Bayesian Quantum Measurement

Bayesian language is extremely useful in quantum measurement, but it must be used with care. Measurement outcomes provide likelihoods, observers update state assignments, and continuous records drive real-time filters. At the same time, quantum measurements are physical operations: they can disturb the system, remove coherence, entangle it with an apparatus, or apply an outcome-dependent transformation.

The useful slogan is:

$$\text{Bayes explains conditioning;} \qquad \text{instruments explain quantum state change.}$$

This page separates two ideas that are often conflated:

• Bayesian inference about a classical hypothesis, parameter, preparation, or device model;
• quantum state update of the measured system after an outcome.

They coincide in important nondemolition and effectively classical limits. They are not the same principle in general.

Classical Bayes Rule

For a classical hypothesis $h$ and observed outcome $m$, Bayes’ rule says

$$P(h\mid m) = \frac{P(m\mid h)P(h)} {\sum_{h'}P(m\mid h')P(h')}.$$

The likelihood $P(m\mid h)$ says how probable the observed outcome would be if $h$ were true. The posterior $P(h\mid m)$ is an updated probability distribution over the same alternatives.

Nothing in this rule changes the physical object being observed. It changes the observer’s probability assignment about $h$.

For the probability-theory background, see Bayes Rule.

Quantum Likelihoods

In quantum measurement, the likelihood of an outcome comes from the Born rule. If a measurement has POVM effects $\{F_m\}$, then

$$p(m\mid\rho) = \operatorname{Tr}(F_m\rho).$$

This formula can be used in ordinary Bayesian inference. For example, if a source prepares one of several candidate states $\rho_h$ with prior probabilities $P(h)$, then outcome $m$ updates the preparation hypothesis by

$$P(h\mid m) = \frac{\operatorname{Tr}(F_m\rho_h)P(h)} {\sum_{h'} \operatorname{Tr}(F_m\rho_{h'})P(h')}.$$

This is classical Bayesian inference with quantum likelihoods. The alternatives $h$ are classical hypotheses about preparation, calibration, parameters, or device settings. The posterior is a probability distribution over those hypotheses.

For a continuous family of candidate states or parameters $\theta$, the same idea becomes

$$\pi(\theta\mid m) = \frac{p(m\mid\theta)\pi(\theta)} {\int d\theta'\,p(m\mid\theta')\pi(\theta')}.$$

This is the statistical structure behind Bayesian state tomography, detector tomography, Hamiltonian learning, and parameter estimation. It is not yet a statement about the post-measurement state of the measured system.

Quantum Conditional State

The state of the measured system after a known outcome is determined by an instrument, not by the POVM effect alone. For an outcome-resolved operation $\mathcal I_m$,

$$\tilde\rho_m = \mathcal I_m(\rho), \qquad p(m\mid\rho) = \operatorname{Tr}[\mathcal I_m(\rho)],$$

and the normalized conditional state is

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

If the outcome operation has one Kraus operator $M_m$, this becomes

$$\rho_m = \frac{M_m\rho M_m^\dagger} {\operatorname{Tr}(M_m^\dagger M_m\rho)}.$$

The denominator is a likelihood. The numerator is a physical state transformation. It can change coherences, purities, correlations with other systems, and future measurement probabilities.

This is why a POVM alone is not a complete measurement-update rule. Two devices can have the same effect $F_m$ and therefore the same outcome probabilities, while leaving different conditional states.

For the instrument formalism, see Quantum Instruments and State Update Rules.

When the Analogy Becomes Exact

The Bayesian analogy becomes exact for the diagonal probabilities of a quantum nondemolition measurement. Let the system have an orthonormal pointer basis $\{|x\rangle\}$, and suppose the prior state is diagonal:

$$\rho = \sum_x p_x |x\rangle\langle x|.$$

Consider a measurement whose outcome $m$ has conditional probabilities $p(m\mid x)$ and does not mix the pointer states. A simple Kraus operator is

$$M_m = \sum_x \sqrt{p(m\mid x)} |x\rangle\langle x|.$$

Then

$$M_m\rho M_m^\dagger = \sum_x p(m\mid x)p_x |x\rangle\langle x|,$$

and normalization gives

$$p_x \longmapsto p(x\mid m) = \frac{p(m\mid x)p_x} {\sum_y p(m\mid y)p_y}.$$

For these diagonal probabilities, the quantum update is exactly Bayes’ rule. Repeating this as a nondemolition measurement also requires that the intervening dynamics preserve the pointer alternatives; a noncommuting Hamiltonian generally mixes them. This is the regime behind many quantum nondemolition readout models, classical pointer-state filters, and weak continuous measurements of a commuting observable.

What Happens to Coherence

If the input state has coherences in the pointer basis,

$$\rho = \sum_{x,y} \rho_{xy}|x\rangle\langle y|,$$

the same measurement transforms the unnormalized matrix elements as

$$\rho_{xy} \longmapsto \sqrt{p(m\mid x)p(m\mid y)}\,\rho_{xy}.$$

After normalization, the diagonal elements follow Bayes’ rule, but the off-diagonal elements are also changed. In the unread ensemble, the off-diagonal element is multiplied by $\sum_m\sqrt{p(m\mid x)p(m\mid y)}\le1$. An individual normalized outcome can instead increase a coherence magnitude by making previously unequal populations more balanced. Ideal one-Kraus conditioning preserves purity, even as the ensemble loses coherence.

This is the simplest place to see the boundary:

$$\text{Bayes updates populations;} \qquad \text{quantum backaction also changes coherences.}$$

The coherence change is not an optional philosophical interpretation. It affects later interference experiments.

Same Likelihood, Different Backaction

Outcome probabilities do not determine the post-measurement state. Suppose a one-Kraus operation has $M_m^\dagger M_m=F_m$. If an apparatus applies an outcome-dependent unitary $U_m$ after the readout, then

$$N_m = U_mM_m$$

has the same effect:

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

Thus $M_m$ and $N_m$ give the same likelihood for outcome $m$, but their conditional states differ:

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

Bayesian likelihoods alone cannot tell these devices apart. Sequential measurements can.

Selective and Nonselective Views

Bayesian language usually refers to selective information: the observer knows the outcome $m$. Quantum measurement theory also needs the nonselective state when the outcome is ignored:

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

The selective state $\rho_m$ is conditioned on a record. The nonselective state $\rho'$ averages over outcomes. Confusing them produces many apparent paradoxes: one observer who has read the detector assigns $\rho_m$, while another observer who knows only that the measurement occurred assigns $\rho'$.

This is the finite-outcome version of the continuous-monitoring distinction between a conditional trajectory and an unconditional master equation.

See Selective and Nonselective Measurements for the canonical treatment.

Weak Measurement

Weak measurements make the Bayesian structure especially visible. A single weak outcome often gives only a small likelihood ratio between alternatives, so the state update is modest. Repeated weak outcomes can nevertheless strongly distinguish the alternatives.

For a two-level nondemolition readout of $\sigma_z$, write the pointer alternatives as $z=\pm1$. If a weak detector output $r$ has likelihood density $p(r\mid z)$, then the diagonal odds update as

$$\frac{p(z=+1\mid r)} {p(z=-1\mid r)} = \frac{p(r\mid z=+1)} {p(r\mid z=-1)} \frac{p(z=+1)} {p(z=-1)}.$$

The log-odds form is additive:

$$\log \frac{p(+\mid r)} {p(-\mid r)} = \log \frac{p(+)}{p(-)} + \log \frac{p(r\mid +)} {p(r\mid -)}.$$

Continuous measurement is the limit of many such small updates. The record provides a stream of incremental likelihood ratios, while the stochastic master equation keeps track of the accompanying quantum backaction.

For the measurement-strength and weak-update viewpoint, see Weak Measurements and Continuous Monitoring.

An exactly solvable continuous readout

For $H=0$, monitor $c=\sqrt\gamma\,\sigma_z$ at unit efficiency. The two eigenstates give $Y_T\sim\mathcal N(\pm2\sqrt\gamma T,T)$. Their Gaussian likelihood ratio is

$$\frac{p(Y_T\mid+)}{p(Y_T\mid-)}=e^{4\sqrt\gamma Y_T}.$$

With equal prior populations, the conditional Bloch coordinate is therefore

$$z_c(T)=\tanh(2\sqrt\gamma Y_T).$$

If the initial state is $\lvert+x\rangle$, an ideal positive diagonal Kraus update also gives $x_c(T)=\operatorname{sech}(2\sqrt\gamma Y_T)$ and $y_c=0$. Hence $x_c^2+z_c^2=1$: population inference and a pure quantum conditional state coexist. These closed expressions are useful checks on a numerical filter. The ideal record distribution is the equal mixture of the two Gaussians, even though the initial quantum state is a coherent superposition.

Continuous Records and Filtering

For a continuous record, the posterior is updated in time. In a diffusive model,

$$dY_t = \mu_t\,dt + dW_t,$$

where $\mu_t$ is the predicted signal from the current conditional state. The likelihood density of a small record increment has the Gaussian form, up to normalization independent of $\mu_t$,

$$p(dY_t\mid\rho_c) \propto \exp \left[ - \frac{(dY_t-\mu_tdt)^2}{2dt} \right].$$

The innovation

$$dW_t = dY_t-\mu_tdt$$

is the new information in the record. A quantum filter uses this innovation to update the conditional state. The update has a Bayesian flavor, but it is implemented by the continuous-time limit of instruments, not by a classical likelihood alone.

For the full recursive estimation viewpoint, see Quantum Filtering.

Bayesian State Tomography Is Different

Bayesian quantum state tomography places a prior distribution over possible preparation states and updates that prior using measurement outcomes. If the unknown preparation state is denoted $\rho_\theta$, then a data set $D$ gives

$$\pi(\theta\mid D) \propto p(D\mid\rho_\theta)\pi(\theta).$$

This posterior is about the source or preparation procedure. It should not be confused with the state of a particular system after being measured. A destructive tomography measurement may leave no usable post-measurement system at all, while still providing excellent information about the preparation ensemble.

This distinction is essential in tomography, benchmarking, and calibration:

$$\text{posterior over models} \ne \text{post-measurement state of one system}.$$

For detector and measurement tomography, see Measurement Tomography.

Exercises

1. A source prepares either $\rho_0$ or $\rho_1$ with prior probabilities $\pi_0$ and $\pi_1$. A measurement with effect $F_m$ gives outcome $m$. Derive the posterior odds.

Solution

The likelihoods are

$$p(m\mid\rho_j) = \operatorname{Tr}(F_m\rho_j), \qquad j=0,1.$$

Bayes’ rule gives

$$\frac{P(0\mid m)} {P(1\mid m)} = \frac{\operatorname{Tr}(F_m\rho_0)} {\operatorname{Tr}(F_m\rho_1)} \frac{\pi_0}{\pi_1}.$$

This posterior is about the preparation hypothesis, not by itself the state left after the measurement.

2. Measure an initial $\lvert+x\rangle$ with $M_0=\lvert0\rangle\langle0\rvert$. A second device uses $N_0=\sigma_xM_0$. Compare the probability of outcome 0 and a subsequent computational-basis measurement.

Solution

Both effects equal $\lvert0\rangle\langle0\rvert$, so outcome 0 has probability $1/2$ for both devices. Conditioned on it, the first leaves $\lvert0\rangle$ and the second leaves $\lvert1\rangle$. A subsequent computational-basis measurement distinguishes the devices with certainty. The likelihood of the first record therefore does not specify its backaction.

3. Inefficient continuous readout. Extend the solvable readout above to efficiency $0\le\eta\le1$. Start in $|+x\rangle$, retain $H=0$ and $c=\sqrt\gamma\sigma_z$, and assume no other channels. The record is $dY=2\sqrt{\eta\gamma}\,z_cdt+dW$. Find the conditional Bloch vector and purity at time $T>0$. Check both efficiency endpoints and the unread coherence. Does inefficient monitoring necessarily prevent eventual purification in this model?

Solution

Split the same physical channel into an observed part $\sqrt{\eta\gamma}\sigma_z$ and an unobserved part $\sqrt{(1-\eta)\gamma}\sigma_z$ (Jacobs and Steck, 2006, Sec. VII). The latter multiplies coherence by $e^{-2(1-\eta)\gamma T}$ while leaving populations unchanged. All these updates commute in this nondemolition model.

The eigenstate likelihoods are $Y_T\mid\pm\sim\mathcal N(\pm2\sqrt{\eta\gamma}T,T)$. Put $u=2\sqrt{\eta\gamma}Y_T$. The positive diagonal observed instrument gives the same population odds as before, with the reduced measurement strength. Combining it with the unobserved dephasing yields

$$z_c=\tanh u,\qquad x_c=e^{-2(1-\eta)\gamma T}\operatorname{sech}u,\qquad y_c=0.$$

Consequently,

$$\operatorname{Tr}\rho_c^2 =1-\frac{1-e^{-4(1-\eta)\gamma T}}{2}\operatorname{sech}^2u.$$

At $\eta=1$ every conditional state is pure. At $\eta=0$, the record gives no information: $z_c=0$ and $x_c=e^{-2\gamma T}$, exactly the unread state. For $\eta=1/2$, $\gamma T=1$ and a balanced record $Y_T=0$, the purity is $(1+e^{-2})/2\simeq0.568$.

For the ensemble check, the physical record density is

$$p(Y_T)=\frac{e^{-Y_T^2/(2T)}}{\sqrt{2\pi T}} e^{-2\eta\gamma T}\cosh u.$$

Multiplying by $x_c$ cancels $\cosh u$, so integration gives $\mathbb E[x_c]=e^{-2\gamma T}$ for every efficiency. Lost information reduces conditioning, not the physical dephasing.

For any fixed $\eta>0$, the two record means separate linearly in $T$ while their widths grow as $\sqrt T$. The record resolves a pointer alternative in the long-time limit, and the conditional purity tends to one with probability one. This conclusion uses the preserved $\sigma_z$ alternatives and absence of other channels; it is not a general purification guarantee for inefficient monitoring with transitions or competing noise.

4. A data-analysis routine estimates a posterior distribution over an unknown detector efficiency $\eta$. Is that posterior the same object as the conditional quantum state of the measured system?

Solution

No. The posterior over $\eta$ is a classical probability distribution over a model parameter. The conditional quantum state is an operator assigned to the measured system, usually updated through an instrument or stochastic master equation. The two can be coupled in a hybrid inference problem, but they are different mathematical objects with different operational meanings.

References

• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer, 2009.
• C. M. Caves, C. A. Fuchs, and R. Schack, “Quantum probabilities as Bayesian probabilities,” Physical Review A 65, 022305, 2002.
• C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, 1976.
• A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, 1982.
• K. Jacobs and D. A. Steck, “A straightforward introduction to continuous quantum measurement,” Contemporary Physics 47, 279–303 (2006), Sec. VII. DOI; author manuscript.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press, 2014.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.