Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, $\hbar$, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of $\hbar$---all orders higher than $\hbar^0$ can be interpreted as quantum corrections to the order $\hbar^0$ term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order $\hbar^0$ terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is equivalent to orders of $\hbar$ as a resource within the WWM formalism. This explains why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres-Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, we find that qubit Pauli observables lack an order $\hbar^0$ contribution in their Weyl symbol and so exhibit contextuality regardless of the state being measured. On the other hand, odd-dimensional qudit observables generally possess non-zero order $\hbar^0$ terms, and higher, in their WWM formulation, and so exhibit contextuality depending on the state measured: odd-dimensional qudit states that exhibit measurement contextuality have an order $\hbar^1$ contribution that allows for the violation of classical bounds while states that do not exhibit measurement contextuality have insufficiently large order $\hbar^1$ contributions.
Simulating quantum contextuality with classical systems requires memory. A fundamental yet open question is which is the minimum memory needed and, therefore, the precise sense in which quantum systems outperform classical ones. Here we make rigorous the notion of classically simulating quantum state-independent contextuality (QSIC) in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite QSIC set. We obtain the minimum memory classical systems need to simulate arbitrary QSIC sets under the assumption that the simulation should not contain any oracular information. In particular, we show that, while classically simulating two qubits tested with the Peres-Mermin set requires $\log_2 24 \approx 4.585$ bits, simulating a single qutrit tested with the Yu-Oh set requires, at least, $5.740$ bits.
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher-symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher-symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher-symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.
Nearly 30 years ago, J.P. Crutchfield and K. Young proposed in Phys. Rev. Lett. \bf 63, 105 (1989) some supposedly novel measures of time series complexity, and their relations to existing concepts in nonlinear dynamical systems. At that time it seemed that the multiple faults of this paper would make it obsolete soon. Since this has not happened, and these faults still infest the literature on what is now called "computational mechanics", I want here to rectify the situation.
Algebraic number theory relates SIC-POVMs in dimension $d>3$ to those in dimension $d(d-2)$. We define a SIC in dimension $d(d-2)$ to be aligned to a SIC in dimension $d$ if and only if the squares of the overlap phases in dimension $d$ appear as a subset of the overlap phases in dimension $d(d-2)$ in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension $d$ there exists an aligned SIC in dimension $d(d-2)$. In all our examples the aligned SIC has lower dimensional equiangular tight frames embedded in it. If $d$ is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If $d-2$ is an odd prime number we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.
In the Majorana or stellar representation of quantum states, an arbitrary pure state of a spin-1 system is represented by a pair of unit vectors. The squared modulus of the inner product of two spin-1 states is given by an expression involving their four Majorana vectors. Starting from this expression, a purely geometrical derivation is given of the MUBs and SIC-POVMs of a spin-1 system. In the case of the MUBs, the additional assumption of time-reversal invariance is required but it confers a number of benefits: it determines the point symmetries of the MUBs (or, more precisely, of their Majorana vectors in ordinary three-dimensional space); it shows that a maximal set of MUBs is unique; and it allows all the unextendible sets of bases to be deduced. Most of the results in this paper are not new and duplicate those obtained earlier by other methods, but the Majorana approach nevertheless illuminates them from a new point of view. In particular, it reveals the MUBs and SICs as symmetrical sets of vectors in ordinary three-dimensional space, in contrast to the usual description of them as vectors in a generally complex and multidimensional Hilbert space. The possible utility of this viewpoint is discussed.
We address the question of whether the set of almost quantum correlations can be predicted by a generalised probabilistic theory. We answer this in the negative whenever the theory satisfies the so-called no-restriction hypothesis, in which the set of possible measurements is the dual of the set of states. The no-restriction hypothesis is an explicit axiom in many reconstructions of quantum theory and is implied by a number of intuitive physical principles. Our results imply that any theory attempting to reproduce the set of almost quantum correlations requires a super-selection-type rule limiting the possible measurements. They also suggest that the no-restriction hypothesis may play a fundamental role in singling out the set of quantum correlations among general non-signalling ones.
We present a conjectured family of SIC-POVMs which have an additional symmetry group whose size is growing with the dimension. The symmetry group is related to Fibonacci numbers, while the dimension is related to Lucas numbers. The conjecture is supported by exact solutions for dimensions d=4,8,19,48,124,323, as well as a numerical solution for dimension d=844.
We reflect on the information paradigm in quantum and gravitational physics and on how it may assist us in approaching quantum gravity. We begin by arguing, using a reconstruction of its formalism, that quantum theory can be regarded as a universal framework governing an observer's acquisition of information from physical systems taken as information carriers. We continue by observing that the structure of spacetime is encoded in the communication relations among observers and more generally the information flow in spacetime. Combining these insights with an information-theoretic Machian view, we argue that the quantum architecture of spacetime can operationally be viewed as a locally finite network of degrees of freedom exchanging information. An advantage -- and simultaneous limitation -- of an informational perspective is its quasi-universality, i.e. quasi-independence of the precise physical incarnation of the underlying degrees of freedom. This suggests to exploit these informational insights to develop a largely microphysics independent top-down approach to quantum gravity to complement extant bottom-up approaches by closing the scale gap between the unknown Planck scale physics and the familiar physics of quantum (field) theory and general relativity systematically from two sides. While some ideas have been pronounced before in similar guise and others are speculative, the way they are strung together and justified is new and supports approaches attempting to derive emergent spacetime structures from correlations of quantum degrees of freedom.