Getting a Handle on a Single Qubit

Andrew Doherty work in collaboration with

Agata Branczyk, Paulo Mendonca (poster), Steve Bartlett, and Alexei Gilchrist

PRAQSYS2006 Boston, 8 August 2006

BackgroundQuantum measurements are more limited that classicalones: •It is not possible to determine the state of a quantum systemby means of a single measurement no matter how precise.Some classical strategies will not perform ideally.

•It is not generally possible to make a measurement on aquantum system without disturbing it.A better sensor won’t always lead to better control performance

Various authors have investigated the effects of this onoptimal quantum feedback control problems.Here I will describe a very simple scenario that I believeillustrates these features fairly well.

Scenario•Choose a state stabilization problem (recover from noise).•Single qubit and single feedback step since it simplest•Two non-orthogonal states since they cannot be perfectlydistinguished

Scenario: details

Objective function: average fidelity

Two input states

Noise process

Effect of noise on Bloch sphere

Initial states and states afternoise process.

X-component of Blochvector shortened.

Orientation means thatdistinguishability of thetwo states is not affected.

Can easily solve for much moregeneral class of noises but we willstick with this.

“Classical strategy”Think of repeaters for optical communication. Intermittentlymeasure optical signal, use electrical output to recreate lightpulse.

Noise Discriminate Resend

Because the two states are imperfectly distinguishable themeasurement step fails with some probability.

Chose measurement to minimize this failure probability.

Optimal measurement is the Helstrom measurement.

Helstrom measurementGiven two states and we make a measurement describedby a two-outcome POVM to distinguish the two.

The probability of error is

is a POVM iff these conditions hold

Known analytically solvable optimization, for our states find that optimal measurement is projective measurement of Z.

Performance of classical scheme

Optimal probability of error.

Resulting fidelity:

Not so fast!

There is no need to replace the state with one of two signalstates!

Noise Discriminate Resend

Because of the error probability there is actually a benefitto resending states with larger overlap than the originalsignal states.

Resulting fidelity is improved

Performance of classical scheme

•Ideal performance for orthogonal (distinguishable) states.•Difference between two classical schemes for less distinguishable states.

General intercept-resend strategiesUp to now we have found the measurement that minimizesthe probability of error in state discrimination and, given that,found the optimal state to resend.

Noise Discriminate Resend

In general we could perform a measurement with anynumber of outcomes and replace with any state.

The resulting class of quantum operations has been studied and are named entanglement breaking channels.

Our scheme is the optimal EB recovery operation. (cf Fuchs,Sasaki)

Performance of classical scheme

•Ideal performance for orthogonal (distinguishable) states.•Difference between two classical schemes for less distinguishable states.

A Second “Classical strategy”

Do nothing!

Since we are going to allow noiseless recovery operations itseems fair to regard this as a classical strategy, definitely notEB channel. Outperforms discriminate-resend for low noise and highlydistinguishable states. (this relationship depends on noise model)

General feedback strategyNow we don’t require the output of measurement stepto be a classical outcome. The conditioned state is availablefor feedback.

Noise Measurement Feedback

Once again I will describe and physically motivate a strategy that turns out to be optimal.

Optimality argument as in previous case is analytic, requireslinearly mapping completely positive trace preserving maps onto certain non-negative matrices and using semidefinite programming techniques. (Choi, Jamiolkowski, Audenaert and deMoor)

Measure the noise and not the state

A common physical picture for quantum error correctionis that it gets around the disturbance due to measurementsby measuring effects of noise directly without learning aboutthe encoded state to be protected. We will use this idea.

Physical picture of noise process is not unique. Can view itas a random rotation about Z-axis with a fixed angle.

A measurement along the Y-axis will help distinguishwhich rotation occurred. Then we may correct by rotatingthe state back to the XZ-plane.

For small rotations, measurementsalong Y-axis determine which noise rotation took place.

We expect that we need to tune the strength of the measurement, e.g.:

If the measurement operators are the same andAre proportional to the identity. If the measurementis projective. is something like a sensitivity, or SNR.

Y-measurements

Circuit for weak measurement

Projective Z measurement

Overall scheme

•Measure Y with some strength •Rotate about Z axis so that conditioned state is in XZ plane

Required rotation angle can be found. Resulting fidelity:

This formula reflects an optimum signal to noise. For weak measurement performance improves, but eventually theeffect of the disturbance due to measurement, leads to reduced performance.

Feedback performance

Performance with measurement strength

do nothing

Optimal DR

Now optimize over measurement strength:

Performance of optimal scheme

Feedback scheme can be shown to be optimal among allrecovery operations, so always outperforms classical strategies.

Advantage of the optimal scheme

Feedback unravellingsThere is a single optimal recovery operation that can be calculated following (Audenaert and deMoor). This is not necessarily a feedback process, but (by accident?) it is realized by choosing the optimal measurement strengthin our scheme.

Despite the (perhaps) compelling motivation for this feedbackscheme we could have proceeded differently. (Combes, Blume-Kohout)

•Weak measurement of Z:

•Followed by rotation about Y•Optimizing over rotation angle gives a curve describing the performance as a function of measurement strength. •Again there is an optimum measurement strength.

Z-measurement version

•Make weak measurement of Z•Based on outcome rotate about Yby optimal rotation angle•Interpolates between Helstrommeasurement and do nothingCf: optimal eavesdropping (Fuchs, Peres and Griffiths, Niu)

Resulting fidelity:

Performance with measurement strength

do nothing

optimal DR

Feedback unravellings

The optimum given here is the same as for our measurethe noise protocol! (Combes, Blume-Kohout)

Thus there are at least two ways of implementing the optimal noise recovery operation as a feedback. One strategy is more “classical” one is more “quantum” bothcould have been found by physical arguments.

Every quantum operation can be realized as a one-stepfeedback (e.g. ACD,Jacobs,Jungman, and Lloyd,Viola) but this is not unique.

Is it possible to characterize feedback strategies realizing a given quantum operation? (In our example doesmeasurement matter)

SummaryOptimal recovery from noise for two non-orthogonal states shows two qualitative effects of quantum measurements on feedback. •It is not possible to determine the state of a quantum systemby means of a single measurement no matter how precise.Optimal discriminate-resend scheme does not work perfectlyeven for no noise.

•It is not generally possible to make a measurement on aquantum system without disturbing it.Optimal sensitivity in feedback stabilization (both Y and Z measurements)

How many ways to skin a cat? Feedback unravellings.

Completely positive maps reviewThe general quantum operation is mathematically atrace preserving completely positive map:

There is a one-to-one and onto map of these to non-negative matrices on such that

The constraints on are:

Reduced density matrix is unique matrix such that for all X:

Optimality Arguments

Maximize fidelity between initial and final states after arbitraryCP map recovery operation following Audenaert, deMoor.Maximize:Subject to: This semidefinite program can be solved analytically with theassistance of some symmetry arguments following Gatermann, Parrilo

For the discriminate-resend models, the completely poisitive mapis “entanglement breaking” For qubits this can be expressed byan additional semidefinite constraint. Maximize:Subject to: