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

Background Quantum measurements are more limited that classical ones: It is not possible to determine the state of a quantum system by 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 a quantum system without disturbing it. A better sensor won’t always lead to better control performance Various authors have investigated the effects of this on optimal quantum feedback control problems. Here I will describe a very simple scenario that I believe illustrates 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 perfectly distinguished

Scenario: details Objective function: average fidelity Two input states Noise process

Effect of noise on Bloch sphere Initial states and states after noise process. X-component of Bloch vector shortened. Orientation means that distinguishability of the two states is not affected. Can easily solve for much more general class of noises but we will stick with this.

“Classical strategy” Think of repeaters for optical communication. Intermittently measure optical signal, use electrical output to recreate light pulse. NoiseDiscriminateResend Because the two states are imperfectly distinguishable the measurement step fails with some probability. Chose measurement to minimize this failure probability. Optimal measurement is the Helstrom measurement.

Helstrom measurement Given two states and we make a measurement described by 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 signal states! NoiseDiscriminateResend Because of the error probability there is actually a benefit to resending states with larger overlap than the original signal 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 strategies Up to now we have found the measurement that minimizes the probability of error in state discrimination and, given that, found the optimal state to resend. NoiseDiscriminateResend In general we could perform a measurement with any number 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 it seems fair to regard this as a classical strategy, definitely not EB channel. Outperforms discriminate-resend for low noise and highly distinguishable states. (this relationship depends on noise model)

General feedback strategy Now we don’t require the output of measurement step to be a classical outcome. The conditioned state is available for feedback. NoiseMeasurementFeedback Once again I will describe and physically motivate a strategy that turns out to be optimal. Optimality argument as in previous case is analytic, requires linearly 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 correction is that it gets around the disturbance due to measurements by measuring effects of noise directly without learning about the encoded state to be protected. We will use this idea. Physical picture of noise process is not unique. Can view it as a random rotation about Z-axis with a fixed angle. A measurement along the Y-axis will help distinguish which rotation occurred. Then we may correct by rotating the state back to the XZ-plane.

For small rotations, measurements along 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 and Are proportional to the identity. If the measurement is 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 the effect 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 all recovery operations, so always outperforms classical strategies.

Advantage of the optimal scheme

Feedback unravellings There 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 strength in our scheme. Despite the (perhaps) compelling motivation for this feedback scheme 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 Y by optimal rotation angle Interpolates between Helstrom measurement and do nothing Cf: 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 measure the 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” both could have been found by physical arguments. Every quantum operation can be realized as a one-step feedback (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 does measurement matter)

Summary Optimal 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 system by means of a single measurement no matter how precise. Optimal discriminate-resend scheme does not work perfectly even for no noise. It is not generally possible to make a measurement on a quantum 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 review The general quantum operation is mathematically a trace 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 arbitrary CP map recovery operation following Audenaert, deMoor. Maximize: Subject to: This semidefinite program can be solved analytically with the assistance of some symmetry arguments following Gatermann, Parrilo For the discriminate-resend models, the completely poisitive map is “entanglement breaking” For qubits this can be expressed by an additional semidefinite constraint. Maximize: Subject to: