Non-negative Wigner Function Research Articles

Verification of joint measurability using phase-space quasiprobability distributions

Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results therefore establish a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions for bosonic systems and Gaussian channels. In particular, these conditions provide useful tools for investigating the effects of errors and imperfections on the incompatibility of measurements in practice. To illustrate our method, we consider all classes of single-mode Gaussian channels. We show that pure lossy channels with 50% or more losses break the incompatibility of all measurements that can be represented by non-negative Wigner functions, which includes the set of Gaussian measurements.

Re-examining a local-realism test by modifying the classical boundary for modular phase-space measurements

In recent work, Arora and Asadian have proposed a sophisticated physical structure for testing the macroscopic Clauser-Horne-Shimony-Holt (CHSH) inequality of two entangled particles using a multislit Young experiment that employs classical-like modular variables and measurements. By imposing the algebraic CHSH constraint on the Wigner representations of these observables, they found that a strong connection exists between the non-negative Wigner function of a quantum state and having the macroscopic CHSH inequality hold. The entangled states with negative Wigner functions violate the macroscopic CHSH inequality when the number of slits is $N\ensuremath{\ge}8$ and the maximum violation, in principle, requires $N\ensuremath{\rightarrow}+\ensuremath{\infty}$. In this paper, we further study this macroscopic local-realism test by analyzing the eigenvalues of the classical-like modular variables when $N$ is finite. Through this analysis, we find that one can modify the classical boundary for the macroscopic CHSH inequality when $N$ is finite, which makes it possible to demonstrate macroscopic Bell nonlocality for entangled states with $N<8$. More interestingly, we find another form of an entangled state that violates the macroscopic CHSH inequality under the modified boundaries when the number of slits is $N=2$. But the violation of the macroscopic CHSH inequality under the modified classical boundaries with a finite $N$ does not necessarily correspond to states with a non-positive-definite Wigner function.

Quantifying the magic of quantum channels

To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

Entanglement with negative Wigner function of three thousand atoms heralded by one photon

Quantum-mechanically correlated (entangled) states of many particles are of interest in quantum information, quantum computing and quantum metrology. Metrologically useful entangled states of large atomic ensembles have been experimentally realized [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], but these states display Gaussian spin distribution functions with a non-negative Wigner function. Non-Gaussian entangled states have been produced in small ensembles of ions [11, 12], and very recently in large atomic ensembles [13, 14, 15]. Here, we generate entanglement in a large atomic ensemble via the interaction with a very weak laser pulse; remarkably, the detection of a single photon prepares several thousand atoms in an entangled state. We reconstruct a negative-valued Wigner function, an important hallmark of nonclassicality, and verify an entanglement depth (minimum number of mutually entangled atoms) of 2910 ± 190 out of 3100 atoms. Attaining such a negative Wigner function and the mutual entanglement of virtually all atoms is unprecedented for an ensemble containing more than a few particles. While the achieved purity of the state is slightly below the threshold for entanglement-induced metrological gain, further technical improvement should allow the generation of states that surpass this threshold, and of more complex Schrödinger cat states for quantum metrology and information processing.

Proposal for a macroscopic test of local realism with phase-space measurements

We propose a new test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representation of these observables are bounded in phase space, and therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase space coordinates, do not reveal non-locality. We construct a class of entangled states resulting in a violation of the inequality, and thus truly demonstrate non-locality in phase space. The states can be realized through grating techniques in space-like separated interferometric setups. The non-locality is verified from the spatial correlation data, collected from the screens.

An Analysis of the Spekkens Toy Theory with Connection to Wootters Discrete Phase Space

The toy model of Spekkens is a formalism which can partially describe quantum mechanics. The theory deals with the (epistemic) states of a spin-1/2 particle, or qubits and it is closely related to the discrete phase space formalism of Wootters and collaborators. One can apply the stabilizer formalism for finding similarities of these two models. Noting that MUB basis vectors are obtained by eigenstates of generalized Pauli operators, the MUB basis vectors are thus the set of stabilizer states. Galvao has characterized the set of states with non-negative Wigner function class; they form the convex hull of the stabilizer states used as the MUB basis vectors. By combining both approaches, one can show epistemic states that are analogous to the convex hull of the stabilizer states (used as basis vectors in the MUB set) always make valid nonmaximal knowledge epistemic states.

Non-negative subtheories and quasiprobability representations of qubits

Negativity in a quasiprobability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude states that are non-negative from exhibiting phenomena typically associated with quantum mechanics - the single qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We seek to determine what other sets of quantum states and measurements for a qubit can be non-negative in a quasiprobability representation, and to identify nontrivial unitary groups that permute the states in such a set. These sets of states and measurements are analogous to the single qubit stabilizer states. We show that no quasiprobability representation of a qubit can be non-negative for more than four bases and that the non-negative bases in any quasiprobability representation must satisfy certain symmetry constraints. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability representation and are also permuted by a nontrivial unitary group. This list includes two families of three bases that both include the single qubit stabilizer states as a special case and a family of four bases whose symmetry group is the Pauli group. For higher dimensions, we prove that there can be no more than 2^{d^2} states in non-negative bases of a d-dimensional Hilbert space in any quasiprobability representation. Furthermore, these bases must satisfy certain symmetry constraints, corresponding to requiring the bases to be sufficiently complementary to each other.

Non-negative Wigner functions for orbital angular momentum states

The Wigner function of a pure continuous-variable quantum state is non-negative if and only if the state is Gaussian. Here we show that for the canonical pair angle and angular momentum, the only pure states with non-negative Wigner functions are the eigenstates of the angular momentum. Some implications of this surprising result are discussed.

Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions

We study the possibility of giving a classical interpretation to quantum projective measurements for a particle described by a pure Gaussian state whose Wigner function is non-negative. We analyze the case of a projective measurement which gives rise to a proper Wigner function\char22{}i.e., taking on, as its values, the eigenvalues of the projector. We find that, despite having this property, this kind of projector produces a state whose Wigner function ceases to be non-negative and hence precludes its interpretation as a classical probability density. We also study the general case in which the projected state has a non-negative Wigner function, but then we find that the Wigner function of the projector is not a proper one. Thus, we conclude that a non-negative Wigner function is inadequate to serve as a hidden-variable model for quantum processes in which projective measurements take place.

Non-negative Wigner functions in prime dimensions

According to a classical result due to Hudson, the Wigner function of a pure, continuous-variable quantum state is non-negative if and only if the state is Gaussian. We have proven an analogous statement for finite-dimensional quantum systems. In this context, the role of Gaussian states is taken on by stabilizer states. The general results have been published in [1]. For the case of systems of odd prime dimension, a different, greatly simplified method of proof can be employed which still exhibits the main ideas. The present paper gives a self-contained account of these methods.

Hudson’s theorem for finite-dimensional quantum systems

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

Bell’s inequality violation with non-negative Wigner functions

A Bell inequality violation (BIQV) allowed by the two-mode squeezed state (TMSS), whose Wigner function is nonnegative, is shown to hold only for correlations among dynamical variables (DV) that cannot be interpreted via a local hidden variable (LHV) theory. Explicit calculations and interpretation are given for Bell's suggestion that the EPR (Einstein, Podolsky and Rosen) state will not allow for BIQV in conjuction with its Wigner representative state being nonnegative. It is argued that Bell's theorem disallowing the violation of Bell's inequality within a local hidden-variable theory depends on the DV's having a definite value --assigned by the LHV-- even when they cannot be simultaneously measured. The analysis leads us to conclude that BIQV is to be associated with endowing these definite values to the DV's and {\it not} with their locality attributes.

Classical phase-space descriptions of continuous-variable teleportation.

The non-negative Wigner function of all quantum states involved in teleportation of Gaussian states using the standard continuous-variable teleportation protocol means that there is a local realistic phase-space description of the process. This includes the coherent states teleported up to now in experiments. We extend the phase-space description to teleportation of non-Gaussian states using the standard protocol and conclude that teleportation of non-Gaussian pure states with a fidelity of 2/3 is a "gold standard" for this kind of teleportation.

Time evolution of the Wigner function

In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.