Symmetric Informationally Complete Research Articles

Generalized Quantum Measurements on a Higher-Dimensional System via Quantum Walks.

Quantum measurements play a fundamental role in quantum mechanics. Especially, generalized quantum measurements provide a powerful and versatile tool to extract information from quantum systems. However, how to realize them on an arbitrary higher-dimensional quantum system remains a challenging task. Here we propose a simple recipe for the implementation of a general positive-operator valued measurement (POVM) on a higher-dimensional quantum system via a one-dimensional discrete-time quantum walk with a two-dimensional coin. Furthermore, using single photons and linear optics, we realize experimentally a symmetric, informationally complete (SIC) POVM on a three-dimensional system with high fidelity. As an application, we realize a qutrit state tomography with SIC-POVM and confirm that the infidelity scaling achieved by the qutrit SIC-POVM is as good as that based on mutually unbiased bases. Our study paves the way to explore physics and information in higher-dimensional quantum systems and finds applications in various quantum information processing tasks that rely on generalized quantum measurements.

Experimental Single-Setting Quantum State Tomography

Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. Measurement and classical computing costs, however, increase exponentially in the system size - a bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVM). We implement these nonorthogonal measurements on an ion trap device by utilizing more energy levels in each ion - without ancilla qubits. More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, providing full tomographic information in every shot. Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination. SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods. The latter enables in-depth entanglement studies, which we experimentally showcase on a 5-qubit absolutely maximally entangled (AME) state. Moreover, the fact that the full tomography information is available in every shot enables online QST in real time. We demonstrate this on an 8-qubit entangled state, as well as for fast state identification. All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.

Bipartite quantum measurements with optimal single-sided distinguishability

We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.

What Are the Minimal Conditions Required to Define a Symmetric Informationally Complete Generalized Measurement?

Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1)every POVM element is rank one, (2)the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3)the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.

Tight frames, Hadamard matrices and Zauner’s conjecture

We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and complex Hadamard matrix in dimension d2, and a pair of ETFs (equiangular tight frames) in dimensions . We also show that a WH (Weyl–Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all .

Separability criteria via some classes of measurements

Mutually unbiased bases (MUBs) and symmetric informationally complete (SIC) positive operator-valued measurements (POVMs) are two related topics in quantum information theory. They are generalized to mutually unbiased measurements (MUMs) and general symmetric informationally complete (GSIC) measurements, respectively, that are both not necessarily rank 1. We study the quantum separability problem by using these measurements and present separability criteria for bipartite systems with arbitrary dimensions and multipartite systems of multi-level subsystems. These criteria are proved to be more effective than previous criteria especially when the dimensions of the subsystems are different. Furthermore, full quantum state tomography is not needed when these criteria are implemented in experiment.

The SIC Question: History and State of Play

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

Nice error frames, canonical abstract error groups and the construction of SICs

Nice error bases are generalisations of the Pauli matrices which have applications in quantum information theory. These orthonormal bases for the d×d matrices Md(C) also generalise the projective action of the Heisenberg group on Cd. Here we extend nice error bases to nice error frames. These are equal-norm tight frames for Md(C) consisting of d×d unitary matrices with a group indexing structure. We show that each nice error frame (irreducible faithful projective representation) is associated with a canonical abstract error group. This is calculated in number of examples, e.g., for all nice error bases for d<14, which then allows us to investigate which nice error bases might give rise to SICs (symmetric informationally complete positive operator valued measures). These results show that the current catalogue of nice error bases over counts. In particular, we give an explicit example of a SIC for d=6 with a nonabelian index group, and show that the Hoggar lines appear for various nice error bases, some of which are subgroups of the Clifford group. Thus all known SICs appear as orbits of subgroups of the Clifford group.

Communication capacity of mixed quantumt-designs

We operationally introduce mixed quantum t-designs as the most general arbitrary-rank extension of projective quantum t-designs which preserves indistinguishability from the uniform distribution for t copies. First, we derive upper bounds on the classical communication capacity of any mixed t-design measurement for t in [1,5]. Second, we explicitly compute the classical communication capacity of several mixed t-design measurements, including the depolarized version of any qubit and qutrit symmetric, informationally complete (SIC) measurement and complete mutually unbiased bases, the qubit icosahedral measurement, the Hoggar SIC measurement, any anti-SIC (where each element is proportional to the projector on the subspace orthogonal to one of the elements of the original SIC), and the uniform distribution over pure effects.

Experimental realization of generalized qubit measurements based on quantum walks

We report an experimental implementation of a single-qubit generalized measurement scenario, the positive-operator valued measure (POVM), based on a quantum walk model. The qubit is encoded in a single-photon polarization. The photon performs a quantum walk on an array of optical elements, where the polarization-dependent translation is performed via birefringent beam displacers and a change of the polarization is implemented with the help of wave plates. We implement: (i) trine POVM, i.e., the POVM elements uniformly distributed on an equatorial plane of the Bloch sphere; (ii) symmetric-informationally-complete (SIC) POVM; and (iii) unambiguous discrimination of two nonorthogonal qubit states.

Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.

Construction of all general symmetric informationally complete measurements

We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.

Tight bounds on accessible information and informational power

The accessible information quantifies the amount of classical information that can be extracted from an ensemble of quantum states. Analogously, the informational power quantifies the amount of classical information that can be extracted by a quantum measurement. For both quantities, we provide upper and lower bounds that depend only on the dimension of the system, and we prove their tightness. In the case of symmetric informationally complete (SIC) ensembles and measurements, stronger bounds are provided and their tightness proved for qubits and qutrits. From our upper bounds, we notice, perhaps surprisingly, that the statistics generated by SIC ensembles or measurements in arbitrary dimension, though optimal for tomographic purposes, in fact never contain more than just one bit of information, the rest being constituted by completely random bits. On the other hand, from our lower bounds, we obtain an explicit strategy beating the so-called pretty-good one for the extraction of mutual information in the case of SIC ensembles and measurements.

Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

Recently there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: (1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the space of density operators as possible; and (2) in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant SIC-sets are intimately related: The latter represent minimum uncertainty states for the former in the sense of Wootters and Sussman. Finally, we contribute to the question of existence by conjecturing a quadratic redundancy in the equations for Weyl-Heisenberg SIC-sets.

Quantum-Bayesian coherence

In the Quantum-Bayesian interpretation of quantum theory (or QBism), the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics.

Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.

Experimental proposal for symmetric minimal two-qubit state tomography

We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The basis of the first measurement and the four bases of the second measurements are pairwise unbiased --- a hint at a possibly profound link between SIC POMs and mutually unbiased bases.

Quantum state tomography with fully symmetric measurements and product measurements

We introduce random-matrix theory to study the tomographic efficiency of a wide class of measurements constructed out of weighted 2-designs, including symmetric informationally complete (SIC) probability operator measurements (POMs). In particular, we derive analytic formulas for the mean Hilbert-Schmidt distance and the mean trace distance between the estimator and the true state, which clearly show the difference between the scaling behaviors of the two error measures with the dimension of the Hilbert space. We then prove that the product SIC POMs, the multipartite analog of the SIC POMs, are optimal among all product measurements in the same sense as the SIC POMs are optimal among all joint measurements. We further show that, for bipartite systems, there is only a marginal efficiency advantage of the joint SIC POMs over the product SIC POMs. In marked contrast, for multipartite systems, the efficiency advantage of the joint SIC POMs increases exponentially with the number of parties.

Properties of QBist State Spaces

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication (Fuchs and Schack, arXiv:0906.2187v1, 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we study how this subset—quantum-state space—might be characterized. Our leading characteristic is that the inner products of the probabilities are bounded, a simple condition with nontrivial consequences. To get quantum-state space something more detailed about the extreme points is needed. No definitive characterization is reached, but we see several new interesting features over those in Fuchs and Schack (arXiv:0906.2187v1, 2009), and all in conformity with quantum theory.

On the quantumness of a hilbert space

We derive an exact expression for the quantumness of a Hilbert space (defined in C. A.Fuchs and M. Sasaki, Quant. Info. Comp. 3, 377 (2003)), and show that in composite Hilbertspaces the signal states must contain at least some entangled states in order to achieve such asensitivity. Furthermore, we establish that the accessible fidelity for symmetric informationallycomplete signal ensembles is equal to the quantumness. Though spelling the most trouble foran eavesdropper because of this, it turns out that the accessible fidelity is nevertheless easyfor her to achieve in this case: Any measurement consisting of rank-one POVM elements is anoptimal measurement, and the simple procedure of reproducing the projector associated withthe measurement outcome is an optimal output strategy.