Repeatability and QND Measurement

Repeatability and quantum nondemolition measurement are related but not identical ideas.

A measurement is repeatable when, after obtaining an outcome, an immediate repetition of the same ideal measurement gives the same outcome with probability one. This is a statement about the post-measurement state and the next measurement.

A measurement is quantum nondemolition (QND) when an observable can be monitored over time without the measurement process randomizing the later values of that same observable. This is a dynamical statement: it involves the system Hamiltonian, the measurement interaction, environmental loss, and the time scale on which repeated predictions are claimed.

Helpful background. Lüders rule supplies immediate repeatability, while measurement backaction explains how other observables can still be disturbed.

An ideal projective measurement can be repeatable and still disturb phases, conjugate variables, or unresolved internal labels. A QND measurement is allowed to disturb other degrees of freedom; it is designed so that the disturbance avoids the monitored observable.

Repeatability Condition

For the finite-dimensional repeatability argument, let a sharp measurement be represented by projectors $\{\Pi_a\}$, with

$$\Pi_a\Pi_b=\delta_{ab}\Pi_a, \qquad \sum_a\Pi_a=I.$$

For the ideal Lüders operation,

$$\mathcal I_a(\rho)=\Pi_a\rho\Pi_a,$$

the conditional state after outcome $a$ is

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

A second immediate measurement of the same projective decomposition gives $a$ with probability one:

$$\operatorname{Tr}(\Pi_a\rho_a) = \frac{\operatorname{Tr}(\Pi_a\Pi_a\rho\Pi_a)} {\operatorname{Tr}(\Pi_a\rho)} =1.$$

This is the simplest repeatability theorem for ideal projective measurements. It assumes no intervening dynamics, no noise, and a post-measurement system in the same Hilbert space.

For a more general outcome operation $\mathcal I_a$, repeatability relative to the sharp question $\{\Pi_a\}$ requires the output of the $a$ operation to lie entirely in the $a$ subspace:

$$\Pi_a\mathcal I_a(\rho)\Pi_a = \mathcal I_a(\rho)$$

for all inputs $\rho$ for which outcome $a$ can occur. Equivalently, when $\operatorname{Tr}\mathcal I_a(\rho)\ne0$,

$$\frac{ \operatorname{Tr}[\Pi_a\mathcal I_a(\rho)] }{ \operatorname{Tr}\mathcal I_a(\rho) } =1.$$

The operation may still disturb other observables. Repeatability fixes only the next answer to the same question.

Repeatability Is Not Nondestructiveness

Repeatability is often misread as “the system was not disturbed.” That is too strong.

An ideal measurement of $\{\Pi_a\}$ can be repeatable while changing the state from $\rho$ to $\rho_a$. If the input had coherence between different outcome subspaces, the selective update removes all components outside the observed subspace. If the outcome is ignored, the nonselective channel

$$\rho \longmapsto \sum_a\Pi_a\rho\Pi_a$$

removes coherence between different eigenspaces.

Degeneracy makes the point sharper. A Lüders measurement of a degenerate observable preserves coherence inside the measured eigenspace, while a refined apparatus can secretly measure additional labels and then report only the coarse outcome. Both devices can be repeatable for the coarse eigenvalue, but they leave different states for later compatible measurements. See Lüders Rule.

Destructive detection is different again. A photodetector click may be a sharp event in the detector record while absorbing the photon. It is not a QND measurement of the photon number remaining in the same mode; the carrier has been removed.

Dynamical Repeatability

Suppose the first measurement prepares the system in the $\Pi_a$ subspace and the system then evolves unitarily under $H_S$ for time $t$:

$$U(t)=\exp(-iH_St/\hbar).$$

The outcome $a$ remains certain at the later time if the subspace is invariant under the evolution:

$$U(t)\Pi_aU^\dagger(t)=\Pi_a.$$

A sufficient time-independent condition is

$$[H_S,\Pi_a]=0$$

for every measured projector. Then

$$\operatorname{Tr} \!\left[ \Pi_aU(t)\rho_aU^\dagger(t) \right] =1.$$

If the measured observable is

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

then $[H_S,A]=0$ implies the spectral projectors are conserved, in finite dimension. For unbounded operators, conservation requires the corresponding spectral commutation or invariance statement; a formal commutator on an unspecified domain is insufficient. This is why energy measurements are the textbook example of repeatability over time in a closed system: ideal energy eigenprojectors commute with the Hamiltonian.

For comparison, take $A=\sigma_z$ and $H_S=\hbar\Omega\sigma_x/2$. After obtaining $+z$, the state evolves to $\cos(\Omega t/2)|0\rangle-i\sin(\Omega t/2)|1\rangle$. The next $+z$ probability is $\cos^2(\Omega t/2)$: immediate repeatability is exact, but finite-time certainty is lost.

The condition can be relaxed. In stroboscopic protocols, one may only require the measured Heisenberg-picture observables at selected times to be mutually compatible:

$$[A(t_i),A(t_j)]=0$$

for the times actually probed. This allows QND-style monitoring of selected oscillator quadratures at special phases, even when $A$ is not conserved at every instant.

Position readouts separated by half a cycle

Consider an isolated harmonic oscillator with $H_S=P^2/(2m)+m\omega^2X^2/2$, $m,\omega>0$, and instantaneous position-measurement pulses. Between pulses,

$$X(t)=X(0)\cos\omega t+\frac{P(0)}{m\omega}\sin\omega t, \qquad [X(t),X(t')]=\frac{i\hbar}{m\omega}\sin[\omega(t'-t)].$$

At times $t_n=n\pi/\omega$, the free evolution gives $X(t_n)=(-1)^nX(0)$. The sign reverses each half cycle, but these sampled observables commute. An instantaneous position measurement changes momentum; that momentum disturbance contributes nothing to position at the next exact half cycle. This is the stroboscopic example discussed by Braginsky, Vorontsov, and Thorne (1980), Eq. (4a). It requires the harmonic dynamics and the specified measurement coupling, not only a choice of sampling times.

To quantify the backaction without preparing an unnormalizable position eigenstate, use the Gaussian position instrument with resolution $\sigma>0$. Its unread channel is equivalent to an independent zero-mean momentum kick $\xi$ with variance $\hbar^2/(4\sigma^2)$. A pulse at $t=0$ therefore adds, relative to evolution without that pulse,

$$\delta\operatorname{Var}X(t) =\frac{\hbar^2}{4m^2\omega^2\sigma^2}\sin^2\omega t.$$

This added variance vanishes at every $t_n$, although it is nonzero between them. The same channel adds mean energy $\hbar^2/(8m\sigma^2)$: nondemolition of the sampled variable does not mean no heating. These are ensemble statements for inputs with finite second moments. Finite-resolution readouts still fluctuate, and the state conditioned on a record still updates; QND does not promise identical raw numbers from a noisy detector. Finite pulse duration, damping, anharmonicity, or a timing error can feed backaction into the next sampled position.

QND Observables

A QND observable is one whose value can be inferred repeatedly without the measurement backaction demolishing the future value being inferred. In a common idealized system-meter model, the interaction has the form

$$H_I(t)=\hbar g(t) A\otimes P_M,$$

where $P_M$ is a meter variable conjugate to the pointer readout. If

$$[A,H_S]=0, \qquad [A\otimes I_M,H_I(t)]=0,$$

then the interaction can correlate the meter with $A$ without directly inducing transitions between different eigenspaces of $A$. The measurement may dephase superpositions of different $A$ values, but it does not change the populations in the $A$ basis in the ideal model.

These commutation relations are best read as sufficient idealizations, not as a complete laboratory definition. Real QND claims must also specify:

1. the Hilbert space and observable being monitored;
2. the Hamiltonian terms retained in the approximation;
3. leakage, loss, relaxation, and detector inefficiency;
4. the time scale over which nondemolition behavior is asserted;
5. which later predictions are protected and which variables are allowed to be disturbed.

A measurement can be QND for $A$ while being strongly demolishing for a conjugate phase or quadrature.

Dephasing Without Population Change

Continuous QND monitoring often appears as measurement-induced dephasing in the monitored basis. For a Hermitian dimensionless observable $A$, an unconditional master equation may contain

$$\dot\rho = -\frac{i}{\hbar}[H,\rho] + \Gamma_m\mathcal D[A]\rho,$$

with

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

If $A|a\rangle=a|a\rangle$ and $H$ is diagonal in the same basis, then

$$\dot\rho_{ab} = -\frac{i}{\hbar}(E_a-E_b)\rho_{ab} - \frac{\Gamma_m}{2}(a-b)^2\rho_{ab}.$$

For $a=b$, the dephasing term vanishes. The populations $\rho_{aa}$ are unchanged by the measurement dephasing, while coherences between different $A$ values decay. This is the unconditional evolution produced by an ideal nondemolition monitor. The same equation can also describe unobserved environmental or classical noise, so dephasing alone does not demonstrate that a usable measurement record was obtained. In a monitored realization, conditional populations change as the record reveals information, even though the ensemble populations remain fixed.

The conditioned version is described by a stochastic master equation. The measurement record gradually reveals information about $A$, while the conditioned state is updated by the observed noise. For the trajectory formalism, see Stochastic Master Equations and Diffusive Trajectories.

Circuit-QED Readout

Dispersive superconducting-qubit readout is a central approximate QND example. In the simplest two-level dispersive model,

$$\frac{H_{\mathrm{disp}}}{\hbar} = \left(\omega_r+\chi\sigma_z\right)a^\dagger a + \frac{1}{2}(\omega_q+\chi)\sigma_z.$$

This Hamiltonian commutes with $\sigma_z$ in the idealized qubit subspace:

$$[H_{\mathrm{disp}},\sigma_z]=0.$$

A microwave pulse sent through the resonator acquires a qubit-state-dependent phase. The outgoing field therefore carries information about $\sigma_z$, while the dominant ideal backaction dephases superpositions of $\lvert g\rangle$ and $\lvert e\rangle$.

The word “approximate” matters. Dispersive readout ceases to be perfectly QND when relaxation, Purcell decay, leakage outside the qubit subspace, nonadiabatic pulse effects, dressed-state transitions, residual thermal photons, or detector-induced excitation become relevant. The platform discussion is Circuit QED.

Cavity-QED Photon Number

In cavity QED, photon-number QND measurements use a probe whose phase depends on the number of photons in a cavity mode without absorbing those photons in the ideal limit. A dispersive atom-field interaction can have the schematic form

$$\frac{H_{\mathrm{int}}}{\hbar} = \chi a^\dagger a\,\sigma_z.$$

Since

$$[a^\dagger a,H_{\mathrm{int}}]=0,$$

the interaction can imprint photon-number information on the probe atom while preserving the photon number of the cavity field in the idealized interaction. The state may still lose phase coherence between different number states, and real cavities have loss. Thus the QND statement is always limited by photon lifetime, imperfect probes, and unwanted absorption or emission channels.

See Cavity QED for the broader open-system setting.

Gravitational-Wave and Optomechanical Contexts

QND ideas entered gravitational-wave detection because a continuous position measurement of a free mass is not automatically nondemolition. For a free mass,

$$X(t)=X(0)+\frac{P(0)t}{m},$$

so

$$[X(t),X(t')] = \frac{i\hbar}{m}(t'-t).$$

Backaction on momentum at one time can feed into position at a later time. The standard quantum limit for simple interferometric position readout reflects this tradeoff between imprecision and radiation-pressure backaction.

QND and backaction-evading strategies change the measurement design: they may monitor a variable whose later value is not contaminated by the backaction, correlate imprecision and backaction noise, use squeezed input states, or measure selected mechanical quadratures. These strategies do not remove quantum backaction from nature; they route it away from the signal variable for a specified task. See Optomechanics for the platform-level context.

Exercises

A timing error returns the backaction

In the oscillator example, the next pulse arrives at $t=\pi/\omega+\epsilon$ with $|\omega\epsilon|\ll1$. Find the leading added position variance from the first pulse. Compare it with the detector variance $\sigma^2$ and obtain a condition on $|\epsilon|$ for this contribution to be negligible. Why does making the first readout sharper make the timing requirement stricter?

Solution

The kick contributes $\delta X=\xi\sin(\pi+\omega\epsilon)/(m\omega)$. Averaging its square gives

$$\delta\operatorname{Var}X =\frac{\hbar^2\epsilon^2}{4m^2\sigma^2} \left[1+O((\omega\epsilon)^2)\right].$$

The extra variance is much smaller than $\sigma^2$ when $|\epsilon|\ll2m\sigma^2/\hbar$, in addition to $|\omega\epsilon|\ll1$. This comparison isolates the first pulse’s backaction; it is not the total prediction error, which also includes the initial state uncertainty and the next detector’s imprecision. Reducing $\sigma$ increases the first pulse’s momentum disturbance and reduces the error budget against which it is compared, so the timing scale falls as $\sigma^2$.

At exact half-cycle spacing this contribution is zero even though the mean energy added by the first pulse remains positive. From $P\mapsto P+\xi$, zero mean and independence remove the cross term, giving $\delta\langle H_S\rangle=\operatorname{Var}(\xi)/(2m)=\hbar^2/(8m\sigma^2)$.

Conserved Projector

Let $U(t)=\exp(-iH_St/\hbar)$ and suppose $[\Pi_a,H_S]=0$. Show that if $\rho_a$ satisfies $\Pi_a\rho_a\Pi_a=\rho_a$, then the later state $U(t)\rho_aU^\dagger(t)$ is still supported in the same subspace.

Solution

The commutator condition implies $U(t)\Pi_a=\Pi_aU(t)$. Then

$$\Pi_aU(t)\rho_aU^\dagger(t)\Pi_a = U(t)\Pi_a\rho_a\Pi_aU^\dagger(t) = U(t)\rho_aU^\dagger(t).$$

Thus a later ideal measurement of the same projector gives outcome $a$ with probability one.

Pure Dephasing as QND Monitoring

For a qubit with $A=\sigma_z$, consider the unconditional master equation

$$\dot\rho = \Gamma_m \left( \sigma_z\rho\sigma_z-\rho \right).$$

Write the equations for the density-matrix elements in the $\sigma_z$ basis.

Solution

Let

$$\rho = \begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix}.$$

Since

$$\sigma_z\rho\sigma_z = \begin{pmatrix} \rho_{00} & -\rho_{01}\\ -\rho_{10} & \rho_{11} \end{pmatrix},$$

the equation gives

$$\dot\rho_{00}=0, \qquad \dot\rho_{11}=0, \qquad \dot\rho_{01}=-2\Gamma_m\rho_{01}, \qquad \dot\rho_{10}=-2\Gamma_m\rho_{10}.$$

The populations in the measured basis are unchanged, while coherence between the two measured values decays. This is the unconditional dephasing associated with ideal $\sigma_z$ monitoring; an observed record is additional information not specified by this master equation alone.

Destructive Photon Counting

Why is ordinary absorption photodetection not a QND measurement of photon number in the same mode?

Solution

An absorption click removes a photon from the mode. In quantum-jump language the click operation is proportional to

$$a\rho a^\dagger.$$

This changes the photon number of the remaining field. The detector record can be sharp and useful, but it demolishes the photon whose presence was detected. A photon-number QND measurement instead tries to infer $a^\dagger a$ through a dispersive probe while preserving the photon number, up to cavity loss and other imperfections.

References

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• V. B. Braginsky, Yu. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209, 547–557 (1980), Eq. (4a) and the discussion of stroboscopic measurements. Author-hosted article.
• V. B. Braginsky and F. Y. Khalili, Quantum Measurement, Cambridge University Press (1992).
• C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Reviews of Modern Physics 52, 341–392 (1980).
• S. Gleyzes et al., “Quantum jumps of light recording the birth and death of a photon in a cavity,” Nature 446, 297–300 (2007).
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• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).
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