Measurement Records

A measurement record is the classical data stream produced by a measurement apparatus. In continuous monitoring, the record is the object on which the quantum state is conditioned. It may be a list of detector-click times, a noisy current, two quadrature streams, a binned voltage trace, or a processed version of a raw laboratory signal.

The record is not the quantum state, and it is not usually the instantaneous value of an observable. It is a stochastic classical process whose probability law is determined by the measurement model and the conditional state.

The compact slogan is:

$$\text{record} \quad\longrightarrow\quad \text{likelihood} \quad\longrightarrow\quad \text{conditional state update}.$$

For the conceptual distinction between the conditional state and the ensemble state, see Continuous Monitoring.

Records and Conditional Probabilities

Over a short interval $dt$, a record increment $dr$ labels an outcome operation $\mathcal I_{dr}^{dt}$. The likelihood density or probability weight for that increment is

$$p(dr|\rho_c(t)) = \operatorname{Tr} \mathcal I_{dr}^{dt}(\rho_c(t)).$$

The conditional state after observing $dr$ is

$$\rho_c(t+dt) = \frac{\mathcal I_{dr}^{dt}(\rho_c(t))} {\operatorname{Tr}\mathcal I_{dr}^{dt}(\rho_c(t))}.$$

A full record over $[0,T]$ is a collection of increments. In a discrete approximation,

$$Y_{[0,T]} = (dr_1,dr_2,\ldots,dr_N), \qquad T=Ndt.$$

The probability law of the whole record is built from the composed instrument operations. The normalized state at time $T$ is the filtered state conditioned on that entire record, not merely on the last data point.

Raw Data Versus Idealized Record

The same experiment can have several layers of records:

• raw electronics samples;
• demodulated in-phase and quadrature samples;
• integrated time-bin values;
• thresholded click or no-click events;
• a dimensionless stochastic process used in the theory;
• a further filtered record used for feedback or state estimation.

The theory usually writes an idealized record such as $dN_t$, $dY_t$, or $I(t)dt$. A laboratory paper must additionally state calibration, bandwidth, sampling time, offsets, detector efficiency, and any filtering applied before the record is compared with the model.

This distinction prevents a common mistake: differentiating a white-noise record as if it were an ordinary smooth function. In diffusive models, increments are well defined; the formal current is a distributional object.

Photon Counts

Photon counting is modeled by a counting process $N_t$. The increment

$$dN_t = N_{t+dt}-N_t$$

is either $0$ or $1$ in an ideal infinitesimal interval:

$$dN_t\in\{0,1\}, \qquad dN_t^2=dN_t.$$

For a monitored collapse operator $c$, one common ideal convention is

$$\mathbb E[dN_t|\rho_c(t)] = \operatorname{Tr}(c^\dagger c\rho_c(t))\,dt.$$

With detector efficiency $\eta$ and dark-count rate $\lambda_{\mathrm d}$, the observed click rate is often modeled schematically as

$$\lambda_{\mathrm{obs}}(t) = \eta\, \operatorname{Tr}(c^\dagger c\rho_c(t)) + \lambda_{\mathrm d}.$$

For independent dark counts that do not disturb the system, the click operation per unit time is

$$\mathcal J_{\mathrm{obs}}(\rho)=\eta c\rho c^\dagger+\lambda_{\mathrm d}\rho, \qquad \rho_{\mathrm{click}}=\frac{\mathcal J_{\mathrm{obs}}(\rho_c)}{\lambda_{\mathrm{obs}}}.$$

Its trace gives the observed intensity. A dark count is mixed with a real emission in the state update; adding a background rate while retaining an ideal jump for every click would be inconsistent. Inefficiency does the complementary damage: some emitted quanta are not included in the observer’s record and remain as unmonitored decoherence.

Counting records are naturally represented either by the cumulative process $N_t$ or by event times

$$0\lt t_1\lt t_2\lt\cdots\lt t_k\le T.$$

In a finite time bin $[t,t+\Delta t]$, the recorded count is

$$\Delta N_t = N_{t+\Delta t}-N_t.$$

For small enough bins $\Delta N_t$ is usually $0$ or $1$, but for larger bins it may contain several events. The correct bin size is a modeling choice tied to detector resolution and system time scales.

For the direct-detection model, see Photon Counting. For the conditioned state dynamics associated with this record, see Quantum Jump Trajectories.

Homodyne Current

Homodyne detection measures one quadrature of an output field by interfering it with a strong local oscillator. In an idealized Markovian normalization, the observed increment has the form

$$dY_t = \sqrt{\eta}\, \left\langle e^{-i\phi}c+e^{i\phi}c^\dagger \right\rangle_c dt + dW_t,$$

where $c$ is the system operator coupled to the monitored output, $\phi$ is the local-oscillator phase, $\eta$ is the total efficiency, and $dW_t$ is a Wiener increment.

The conditional mean of the increment is

$$\mathbb E[dY_t|\rho_c(t)] = \sqrt{\eta}\, \left\langle e^{-i\phi}c+e^{i\phi}c^\dagger \right\rangle_c dt,$$

while its conditional variance is

$$\operatorname{Var}(dY_t|\rho_c(t)) = dt.$$

Thus the short-time record is noisy even in an ideal detector. The state update is driven by the innovation,

$$dW_t = dY_t - \mathbb E[dY_t|\rho_c(t)].$$

The local-oscillator phase changes which quadrature is monitored. Changing $\phi$ changes the individual record and the conditioned trajectory, even when the same unconditional Lindblad master equation is recovered after averaging.

The connection between $c$ and the measured output field is part of Input–Output Theory. The detection model is developed in Homodyne Detection, and the stochastic state update is treated in Diffusive Trajectories.

Heterodyne Current

A balanced two-quadrature receiver produces

$$\begin{aligned} dY_x&=\sqrt{\eta/2}\,\langle c+c^\dagger\rangle_c\,dt+dW_x,\\ dY_y&=\sqrt{\eta/2}\,\langle-i(c-c^\dagger)\rangle_c\,dt+dW_y, \end{aligned}$$

with $dW_x^2=dW_y^2=dt$ and $dW_xdW_y=0$. Define $dZ=(dY_x+i\,dY_y)/\sqrt2$ and $d\zeta=(dW_x+i\,dW_y)/\sqrt2$. Then

$$dZ=\sqrt\eta\,\langle c\rangle_c\,dt+d\zeta, \qquad d\zeta\,d\zeta^*=dt,\quad d\zeta^2=0.$$

The split reduces the signal in each real quadrature by $1/\sqrt2$ relative to homodyne detection at the same total efficiency. It does not mean that half the output is unobserved when both records are retained. Heterodyne Detection derives the added-noise interpretation and conditional update.

Qubit Readout Voltage

In dispersive superconducting-qubit readout, the raw laboratory record is often a voltage trace after amplification and demodulation. After calibration and normalization, a simple diffusive model for a measurement of $\sigma_z$ is

$$dY_t = 2\sqrt{\eta\Gamma_{\mathrm m}}\, \langle\sigma_z\rangle_c\,dt + dW_t.$$

This convention corresponds to $c=\sqrt{\Gamma_{\mathrm m}}\sigma_z$ and ensemble coherence-decay rate $2\Gamma_{\mathrm m}$. Before normalization, one might write a schematic voltage model

$$V(t) = V_0 + A\,\langle\sigma_z\rangle_c + \xi(t),$$

where $V_0$ is an offset, $A$ is a calibrated response amplitude, and $\xi(t)$ is amplifier and shot-noise dominated readout noise. The stochastic model is obtained only after specifying how the voltage is integrated, demodulated, and scaled.

For a finite integration time $T$, the averaged readout

$$\overline V_T = \frac1T \int_0^T V(t)\,dt$$

has a noise width that decreases with $T$, while the two mean averaged voltages remain separated by the calibrated response. Their separation relative to the shrinking noise width grows with integration time. Short integrations are weak and ambiguous; long integrations can become effectively projective, provided relaxation, leakage, and calibration drift are controlled.

This kind of record is central in circuit QED. The broader platform setting is discussed in Circuit QED.

Time Bins and Bandwidth

Continuous-time notation is an idealization. Actual records are sampled or integrated over finite time bins. For a diffusive record,

$$\Delta Y_k = Y_{t_{k+1}}-Y_{t_k} = \int_{t_k}^{t_{k+1}}\mu_t\,dt + \Delta W_k.$$

If the bin width is $\Delta t$, then

$$\operatorname{Var}(\Delta W_k)=\Delta t.$$

The corresponding bin-averaged current

$$I_k = \frac{\Delta Y_k}{\Delta t}$$

has noise variance proportional to $1/\Delta t$. Making bins shorter does not produce a smooth, more accurate instantaneous current; it produces a noisier current sample. The integrated increment is the better-behaved object.

Finite detector bandwidth modifies this idealization. A measured current may be a filtered version of the ideal record:

$$I_{\mathrm{meas}}(t) = \int_{-\infty}^{t} h(t-s)I_{\mathrm{ideal}}(s)\,ds + \text{added noise},$$

where $h$ is a detector response function. If the filter time is comparable to system dynamics, the Markovian white-noise record may no longer be an adequate description without enlarging the model.

Likelihood and Innovations

The record determines a likelihood for candidate states or parameters. In a diffusive model

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

the innovation is

$$dW_t=dY_t-\mu_t\,dt.$$

In a well-specified model, the innovations are conditionally zero mean with the expected variance:

$$\mathbb E[dW_t|\rho_c(t)]=0, \qquad \mathbb E[dW_t^2|\rho_c(t)]=dt.$$

This is more than notation. Innovation diagnostics are a practical way to check a filter: if the residuals have systematic drift, wrong variance, or unexpected correlations, the measurement strength, phase, efficiency, Hamiltonian, detector bandwidth, or noise model may be wrong.

For counting records, the analogous diagnostic compares the observed event rate with the conditional intensity $\lambda_t$. A good model should not consistently predict clicks where none occur, or miss bursts that appear in the data, except within the statistical fluctuations expected for the point process.

Coarse Graining and Lost Records

Records can be deliberately or accidentally coarse grained:

• exact event times may be replaced by counts per bin;
• analog currents may be thresholded into binary outcomes;
• high-bandwidth traces may be low-pass filtered;
• one quadrature may be kept while another is discarded;
• some channels may be unobserved because of finite efficiency.

Coarse graining changes the information available for conditioning. It does not necessarily remove the physical backaction. If a detector interaction occurred but the detailed record is lost, the state should be averaged over the lost alternatives. This is the continuous-time version of the selective/nonselective distinction.

The general warning is:

$$\text{same interaction} \ne \text{same conditioned state}$$

unless the same record is retained with the same resolution.

Exercises

1. For a counting process with conditional intensity $\lambda_t$, suppose $\lambda_t=\lambda$ is constant over a short interval $\Delta t$. Show that the probability of at least one count is $\lambda\Delta t+O(\Delta t^2)$.

Solution

For a Poisson process with constant rate $\lambda$ over the interval, the probability of no count is

$$P(0)=e^{-\lambda\Delta t}.$$

Therefore

$$P(\Delta N\ge1) = 1-e^{-\lambda\Delta t} = \lambda\Delta t+O(\Delta t^2).$$

This is why an infinitesimal counting increment has only the alternatives $0$ and $1$ to first order in $dt$.

2. Let a diffusive record obey $dY_t=\mu\,dt+dW_t$ with constant $\mu$ during one bin of width $\Delta t$. Define $I=\Delta Y/\Delta t$. Find the mean and variance of $I$.

Solution

The integrated increment is

$$\Delta Y = \mu\Delta t+\Delta W,$$

with

$$\mathbb E[\Delta W]=0, \qquad \operatorname{Var}(\Delta W)=\Delta t.$$

Thus

$$\mathbb E[I] = \frac{\mathbb E[\Delta Y]}{\Delta t} = \mu,$$

and

$$\operatorname{Var}(I) = \frac{\operatorname{Var}(\Delta W)}{\Delta t^2} = \frac1{\Delta t}.$$

Shorter bins have noisier current samples, even though the integrated increments remain well behaved.

3. A photon counter observes a channel with system rate $\gamma p_e(t)$, efficiency $\eta$, and dark-count rate $\lambda_{\mathrm d}$. Write the observed conditional intensity and explain what happens when $\lambda_{\mathrm d}$ is large.

Solution

The observed intensity is

$$\lambda_{\mathrm{obs}}(t) = \eta\gamma p_e(t) + \lambda_{\mathrm d}.$$

When $\lambda_{\mathrm d}$ is large compared with the system-dependent term, a click carries little information about $p_e(t)$ because many clicks are likely to be unrelated to the monitored system transition. The record still contains events, but the state update associated with each event should be weaker or mixed with a dark-count branch in the instrument.

4. In a homodyne model, the predicted increment is $dY_t=\mu_tdt+dW_t$. A filter produces residuals $dR_t=dY_t-\mu_tdt$ whose variance is consistently $4dt$. Name two possible modeling errors.

Solution

In the normalized model the residual should have variance $dt$. A variance of $4dt$ could mean, for example, that the record normalization is off by a factor of $2$, the shot-noise calibration is wrong, unmodeled amplifier noise has been omitted, the bin width has been misread, or detector bandwidth has changed the ideal white-noise model. The residual diagnostic does not by itself identify the cause, but it shows that the record model and data are not matched.

References

• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer, 2009.
• A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Reviews of Modern Physics 93, 025005, 2021.
• L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM Journal on Control and Optimization 46, 2199–2241, 2007.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press, 2014.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.