Optimal Pure-State Qubit Tomography via Sequential Weak Measurements


A sequence of weak isotropic measurements can yield the optimal measurement of a collection of identical qubits — a POVM over the continuous set of spin-coherent states. We show here a numerical simulation for an independent sequence of collective isotropic weak measurements on an ensemble of 50 qubits. The vertical axis is a measure how close a POVM element is to a spin-coherent state projector with the value the “coherency” C=1 corresponding to a spin-coherent state. The olive region consists of the coherency for 50 samples of the measurement record as a function of time. The spheres are Husimi distributions of the POVM element (left) and the post-measurement state (right) for various times of the black sample trajectory. The POVM converges asymptotically, extracting the estimated direction of the unknown spin, whilst the state does not change drastically, because the measurements are weak.

 

It’s been known since the early days of quantum information science that the optimal way to gain information about a quantum state given many copies is to do a joint measurement of the whole ensemble. An example is seen in quantum state tomography. In 1995, Massar and Popescu proved that the optimal measurement for estimating an unknown direction of N copies of a pure qubit is a collective measurement. The realization for every N is the so-called spin-coherent-state POVM. After various attempts, by 2002 many researchers had come to the conclusion that this measurement was not practically realizable for large N. However, in recent work published in Physical Review Letters (give link) researchers at CQuIC showed that an implementation of the spin-coherent-state POVM could indeed be implemented via a sequence of collective and isotropic weak measurements without feedback or adaptivity.

Spin-coherent states are quantum states that transform in the same way as a vector under 3-dimensional rotations.This vector can be a literal direction in 3D real space but can also be an abstract direction such as in the Bloch sphere of a qubit. For N copies of a qubit, these directions are represented by states which sit in an irreducible subspace that spans N+1 of the 2^N dimensions of the total Hilbert space. This subspace is also known as the spin-J=N/2 representation of SU(2). The spin-coherent-state measurement of a spin-J system is a measurement where the outcomes are the continuum of directions which is represented by a POVM with elements that are projectors onto the spin-coherent states. The weak measurement record eventually converges on the estimated direction.

This protocol could be implemented on currently existing platforms, such can be done using the Faraday interaction to measure the collective spin projection of an atomic ensemble with a continuous laser probe.

With this new breakthrough, CQuIC graduate student Ezad Shojaee and postdoc Chris Jackson, together with Prof. Ivan Deutsch are working on applications for more generalized coherent-state measurements such as those with SU(1,1) symmetry and other semi-simple Lie group symmetries.

This work was published in the September 26, 2018 issue of Physical Review Letters:

Optimal Pure-State Qubit Tomography via Sequential Weak Measurements
Ezad Shojaee, Christopher S. Jackson, Carlos A. Riofrío, Amir Kalev, and Ivan H. Deutsch
Phys. Rev. Lett. 121, 130404

The full article can be found online.

 

Becerra’s group publishes article on robust measurements at low powers by counting photons

Dr. Elohim Becerra Chavez’s research group recently published Robust Measurement for the Discrimination of Binary Coherent States in Physical Review Letters.

Optical communication uses light to encode and transmit information over long distances, such as in optical fibers with losses, and requires reliable detection schemes to read out information from low levels of light. However, noise and imperfections in real-world devices and detectors severely affect the measurement fidelity and ultimately limits the amount of information that we can communicate. Recent work in the quantum-optics group led by Elohim Becerra at UNM demonstrated an optimized measurement capable of overcoming these imperfections by counting the number of photons in pulses of light carrying information in their optical phase. This measurement reads off the information contained in the light by first combining the light pulse with a reference pulse to compare their relative phase, and then counting the number of photons in the combined pulse. Previous work in ultra-sensitive measurements has focused instead on detecting no light vs. any amount of light in the combined pulse, and their sensitivities have been limited by noise and imperfections. This novel measurement scheme allows for a high degree of robustness to these imperfections while reaching high measurement sensitivities. Moreover, due to its simplicity, this measurement is inherently compatible with high-bandwidth communication technologies to accommodate the high rates of information transfer in today’s optical communication networks.

This work was published in the July 2 issue of Physical Review Letters:

Robust Measurement for the Discrimination of Binary Coherent States
M. T. DiMario and F. E. Becerra
Phys. Rev. Lett. 121, 023603 (2018)

The full article can be found online.

Symmetric Phases of Universal Quantum Computation

Jacob Miller and Akimasa Miyake have recently published a paper in Physical Review Letters giving strong evidence that certain forms of symmetric topological quantum matter can be utilized ubiquitously to power quantum computation. Their work is carried out within measurement-based quantum computation, where computation is extracted from a fixed quantum “resource state” using local measurements. In this setting, the power of computation attributes the physical properties of the resource state, but the properties which guarantee a state can carry out universal quantum computation are still unknown.
In their work, the authors study a model of symmetric topological matter and identify special states in each phase which enable universal quantum computation precisely when they possess nontrivial quantum order. This gives an infinite family of new universal resource states whose structure perfectly mirrors a recent classification of symmetric quantum order coming from condensed matter physics. These special resource states are distinguished by their “fractional symmetry”, a property already noticed in previous universal resource states, but which hadn’t been investigated systematically. Overall, the work provides a concrete research program for identifying phases of universal quantum computation within the setting of symmetric quantum matter.
The full article can be found online at Phys. Rev. Lett. 120, 170503 (2018).
Classification of special resource states with fractional symmetry.

Becerra’s group publishes article on multiple coherent states in Journal of Optical Society of America B

Dr. Elohim Becerra Chavez’s research group recently published Implementation of a single-shot receiver for quaternary phase-shift keyed coherent states in Journal of Optical Society of America B

Measurement strategies for multiple coherent states based on single-shot measurements with photon counting can be useful for high bandwidth communications with high spectral efficiency. The quantum-optics group led by Elohim Becerra at UNM investigated implementations of optimized multi-state discrimination strategies based on single-shot measurements extending previous work to include realistic situations with noise and imperfections, which impact the achievable performance of the measurement. The implementation with noise and imperfections allows us to identify the experimental requirements to outperform the sensitivity limit of an ideal heterodyne measurement and can guide future demonstrations of these measurements with high efficiency single-photon detectors surpassing the heterodyne limit.

The full article can be found online.