Super G-spaces

We review the basic theory of super $G$-spaces. We prove a theorem relating the action of a super Harish-Chandra pair $(G_0, \mathfrak{g})$ on a supermanifold to the action of the corresponding super Lie group $G$. The theorem was stated in [DM99] without proof. The proof given here does not use Frobenius theorem but relies on Koszul realization of the structure sheaf of a super Lie group (see [Kosz83]). We prove the representability of the stability subgroup functor

Covariant mutually unbiased bases

The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case their equivalence class is actually unique. Despite this limitation, we show that in even-prime power dimension covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.Comment: 44 pages, some remarks and references added in v

Minimal covariant observables identifying all pure states

It has been recently shown that an observable that identifies all pure states of a d-dimensional quantum system has minimally 4d-4 outcomes or slightly less (the exact number depending on the dimension d). However, no simple construction of this type of observable with minimal number of outcomes is known. In this work we investigate the possibility to have a covariant observable that identifies all pure states and has minimal number of outcomes for this purpose. It is shown that the existence of these kind of observables depends on the dimension of the Hilbert space. The fact that these kind of observables fail to exist in some dimensions indicates that the dual pair of observables -- pure states lacks the symmetry that the dual pair of observables -- states has

Tasks and premises in quantum state determination

The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (POVMs) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.Comment: minor changes in v

Probing quantum state space: does one have to learn everything to learn something?

Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement setup that reveals this property with as little effort as possible. Here we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.Comment: 23 pages, 4 figure

Commutative POVMs and Fuzzy Observables

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements.Comment: Contribution to the Pekka Lahti Festschrif

Extremal covariant positive operator valued measures: the case of a compact symmetry group

Given a unitary representation U of a compact group G and a transitive G-space $\Omega$, we characterize the extremal elements of the convex set of all U-covariant positive operator valued measures.Comment: minor corrections in version

Optimal covariant measurements: the case of a compact symmetry group and phase observables

We study various optimality criteria for quantum observables. Observables are represented as covariant positive operator valued measures and we consider the case when the symmetry group is compact. Phase observables are examined as an example

Efficient Pure State Quantum Tomography from Five Orthonormal Bases

For any finite dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state dependent) scheme proposed by Goyeneche et al. in [Phys. Rev. Lett. 115, 090401 (2015)]. We show that our scheme is robust with respect to noise in the sense that any measurement scheme which approximates these measurements well enough is equally suitable for pure state tomography. Finally, we present two convex programs which can be used to reconstruct the unknown pure state from the measurement outcome distributions.Comment: 5 pages, 2 figures, 1 page of supplemental materia