Measurement Backaction

Measurement backaction is the state change caused by the physical process that obtains information from a quantum system. It is not always a literal mechanical kick, although it can be one. More generally, backaction is the part of a measurement description that answers:

$$\text{What state is left for future predictions?}$$

The central lesson is operational:

$$\text{outcome probabilities are not enough;} \qquad \text{backaction lives in the instrument.}$$

A POVM tells how likely each outcome is. A quantum instrument tells both the outcome probabilities and the state transformation associated with each outcome. For the general framework, see Quantum Instruments.

Backaction as an Operation

Let a measurement have outcome-resolved operations $\{\mathcal I_m\}$. For input state $\rho$,

$$p(m) = \operatorname{Tr}\mathcal I_m(\rho).$$

If outcome $m$ is known, the conditional post-measurement state is

$$\rho_m = \frac{\mathcal I_m(\rho)} {\operatorname{Tr}\mathcal I_m(\rho)}.$$

If the outcome is ignored, the nonselective state is

$$\rho' = \sum_m\mathcal I_m(\rho).$$

The maps $\mathcal I_m$ and their sum encode the measurement backaction. Two devices can have the same outcome probabilities for every input state and still have different backaction.

In Kraus form,

$$\mathcal I_m(\rho) = \sum_\alpha K_{m\alpha}\rho K_{m\alpha}^\dagger.$$

The associated POVM effect is

$$E_m = \sum_\alpha K_{m\alpha}^\dagger K_{m\alpha}.$$

The effect $E_m$ determines $p(m)=\operatorname{Tr}(E_m\rho)$, but the operators $K_{m\alpha}$ determine the state left behind. That is where backaction enters.

Projective Backaction

For an ideal projective measurement with projectors $\{P_a\}$, the selective operation is

$$\mathcal I_a(\rho)=P_a\rho P_a.$$

The conditional state for outcome $a$ is

$$\rho_a = \frac{P_a\rho P_a} {\operatorname{Tr}(P_a\rho)}.$$

The unread measurement is the channel

$$\mathcal M(\rho) = \sum_aP_a\rho P_a.$$

Writing the input in blocks,

$$\rho = \sum_{a,b}P_a\rho P_b,$$

the unread channel removes off-diagonal blocks:

$$\rho \mapsto \sum_aP_a\rho P_a.$$

Thus even when no one uses the outcome, the measurement interaction can disturb future interference experiments by dephasing the measured decomposition. This is the simplest form of measurement backaction.

For the degenerate ideal case and the meaning of minimal disturbance inside an eigenspace, see Lüders Rule.

Noncommuting Observables

Backaction is most visible when a later observable does not commute with the measured one. Suppose an unread measurement $\{P_a\}$ is followed by a projective measurement $\{Q_b\}$. The probability of $b$ after the unread first measurement is

$$p'(b) = \operatorname{Tr} \!\left[ Q_b\sum_aP_a\rho P_a \right].$$

Without the first measurement, the probability would be

$$p(b)=\operatorname{Tr}(Q_b\rho).$$

If every $Q_b$ commutes with every $P_a$, then $p'(b)=p(b)$ for all $\rho$. If they do not commute, the first measurement can change the statistics of the second.

For a spin-$1/2$ example, prepare

$$\lvert +x\rangle = \frac{\lvert +z\rangle+\lvert -z\rangle}{\sqrt2}.$$

If $\sigma_x$ is measured immediately, outcome $+x$ has probability one. If an unread $\sigma_z$ measurement is inserted first, the state becomes

$$\rho' = \frac{1}{2} \lvert+z\rangle\langle+z\rvert + \frac{1}{2} \lvert-z\rangle\langle-z\rvert = \frac{I}{2}.$$

The later probability for $+x$ is then

$$\operatorname{Tr} \!\left[ \lvert+x\rangle\langle+x\rvert \frac{I}{2} \right] = \frac{1}{2}.$$

The unread $\sigma_z$ measurement has disturbed the phase information needed to predict $\sigma_x$.

Pointer Correlations

Backaction is easiest to see in an explicit apparatus model. Suppose a system basis $\{\lvert a\rangle\}$ is correlated with pointer states:

$$\lvert a\rangle\lvert R\rangle \mapsto \lvert a\rangle\lvert M_a\rangle.$$

For an input

$$\lvert\psi\rangle = \sum_a c_a\lvert a\rangle,$$

the joint state becomes

$$\sum_a c_a \lvert a\rangle\lvert M_a\rangle.$$

If the pointer is ignored, the system state is

$$\rho_S' = \sum_{a,b} c_a c_b^* \langle M_b|M_a\rangle \lvert a\rangle\langle b\rvert.$$

The pointer overlaps $\langle M_b|M_a\rangle$ control the disturbance. Orthogonal pointer states carry fully distinguishable outcome information and remove coherences between different $a$. Nearly parallel pointer states carry little information and cause only weak dephasing.

This is the operational core of the information-disturbance tradeoff. The disturbance is not added after the measurement; it is the reduced-system effect of correlating the system with degrees of freedom that store information.

See von Neumann Measurement Model for the unitary measurement model behind this calculation.

Same POVM, Different Backaction

Let an outcome $m$ have a single measurement operator

$$M_m.$$

For any unitary $U_m$, the operator

$$N_m=U_mM_m$$

has the same effect:

$$N_m^\dagger N_m = M_m^\dagger U_m^\dagger U_mM_m = M_m^\dagger M_m.$$

Therefore $M_m$ and $N_m$ give the same outcome probability for every input state. But the conditional states are generally different:

$$\frac{M_m\rho M_m^\dagger} {\operatorname{Tr}(M_m\rho M_m^\dagger)} \quad \text{versus} \quad \frac{U_mM_m\rho M_m^\dagger U_m^\dagger} {\operatorname{Tr}(M_m\rho M_m^\dagger)}.$$

The second device adds an outcome-dependent unitary kick after obtaining the same information. This is why POVMs alone do not describe backaction.

Weak Measurement

A weak measurement obtains only a small amount of information in one shot and causes correspondingly small conditional disturbance per shot. A schematic two-outcome weak measurement of a qubit in the $\sigma_z$ basis may use diagonal measurement operators such as

$$M_\pm = \sqrt{\frac{1}{2}} \left( \sqrt{1\pm\epsilon}\,P_+ + \sqrt{1\mp\epsilon}\,P_- \right), \qquad 0\le\epsilon\ll1.$$

The associated effects are

$$E_\pm = \frac{1}{2} \left( I\pm\epsilon\sigma_z \right).$$

For small $\epsilon$, each outcome only slightly biases the state toward one $\sigma_z$ eigenspace. Repeating many weak measurements can build up strong information and substantial cumulative backaction.

If outcomes are ignored, weak measurements still dephase the measured basis, but only gradually. Continuous Monitoring is the limit in which many weak updates are taken over small time steps and the state is conditioned on a time-dependent record.

Position Measurement and Momentum Disturbance

A finite-resolution position measurement can be modeled by Gaussian measurement operators

$$M_y = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp\!\left[ -\frac{(y-X)^2}{4\sigma^2} \right],$$

where $y$ is the reported position and $\sigma$ is the resolution. The outcome density is

$$p(y)=\operatorname{Tr}(M_y^\dagger M_y\rho).$$

The nonselective map suppresses spatial coherence:

$$\rho(x,x') \mapsto \exp\!\left[ -\frac{(x-x')^2}{8\sigma^2} \right] \rho(x,x').$$

In this convention, the corresponding random momentum disturbance has variance

$$\Delta p_{\text{ba}}^2 = \frac{\hbar^2}{4\sigma^2}.$$

A sharper position readout has smaller $\sigma$ and produces stronger suppression of long-range spatial coherence, along with larger momentum disturbance. This is a measurement-model statement, not merely a slogan about uncertainty.

Photon Counting

Photon counting gives a clean example where backaction is not just dephasing. For a cavity mode with annihilation operator $a$, a detected photon in a short interval is represented by the jump operation

$$\mathcal I_{\text{click}}(\rho) = \kappa\,dt\,a\rho a^\dagger.$$

Conditioned on a click, the state becomes

$$\rho_{\text{click}} = \frac{a\rho a^\dagger} {\operatorname{Tr}(a^\dagger a\rho)}.$$

The click lowers photon number and changes the field state. The absence of a click also carries information. To first order in $dt$, a no-click Kraus operator has the form

$$K_0 \approx I-\frac{\kappa\,dt}{2}a^\dagger a,$$

apart from Hamiltonian evolution. Thus even “nothing happened” in the detector can update the conditional state because it changes what is inferred about the photon number.

Stern–Gerlach Backaction

In a Stern–Gerlach device, a magnetic-field gradient correlates spin with spatial path. If the path is read out, the spin state is conditioned on the corresponding spin projection. If the path information is ignored, the spin state is dephased in the measured spin basis once the paths become distinguishable.

This example contains both a mechanical effect and an information effect. The magnetic gradient exerts forces that separate wavepackets, but the essential measurement backaction is the entanglement of spin with path and apparatus degrees of freedom. See Stern–Gerlach Revisited for the spin-measurement context.

Backaction Evasion

Backaction cannot be made to disappear from arbitrary quantum measurement. It can, however, be directed into variables that are not the target of the measurement.

A quantum nondemolition-style measurement aims to monitor an observable $A$ whose future values remain predictable. A simplified sufficient condition is that the measured observable commute with the free Hamiltonian and with the interaction in a way that does not randomize $A$:

$$[A,H_S]=0, \qquad [A,H_I]=0.$$

Then the measurement may dephase superpositions of different $A$ values while preserving the value of $A$ itself. The backaction is pushed into conjugate observables or phases rather than into the repeated readout variable.

Backaction evasion is therefore not “measurement without disturbance” in an absolute sense. It is measurement designed so that the disturbance avoids the variable one wants to keep predicting.

Measurement-Disturbance Relations

The textbook slogan that measuring position disturbs momentum captures an important idea, but precise measurement-disturbance relations are subtler than the preparation uncertainty relation

$$\Delta X\,\Delta P\ge\frac{\hbar}{2}.$$

Preparation uncertainty concerns spreads in a single quantum state. Measurement disturbance concerns how a measurement apparatus changes later statistics. Different formal definitions of measurement error and disturbance lead to different inequalities.

For this volume, the practical rule is:

Do not infer the backaction from the POVM alone. Specify the instrument or the physical measurement model.

Later specialized pages can develop rigorous error-disturbance inequalities. Here the important point is that disturbance is a property of the measurement dynamics, not only of the observable being measured.

Common Mistakes

Treating backaction as optional bookkeeping

Backaction determines future predictions. It is not extra interpretation placed on top of outcome probabilities.

Equating disturbance with force

Some measurements involve mechanical impulses, but disturbance can also be pure dephasing, conditional filtering, loss, reset, or an outcome-dependent unitary.

Assuming unread means undisturbed

If the measurement interaction occurred and left which-outcome information somewhere, ignoring the record gives a nonselective channel. It does not restore the original coherent state.

Assuming strong means projective

A strong measurement may be destructive, inefficient, or coarse-grained. A projective Lüders update is an ideal instrument, not a synonym for every high-signal detector.

Ignoring no-click backaction

In photodetection and continuous monitoring, absence of a click can be informative and can update the conditional state.

Exercises

Unread spin measurement

A qubit is prepared in $\lvert+x\rangle$. An unread $\sigma_z$ measurement is performed, followed by a $\sigma_x$ measurement. What is the probability of obtaining $+x$ at the end?

Solution

The unread $\sigma_z$ measurement maps

$$\lvert+x\rangle\langle+x\rvert \mapsto \frac{1}{2} \lvert+z\rangle\langle+z\rvert + \frac{1}{2} \lvert-z\rangle\langle-z\rvert = \frac{I}{2}.$$

Therefore

$$p(+x) = \operatorname{Tr} \!\left[ \lvert+x\rangle\langle+x\rvert\frac{I}{2} \right] = \frac{1}{2}.$$

The unread $\sigma_z$ measurement has removed the phase coherence that made the original state a definite $\sigma_x$ eigenstate.

Same effect, different operation

Let $M$ be a measurement operator for an outcome and let $N=UM$, where $U$ is unitary. Show that $M$ and $N$ give the same probability but generally different conditional states.

Solution

The effects are the same:

$$N^\dagger N = M^\dagger U^\dagger UM = M^\dagger M.$$

Thus

$$\operatorname{Tr}(N^\dagger N\rho) = \operatorname{Tr}(M^\dagger M\rho)$$

for every $\rho$. But the conditional states are

$$\rho_M = \frac{M\rho M^\dagger} {\operatorname{Tr}(M\rho M^\dagger)}$$

and

$$\rho_N = \frac{UM\rho M^\dagger U^\dagger} {\operatorname{Tr}(M\rho M^\dagger)} = U\rho_MU^\dagger.$$

Unless $U$ acts trivially on $\rho_M$, the backaction is different.

Pointer overlap

In the pointer model, what happens to the coherence $\lvert a\rangle\langle b\rvert$ when the pointer states have overlap $\langle M_b|M_a\rangle=\gamma$?

Solution

After tracing out the pointer, the coefficient of $\lvert a\rangle\langle b\rvert$ is multiplied by $\gamma$:

$$\rho_{ab} \mapsto \gamma\rho_{ab}.$$

If $\gamma=1$, the pointer carries no distinguishing information between $a$ and $b$, and that coherence is preserved. If $\gamma=0$, the pointer states are orthogonal, and the coherence is fully removed in the reduced system state.

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• V. B. Braginsky and F. Y. Khalili, Quantum Measurement, Cambridge University Press (1992).
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic (1995).
• M. Ozawa, “Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement,” Physical Review A 67, 042105 (2003).
• P. Busch, P. Lahti, and R. F. Werner, “Colloquium: Quantum root-mean-square error and measurement uncertainty relations,” Reviews of Modern Physics 86, 1261–1281 (2014).