}(t) = g(t)A\otimes P_Q.$$

Define the integrated coupling

$$\kappa=\int dt\,g(t).$$

The corresponding unitary is

$$U = \exp \!\left( -\frac{i}{\hbar}\kappa A\otimes P_Q \right).$$

If the system is in an eigenstate $\lvert a\rangle$ and the initial pointer wavefunction is $\phi(q)$, the interaction shifts the pointer:

$$\lvert a\rangle\phi(q) \longmapsto \lvert a\rangle\phi(q-\kappa a).$$

The measurement is weak when the shifts between relevant eigenvalues are small compared with the pointer width:

$$|\kappa(a-b)| \ll \Delta Q.$$

Then the pointer distributions overlap strongly. A single readout gives little information about $a$, and the conditional state changes only slightly.

Kraus Density for a Pointer Readout

If the pointer coordinate is read and the outcome is $q$, the induced system measurement operator is formally

$$M(q) = \langle q|U|\phi\rangle = \phi(q-\kappa A),$$

where the last expression is functional calculus for the observable $A$. The probability density is

$$p(q) = \operatorname{Tr} \!\left[ M(q)^\dagger M(q)\rho \right].$$

The conditional state after observing $q$ is

$$\rho_q = \frac{M(q)\rho M(q)^\dagger} {p(q)}.$$

When $\kappa$ is small relative to the pointer width, nearby eigenvalues of $A$ produce heavily overlapping likelihoods. The measurement is informative only statistically over many repetitions or after combining it with other conditioning.

Repetition and Accumulation

Weak measurements are weak per step, not weak in total. If the same weak unread square-root qubit measurement is repeated $N$ times and the outcomes are ignored, the coherence factor is

$$\left(\sqrt{1-\epsilon^2}\right)^N \approx \exp\!\left(-\frac{N\epsilon^2}{2}\right)$$

for small $\epsilon$. A large number of weak steps can therefore produce strong dephasing.

The record also accumulates information. For a small bias $\epsilon$, the signal-to-noise ratio between the two $\sigma_z$ eigenstates grows like

$$\epsilon\sqrt{N}.$$

Thus $N$ of order $1/\epsilon^2$ weak trials can reveal order-one information. The cumulative backaction appears on the same scale.

Continuous Measurement Limit

Continuous measurement is the limit of many weak instruments applied over short time steps. A common scaling is

$$\epsilon^2 \sim \Gamma_m\,dt,$$

where $\Gamma_m$ is a measurement rate. Individual steps become infinitesimal, but the accumulated record over finite time carries information.

For a Hermitian monitored observable $A$, one common diffusive-record convention has a record increment

$$dr_t = 2\sqrt{\Gamma_m}\, \langle A\rangle_c\,dt + dW_t,$$

where $dW_t$ is a Wiener increment and $\langle A\rangle_c=\operatorname{Tr}(A\rho_c)$. The conditioned state obeys a stochastic master equation whose measurement part has the schematic form

$$d\rho_c = \Gamma_m\mathcal D[A]\rho_c\,dt + \sqrt{\Gamma_m}\, \mathcal H[A]\rho_c\,dW_t,$$

with

$$\mathcal D[A]\rho = A\rho A - \frac{1}{2} \{A^2,\rho\},$$

and, for Hermitian $A$,

$$\mathcal H[A]\rho = A\rho+\rho A - 2\operatorname{Tr}(A\rho)\rho.$$

Different communities use different factors of $2$ in $\Gamma_m$ and in the record. The invariant point is the scaling: weak information per short interval plus stochastic conditioning yields continuous monitoring. See Stochastic Master Equations and Diffusive Trajectories.

Weak Values Preview

Weak values arise when a weak measurement is combined with preselection and postselection. Prepare an initial state $\lvert\psi_i\rangle$, weakly couple a pointer to observable $A$, and later keep only runs in which a final measurement finds $\lvert\psi_f\rangle$.

The weak value is

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

provided the denominator is nonzero.

In the weak-coupling limit, the mean pointer shift in the postselected ensemble is proportional to $\operatorname{Re}A_w$, while the conjugate pointer shift is related to $\operatorname{Im}A_w$ in standard Gaussian-pointer conventions.

Weak values can be complex and can lie outside the spectrum of $A$. That is not a single-shot detector reading and not direct evidence that the system possessed an anomalous pre-existing value. It is a conditional statistical quantity arising from weak coupling plus postselection. The canonical treatment of the weak-value quantity itself is Weak Values.

Why Postselection Matters

Postselection changes the ensemble. A weak measurement without postselection estimates ordinary expectation values. A weak measurement with postselection estimates a conditional response of the pointer in the retained subensemble.

If $\langle\psi_f|\psi_i\rangle$ is small, the weak value can become large. But the postselection probability is then small, so statistical uncertainty and experimental imperfections become crucial. Large weak values are not free amplification; they come with sampling costs and sensitivity to background errors.

This is the main caution:

$$\text{weak-value formulas} \quad \text{are statistical response formulas,} \quad \text{not licenses to ignore measurement modeling.}$$

What Is Actually Disturbed

A weak measurement can be gentle in the measured trial, but it still correlates system and pointer. If the pointer is ignored, the system experiences a nonselective channel. If the pointer is read, the system is conditioned on a noisy outcome. If a postselection is then applied, the retained ensemble is biased by both the weak readout and the final selection.

Disturbance can be small enough to justify a perturbative expansion, but it is not absent. The validity of a weak-measurement calculation depends on:

1. the coupling strength relative to pointer width;
2. the number of repetitions or total monitoring time;
3. whether the pointer record is retained or ignored;
4. whether postselection is rare;
5. whether higher-order backaction terms are negligible for the claimed result.

Common Mistakes

Weak means no disturbance

Weak per step does not mean disturbance-free. Backaction can accumulate over many repetitions or become important after postselection.

Weak means noisy projective measurement

A weak measurement is not merely a sharp measurement with bad classical readout. It requires a weak system-pointer correlation or an instrument with small per-step state change.

Reading weak values as eigenvalues

Weak values can be outside the eigenvalue range and can be complex. They are conditional pointer-response quantities, not ordinary eigenvalues revealed in a single run.

Ignoring the postselection probability

Large weak values often occur when the postselected state is nearly orthogonal to the preselected state. The retained data set can then be rare and statistically fragile.

Forgetting the continuous limit scaling

In continuous measurement, both the information gain and the noise scale with $dt$. Treating the record increment like an ordinary deterministic small number gives the wrong stochastic equation.

Exercises

Weak Qubit Effects

For

$$M_\pm = \sqrt{\frac{1}{2}} \left( \sqrt{1\pm\epsilon}\,P_+ + \sqrt{1\mp\epsilon}\,P_- \right),$$

show that the effects are $E_\pm=(I\pm\epsilon\sigma_z)/2$.

Solution

Since $P_+P_-=0$ and $P_\pm^2=P_\pm$,

$$M_+^\dagger M_+ = \frac{1+\epsilon}{2}P_+ + \frac{1-\epsilon}{2}P_-.$$

Using $P_\pm=(I\pm\sigma_z)/2$, this becomes

$$M_+^\dagger M_+ = \frac{1}{2} \left( I+\epsilon\sigma_z \right).$$

Similarly,

$$M_-^\dagger M_- = \frac{1}{2} \left( I-\epsilon\sigma_z \right).$$

Accumulated Dephasing

If each unread weak step multiplies $\rho_{01}$ by $\sqrt{1-\epsilon^2}$, show that after $N$ identical steps the factor is approximately $\exp(-N\epsilon^2/2)$ for small $\epsilon$.

Solution

After $N$ steps, the factor is

$$\left(\sqrt{1-\epsilon^2}\right)^N = \exp \!\left[ \frac{N}{2} \log(1-\epsilon^2) \right].$$

For small $\epsilon$,

$$\log(1-\epsilon^2)\approx-\epsilon^2.$$

Therefore

$$\left(\sqrt{1-\epsilon^2}\right)^N \approx \exp\!\left(-\frac{N\epsilon^2}{2}\right).$$

Weak Pointer Condition

In the pointer model, explain why $|\kappa(a-b)|\ll\Delta Q$ is the weak-measurement condition for eigenvalues $a$ and $b$.

Solution

The coupling shifts the pointer wavepacket by $\kappa a$ for eigenvalue $a$ and by $\kappa b$ for eigenvalue $b$. The separation between those pointer packets is $|\kappa(a-b)|$. If this separation is much smaller than the initial pointer width $\Delta Q$, the distributions overlap strongly. A single pointer reading then gives little information about which eigenvalue was present, and the measurement is weak for that pair of alternatives.

An Anomalous Weak Value

Let $A=\sigma_z$, $\lvert\psi_i\rangle=\lvert+x\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$, and

$$\lvert\psi_f\rangle = \frac{\lvert0\rangle-\alpha\lvert1\rangle} {\sqrt{1+\alpha^2}}, \qquad 0\le\alpha\lt1.$$

Compute $A_w$ and explain why it can exceed $1$.

Solution

The denominator is

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

The numerator is

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

Therefore

$$A_w = \frac{1+\alpha}{1-\alpha}.$$

As $\alpha$ approaches $1$, the preselected and postselected states become nearly orthogonal, the denominator becomes small, and $A_w$ can exceed the eigenvalue range $[-1,1]$. This is a postselected weak-response quantity, not a single-shot eigenvalue.

Cross-Links

• Unsharp Measurements
• POVMs
• Kraus Operators
• Quantum Instruments
• Weak Values
• Protective Measurements
• Measurement Backaction
• Von Neumann Measurement Model
• Compatible, Incompatible, and Sequential Measurements
• Stochastic Master Equations
• Diffusive Trajectories
• 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).
• 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).
• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer (2016).