Weak Measurements

A weak measurement uses a coupling or instrument that gains little information and causes a small state change in a typical individual step. Rare selected outcomes can behave differently, so the weak approximation must be checked for the ensemble being discussed. Repeating the measurement many times, or taking a continuous-time limit, can still produce strong information and substantial cumulative disturbance.

The useful comparison is between one step and the accumulated measurement:

$$\text{weak} \quad \text{means low signal-to-noise per shot,} \quad \text{with small typical backaction.}$$

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.

Helpful background. Unsharp measurements give the qubit example; the pointer model explains how a coupling produces a readout.

Weak qubit instruments

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 $P_Q$. Assume a fixed initially uncorrelated pointer and neglect free evolution during the interaction. Then

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

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:

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

For a smooth real Gaussian pointer, with no position–momentum correlation, weak distinguishability and small typical backaction follow when the shifts between relevant eigenvalues are small compared with its position standard deviation:

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

Then the pointer distributions overlap strongly. For a Gaussian of position variance $\sigma^2=(\Delta Q)^2$, an unread coherence between $a$ and $b$ is multiplied by

$$\exp\!\left[-\frac{\kappa^2(a-b)^2}{8\sigma^2}\right].$$

Typical pointer readings therefore disturb that coherence only slightly. Width alone is not sufficient for arbitrary pointer states: rapid phase variation can give a large momentum spread and appreciable backaction despite broad position distributions. Rare far-tail outcomes also need a separate conditioning check.

Kraus Density for a Pointer Readout

For a normalized pure pointer, $\int M(q)^\dagger M(q)\,dq=I$. If its 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 the relevant displacements $|\kappa(a-b)|$ are 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. With fresh identical apparatuses, no intervening dynamics, and the same measured axis, $N$ unread square-root qubit measurements give the coherence factor

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

The omitted term in the exponent is $O(N\epsilon^4)$, so small relative error requires $N\epsilon^4\ll1$. For $\epsilon=0.05$ and $N=400$, the exact factor is about $0.60615$, while $e^{-1/2}\approx0.60653$. Each weak step is gentle, but the accumulated loss of coherence is appreciable.

For a fixed $\sigma_z$ eigenstate, the independent labels $s_j=\pm1$ have mean $\pm\epsilon$ and variance $1-\epsilon^2$. The sum has mean $\pm N\epsilon$ and standard deviation $\sqrt{N(1-\epsilon^2)}$, so its signal-to-noise ratio relative to zero 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 $\Gamma_m$ sets an inverse-time scale. Individual steps become infinitesimal, but the accumulated record over finite time carries information.

For a dimensionless Hermitian observable $A$, unit detection efficiency, and no other dynamics, fix the following convention:

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

where $dW_t$ is a Wiener increment and $\langle A\rangle_c=\operatorname{Tr}(A\rho_c)$. The conditioned state then obeys the Itô stochastic master equation

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

with

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

and, for Hermitian $A$,

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

Here $\mathbb E[dW_t]=0$ and $dW_t^2=dt$: the noise is of order $\sqrt{dt}$, while drift is of order $dt$. For the binary example with $A=\sigma_z$, this precise convention requires $\epsilon^2=4\Gamma_m dt$, because the unread off-diagonal decay rate is $2\Gamma_m$. Different definitions of measurement rate move these factors of two; the record and state equation must use the same convention. 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 proof that the system possessed an anomalous pre-existing value. It is a conditional statistical quantity arising from weak coupling plus postselection. Weak values derives the pointer response and its finite-coupling limitations.

Postselection can invalidate a weak expansion

Nearly orthogonal preselection and postselection make $A_w$ large. For a real Gaussian pointer with width $\sigma$, even $|\kappa|\,\|A\|\ll\sigma$ is then insufficient for the simplest conditional response formula: one also needs $|\kappa A_w|\ll\sigma$ and control of the higher conditional moments. The exact calculation in weak values shows how the apparent divergence disappears at fixed coupling.

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

For a smooth real Gaussian pointer, 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 prepared. For a smooth real Gaussian pointer this also means small typical backaction for that pair. Broad position width alone would not control the backaction of an arbitrarily phase-modulated pointer.

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.

References

• 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). Article.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014). Publisher.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010). Publisher.