# 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**

# Motivation

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**

Monitored qubit Z operator:

causally generated readout

**Signal**

**Observ. Exp. Value**

**Gaussian Noise**

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

**Filter independent of trajectory model**

# 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

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

We recover approximate Gaussian noise, as expected:

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

Optimally filtering/smoothing a **single collected readout**

will remove the Gaussian noise, and recover the

**shifted** observable value, **not the expectation value**

**Weak regime**

**Strong regime**

**Smoothed** (shifted) observable mean:

Depends on a **weak value** and a quadratric **correction**:

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

Discriminator:

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

**Strong regime**

**Weak regime**

**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

Discriminator:

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

**Strong regime**

**Weak regime**

**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) ]

# Conclusions

- 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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