What does a continuously monitored qubit readout really show?

Luis Pedro García-Pintos

  Justin Dressel


APS March Meeting 2017

How do we know whether individual quantum state trajectories that we reconstruct from experimental data are correct, using the readout alone?

Current approach:

Perform a random tomographic pulse at the end of each data run

Spot check subensembles of data to verify tomography of final state


Murch et al., Nature 502, 211 (2013)

Hacohen-Gourgy et al., Nature 538, 491 (2016)

If the collected stochastic signal noisily tracks an observable of the qubit, can we filter the signal to estimate that observable trajectory independently?

Idea: Filter the Readout

Classical signals can remove Gaussian noise either:

1) Causally (no future signal), with a filter (e.g. Weiner, Kalman)

2) Non-causally (using future signal), with a smoother


For already collected data, smoothers work best

r(t) = z(t) + \sqrt{\tau}\,\xi(t)

Monitored qubit Z operator:

causally generated readout


Observ. Exp. Value

Gaussian Noise

Structure of collected qubit signal seems amenable to such a filtering technique


Filter independent of trajectory model

z(t) = \mathrm{Tr}[Z\,\rho_{\vec{r}_{\mathrm{past}}}]

Evidence : Qubit Z

Simple single pole filter

Simple single pole smoother

Strong (Zeno) regime: tracking jumps

Weak regime: tracking noisy Rabi oscillations

 Trend : stronger measurements yield more information

--> better fidelity, but more perturbed evolution

Reasonable tracking

Noise harder to remove

Evidence: Qubit X + Z

Simple single pole filter

Simultaneous (here X+Z) measurements work well

(with stronger measurements)

One can track multiple observables simultaneously for a single trajectory with reasonable (~90%) fidelity, without a master equation

Filtering idea works, but fidelity is limited. What is the theoretical limit?

see also Ruskov et al. PRL 105 100506 (2010): >90% state fidelity with 3 axis monitoring + filtering

Hacohen-Gourgy et al., Nature 538, 491 (2016)

Consider a single collected readout r(t), but omit one point at t=t1.

What distribution P[r(t1)] describes the likelihood of the omitted point?

Readout Distribution

P(r\, |\, \vec{r}_{\mathrm{past}},\, \vec{r}_{\mathrm{future}}) = \frac{P(r,\, \vec{r}_{\mathrm{future}}\,|\,\vec{r}_{\mathrm{past}})}{\sum_r P(r,\, \vec{r}_{\mathrm{future}}\,|\,\vec{r}_{\mathrm{past}})} \approx \frac{\exp\left[-\frac{dt}{2\tau}(r-z_s)^2\right]}{\sqrt{2\pi\tau/dt}}

Discretize time into bins of size dt - assume Markovian Gaussian measurements:

We recover approximate Gaussian noise, as expected:

r(t) = z_s(t) + \sqrt{\tau}\,\xi(t)

However, the collected readout follows a shifted mean value

(Consequence of the measurement backaction producing non-Markovian correlations)

The mean is the expectation value of Z only on the boundary, with unknown future record (as appropriate for simulation)

Smoothed Observable Estimate

r(t) = z_s(t) + \sqrt{\tau}\,\xi(t)

Optimally filtering/smoothing a single collected readout

will remove the Gaussian noise, and recover the

shifted observable value, not the expectation value

z_s = z_w\, \left[\frac{2}{(1+e^{-dt/2\tau}) + z_c\,(1-e^{-dt/2\tau})}\right] \approx z_w

Weak regime

Strong regime

\displaystyle z_w = \mathrm{Re}\frac{\mathrm{Tr}[E_{\vec{r}_{\mathrm{future}}}\,Z\,\rho_{\vec{r}_{\mathrm{past}}}]}{\mathrm{Tr}[E_{\vec{r}_{\mathrm{future}}}\,\rho_{\vec{r}_{\mathrm{past}}}]}

Smoothed (shifted) observable mean:

Depends on a weak value and a quadratric correction:

\displaystyle z_c = \frac{\mathrm{Tr}[E_{\vec{r}_{\mathrm{future}}}\,Z\,\rho_{\vec{r}_{\mathrm{past}}}\,Z]}{\mathrm{Tr}[E_{\vec{r}_{\mathrm{future}}}\,\rho_{\vec{r}_{\mathrm{past}}}]}
\rho_{\vec{r}_{\mathrm{past}}} = \mathcal{M}_{\vec{r}_{\mathrm{past}}}[\rho_0]
E_{\vec{r}_{\mathrm{future}}} = \mathcal{M}^*_{\vec{r}_{\mathrm{future}}}[1]

Non-Markovian dependence on both

past state                          and                 future effect matrix:

Consistent with:

Aharonov PRL 60, 1351 (1988), Wiseman PRA 65, 032111 (2002), Tsang PRL 102, 250403 (2009), Dressel PRL 104, 240401 (2010), Dressel PRA 88, 022107 (2013), Mølmer PRL 111, 160401 (2013)

( No additional ad hoc postselection)

Reduced RMS Error from Readout

Is this really true? This smoothed value seems strange - can exceed eigenvalue range

Verification 1: look at relative RMS error of
both estimates compared to raw readout

Optimal filters/smoothers (Weiner, Kalman, etc.) are often defined to minimize the RMS error between a smooth dynamical estimate and the raw noisy signal

\displaystyle \log(Q) \equiv \log\frac{||z - r||}{||z_s - r||}

(>0 implies smoothed value follows readout better than expectation value)

T_R : \mathrm{Rabi\, period}
\Delta^2 : \tau_z/dt

Strong regime

Weak regime

T : \mathrm{traj.\,length}

smoothed better

Yes. The smoothed value is objectively better by the same metric used for finding classical filters/smoothers.

Hypothesis Test from Readout

Verification 2: look at relative log-likelihood of
generating the raw readout from adding Gaussian noise to the two estimates - equivalent to a hypothesis test

\displaystyle \log(R) \equiv \log\frac{P(r|z)}{P(r|z_s)}

(>0 implies smoothed value more likely than expectation value to generate readout)

T_R : \mathrm{Rabi\, period}
\Delta^2 : \tau_z/dt

Strong regime

Weak regime

T : \mathrm{traj.\,length}

smoothed better

Yes. The smoothed value is objectively more likely to generate the observed readout from additive noise


Corroboration by Third Party

Variation: Suppose Bob is weakly monitoring a different observable (X) at the same time


If Alice is more strongly monitoring (Z) and has no access to Bob's record, does her smoothed estimate of X (not measured by her) still correspond to Bob's record?


Yes. Even without access to Bob's record, Alice can construct a smoothed estimate from her record that fits Bob's observed record better than the expectation value (computed either from Alice's subjective state or the most pure state with perfect knowledge of both records)


Test verifies that a better model of the relevant dynamics produces an objectively closer fit to the collected record


Smoothed estimate is operationally meaningful

RMS Error Test

Hypothesis Test

Blue : Alice does not know Bob's record

Pink : Alice knows both records

smoothed better

[Similar question to Guevara, Wiseman PRL 115, 180407 (2015) ]


  • The mean of a single collected readout tracks a smoothed observable estimate (~ a weak value) due to backaction-correlations
  • Both RMS error and hypothesis tests confirm this statement, even for a second party that is simultaneously monitoring the system

Practical Implication:

  • Filtering/smoothing the readout can only approximate a usual state trajectory
  • Better idea: construct smoothed observable estimate from raw record and then verify it with the filtered/smoothed record -- potential route forward for trajectory model verification without additional tomography data

Thank you!

What does a continuously monitored qubit readout really show?

By Justin Dressel

What does a continuously monitored qubit readout really show?

For continuous measurements of a quantum observable it is widely recognized that the measurement output approximates the expectation value of the observable, hidden by additive white noise. Filtering the measurement readout can thus approximately uncover the dynamics of the expectation value, during a single realization. However, using information from the entire output history yields a different, smoothed, observable estimate. We derive the form of this smoothed estimate and show that the observed readout quantitatively tracks it with higher fidelity than the expectation value during a single realization, making it an objectively meaningful quantity. In the weak measurement limit this smoothed estimate approximates a weak value, with no need for additional postselection.

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