Equiprobable States Research Articles

Shannon’s entropy and its bounds for some a priori known equiprobable states

AbstractIt is known that Shannon’s entropy is nonnegative and its maximum value is reached for equiprobable events. Adding or removing impossible events does not affect Shannon’s entropy. However, if we increase the number of events and consider not necessarily all of them equiprobable, but at least as many of them as the initial number of equiprobable events, how does Shannon’s entropy change? We study the lower bound of the interval where the probability value of the a priori assumed equiprobable states must belong when the entropy increases.

Budget-Constrained Linear Utility Maximization and the Hurwicz (1951) Criterion

We show that for a trader displaying state-dependent risk-neutrality, budget constrained maximization based on the Hurwicz criterion for “non-probabilistic” uncertainty with the degree of pessimism being less than half, reduces to expected utility maximization under “the equal ignorance principle”, with all wealth being invested in positive return being available in just one state of nature and nothing at all in other states of nature. Such a portfolio is an extreme point of the set of budget-constrained expected utility maximizing portfolios, with equiprobable states of nature. With more than two uncertain states of nature, the above result extends to the case where the degree of pessimism is equal to half. We also show that if for a degree of pessimism, greater than or equal to half, an optimal solution to budget constrained maximization based on the Hurwicz criterion is not of this type, then it must be a “budget-constrained max-min utility maximizing” portfolio.

High-frequency oscillations in the ripple bands and amplitude information coding: Toward a biomarker of maximum entropy in the preictal signals.

Intracranial electroencephalography (iEEG) can directly record local field potentials (LFPs) from a large set of neurons in the vicinity of the electrode. To search for possible epileptic biomarkers and to determine the epileptogenic zone that gives rise to seizures, we investigated the dynamics of basal and preictal signals. For this purpose, we explored the dynamics of the recorded time series for different frequency bands considering high-frequency oscillations (HFO) up to 240 Hz. We apply a Hilbert transform to study the amplitude and phase of the signals. The dynamics of the different frequency bands in the time causal entropy-complexity plane, H × C, is characterized by comparing the dynamical evolution of the basal and preictal time series. As the preictal states evolve closer to the time in which the epileptic seizure starts, the, H × C, dynamics changes for the higher frequency bands. The complexity evolves to very low values and the entropy becomes nearer to its maximal value. These quasi-stable states converge to equiprobable states when the entropy is maximal, and the complexity is zero. We could, therefore, speculate that in this case, it corresponds to the minimization of Gibbs free energy. In this case, the maximum entropy is equivalent to the principle of minimum consumption of resources in the system. We can interpret this as the nature of the system evolving temporally in the preictal state in such a way that the consumption of resources by the system is minimal for the amplitude in frequencies between 220-230 and 230-240 Hz.

Optimizing voting order on sequential juries: a median voter theorem and beyond

We consider an odd-sized “jury”, which votes sequentially between two equiprobable states of Nature (say A and B, or Innocent and Guilty), with the majority opinion determining the verdict. Jurors have private information in the form of a signal in [-1,+1], with higher signals indicating A more likely. Each juror has an ability in [0, 1], which is proportional to the probability of A given a positive signal, an analog of Condorcet’s p for binary signals. We assume that jurors vote honestly for the alternative they view more likely, given their signal and prior voting, because they are experts who want to enhance their reputation (after their vote and actual state of Nature is revealed). For a fixed set of jury abilities, the reliability of the verdict depends on the voting order. For a jury of size three, the optimal ordering is always as follows: middle ability first, then highest ability, then lowest. For sufficiently heterogeneous juries, sequential voting is more reliable than simultaneous voting and is in fact optimal (allowing for non-honest voting). When average ability is fixed, verdict reliability is increasing in heterogeneity. For medium-sized juries, we find through simulation that the median ability juror should still vote first and the remaining ones should have increasing and then decreasing abilities.

Interacting jammed granular systems.

More than 30 years ago Edwards and co-authors proposed a model to describe the statistics of granular packings by an ensemble of equiprobable jammed states. Experimental tests of this model remained scarce so far. We introduce a simple system to analyze statistical properties of jammed granular ensembles to test Edwards theory. Identical spheres packed in a nearly two-dimensional geometrical confinement were studied in experiments and numerical simulations. When tapped, the system evolves toward a ground state, but due to incompatible domain structures it gets trapped. Analytical calculations reproduce relatively well our simulation results, which allows us to test Edwards theory on a coupled system of two subsystems with different properties. We find that the joint system can only be described by the Edwards theory if considered as a single system due to the constraints in the stresses. The results show counterintuitive effects as in the coupled system the change in the order parameter is opposite to what is expected from the change in the compactivity.

A functional equation of tail-balance for continuous signals in the Condorcet Jury Theorem

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability p, then the probability of a correct verdict tends to one as the jury size tends to infinity (Marquis de Condorcet in Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Imprim. Royale, Paris, 1785). Recently, Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) developed a model where jurors sequentially receive independent signals from an interval according to a distribution which depends on the state of Nature and on the juror’s “ability”, and vote sequentially. This paper shows that, to mimic Condorcet’s binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio alpha (t) of the probability that a mean-zero random variable satisfies X>t given that |X|>t. In particular, we show that under natural symmetry assumptions the tail-balances alpha (t) uniquely determine the signal distribution and so the distributions assumed in Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) are uniquely determined for alpha (t) linear.

On the Anisotropic Trigger Electrical Properties of Two-Dimensional Superlattices

The problem of the spontaneous generation of a transverse electric field with a variation in the pulling field in a two-dimensional superlattice with an additive spectrum is considered. It is shown that if the miniband width is the same in the longitudinal and transverse direction, two equiprobable states of the system corresponding to the appearance of a transverse field of different signs appear upon reaching a certain threshold value by the pulling field, which leads to a knee in the longitudinal current–voltage characteristic. If the miniband widths deviate from equality, one of these states becomes more energetically favorable, but the system can transfer into the second state under a certain pulsed effect, which leads to a jump-like change in the longitudinal current in the superlattice.

Manipulating the Flipping of Water Dipoles in Carbon Nanotubes**Supported by the National Natural Science Foundation of China under Grant Nos 11875236, 61575178, 11574272 and U1832150, the Zhejiang Provincial Natural Science Foundation under Grant Nos LY16A040014 and LY18A040001, and the Zhejiang Provincial Science and Technology Project under Grant No LGN18C200017.

Flipping of water dipoles in carbon nanotubes is of great importance in many physical and biological applications, such as signal amplification, molecular switches and nano-gates. Ahead of these applications, understanding and inhibiting the non-negligible thermal noise is essential. Here, we use molecular dynamics simulations to show that the flipping frequency of water dipoles increases with the rising temperature, and the thermal noise can be suppressed by imposed charges and external uniform electric fields. Furthermore, the water dipoles flip periodically between two equiprobable and stable states under alternating electric fields. These two stable states may be adopted to store 0 and 1 bits for memory storage or molecular computing.

Constructive Method for Detecting the Information Backflow of Non-Markovian Dynamics.

We investigate the relation between two approaches to the characterization of quantum Markovianity, divisibility and lack of information backflow. We show that a bijective dynamical map is completely positive divisible if and only if a monotonic nonincrease of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover, our proof is constructive: given any such map that is not completely positive divisible, we give an explicit construction of two states that, when taken with the same apriori probability, exhibit information backflow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of bijective maps without using any entanglement between the system and ancilla.

Minimum error discrimination of equi-coherent bosonic states of n-qubit by separable measurement

In this paper, local distinguishability of the multipartite equi-coherent quantum states is studied in the bosonic subspace. A method is proposed to give an upper bound on the optimal success probability in terms of quantum coherence for equiprobable states. Then a necessary and sufficient condition to saturate this upper bound is presented. This condition enables us to give optimal measurements and success probabilities for several examples, including general pure states and n-qubit mixed states which may be linearly dependent. Perfect discrimination is possible for linearly independent states that have maximum quantum coherence. Finally, discrimination of these states when restricted to separable measurements is considered. To this aim an appropriate transformation is proposed to obtain separable measurements equivalent to the optimal measurement, in which the success probability for both is equal. So the upper bound is also valid for separable and LOCC protocol. The important advantage is that for the bosonic measurements it is always possible to obtain equivalent separable operators, not only for minimum error discrimination but also for all strategies of discrimination.

Multistability in the mixtures of smectic-C^{*} materials with compensated twisting power.

The effect of multistability in the mixtures of smectic-C^{*} materials with compensated twisting power, i.e., the existence of a large number of almost equiprobable states in the same mixture under the same conditions, is analyzed in the framework of elastic continuum theory. A simple molecular model is also considered. It is shown that multistability can follow from the bulk properties of the smectic-C^{*} materials with compensated twisting power, but with high spontaneous polarization. Multistability leads to the formation of ferroelectric domains, in which the director oscillates in space. The length and amplitude of this oscillation is tunable smoothly by an electric field. Theoretical results for the domain length agree completely with the experimental data. A suggestion is made as to why each domain structure is remembered without the energy consumption, when the electric field is abruptly switched off. The structural dependence on material parameters, such as the spontaneous polarization, the elastic constant, and the equilibrium wave number, is predicted.

Optimal measurements for the discrimination of quantum states with a fixed rate of inconclusive results

We study the discrimination of $N$ mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of $N$ states in a $d$-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating $N$ equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in $d$ dimensions. In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.

A Shared and Programmable Maximum-Confident Discrimination among Linearly Dependent Symmetric Qubit States

We show how a shared and programmable maximum-confidence discrimination (SPMCD) can be implemented by two remote parters Alice and Bob. Here Bob is given a qubit prepared in one of N linearly dependent symmetric equiprobable states. Alice has the knowledge of Bob’s signal states, but Bob has not. We build a quantum network that would be able to perform various desired maximum-confidence discrimination among Bob’s measured (data) states depending on Alice’s auxiliary (program) state. The SPMCD can be thought of as a two-step process, in which a two-outcome shared and programmable probability operator measure (POM) performed on data qubit B is firstly implemented by Alice and Bob followed by a N-outcome local POM on B implemented by Bob. We explicitly construct the required POMs. The fact that the nonlocal data-program conditional evolution, which induces the shared and programmable POM, can be realized deterministically using only two partially entangled qubit pairs is notable. The successful probability of implementing this SPMCD is optimal only for one program setting. However, for a relatively large set of program settings it can be very close to the optimal value in an ordinary, local, maximum-confidence discrimination. This protocol is feasible for current experimental technology.

A GLASS HALF-FULL: BRIAN SKYRMS'SSIGNALS

Brian Skyrms'sSignalshas the virtues familiar from hisEvolution of the Social Contract(1996) andThe Stag Hunt(2003). He begins with a very simple model of agents in interaction, and in a series of brief and beautifully clear chapters, this model and its successors are explored, elaborated, connected and illustrated through biological theory and the social sciences.Signalsborrows its core model from David Lewis: it is Lewis's signalling game. In this game, two agents interact. One agent can observe which of two equi-probable states the world is in, but that agent cannot act directly and profitably on that information. However, the informed agent can act in a way that will be perceptually salient to a second agent: say, by raising a red or a green flag. The second agent does have the capacity to respond appropriately to each state of the world. If that second agent chooses the right option, given the state of the world, both are rewarded. If the second agent fails to choose the right action, neither are. Obviously, the two agents are best off if they have a practice in which the informed agent regularly chooses a distinct, salient cue in response to each of the two world states, and in which the powerful agent uses that cue to select the rewarding act. Less obviously, agents with simple trial and error learning capacities can learn to signal and respond: neither explicit negotiation nor cognitive sophistication are required. Likewise, if individual agents do not have the capacity to learn, but if they breed true but with some variation, the evolutionary version of trial and error learning can take a population to one of the signalling system equilibria.

Maximum-confidence discrimination among symmetric qudit states

We study the maximum-confidence (MC) measurement strategy for discriminating among nonorthogonal symmetric qudit states. Restricting to linearly dependent and equally likely pure states, we find the optimal positive operator valued measure (POVM) that maximizes our confidence in identifying each state in the set and minimizes the probability of obtaining inconclusive results. The physical realization of this POVM is completely determined and it is shown that after an inconclusive outcome, the input states may be mapped into a new set of equiprobable symmetric states, restricted, however, to a subspace of the original qudit Hilbert space. By applying the MC measurement again onto this new set, we can still gain some information about the input states, although with less confidence than before. This leads us to introduce the concept of "sequential maximum-confidence" (SMC) measurements, where the optimized MC strategy is iterated in as many stages as allowed by the input set, until no further information can be extracted from an inconclusive result. Within each stage of this measurement our confidence in identifying the input states is the highest possible, although it decreases from one stage to the next. In addition, the more stages we accomplish within the maximum allowed, the higher will be the probability of correct identification. We will discuss an explicit example of the optimal SMC measurement applied in the discrimination among four symmetric qutrit states and propose an optical network to implement it.

Minimum error discrimination between similarity-transformed quantum states

Using the well-known necessary and sufficient conditions for minimum error discrimination (MED), we extract an equivalent form for the MED conditions. In fact, by replacing the inequalities corresponding to the MED conditions with an equivalent but more suitable and convenient identity, the problem of mixed state discrimination with optimal success probability is solved. Moreover, we show that the mentioned optimality conditions can be viewed as a Helstrom family of ensembles under some circumstances. Using the given identity, MED between $N$ similarity transformed equiprobable quantum states is investigated. In the case that the unitary operators are generating a set of irreducible representation, the optimal set of measurements and corresponding maximum success probability of discrimination can be determined precisely. In particular, it is shown that for equiprobable pure states, the optimal measurement strategy is the square-root measurement (SRM), whereas for the mixed states, SRM is not optimal. In the case that the unitary operators are reducible, there is no closed-form formula in the general case, but the procedure can be applied in each case in accordance to that case. Finally, we give the maximum success probability of optimal discrimination for some important examples of mixed quantum states, such as generalized Bloch sphere $m$-qubit states, spin-$j$ states, particular nonsymmetric qudit states, etc.

The minimum-error discrimination via Helstrom family of ensembles and convex optimization

Using the convex optimization method and Helstrom family of ensembles introduced in Ref. Kimura et al. in Phys Rev A, 79:062306 (2009), we discuss optimal ambiguous discrimination in qubit systems for N known quantum states. We obtain optimal success probability (OSP) and optimal measurement (OM) for N(≥ 4) known equiprobable quantum states where all these states are distributed at the same distance from the center of Bloch ball which all the states do not lie on the same plane. After reproducing known results for distinguishing between two quantum states, we also exactly discuss discrimination of three quantum state where OSP and OM are calculated explicitly. In particular examples of three states, for a numerical case, mirror symmetric states, and particularly chosen coplanar pure states, previously obtained results are reproduced. We also obtain OSP and OM for a particular case of states considered on the line, circumference of a circle, and states that is defined by vertices of a Platonic solid. In addition, OSP is presented for a special case of four states.

Minimum error discrimination problem for pure qubit states

The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among $N$ pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and $N=2,3,4$ pure qubit states given with different prior probabilities.

Optimal positive-operator-valued measures for unambiguous state discrimination

Optimization of the mean efficiency for unambiguous (or error free)discrimination among $N$ given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit $N$-sphere for equiprobable states and on an $N$-ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a POVM requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 till $N$ and the Neumark's extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system + ancilla as the tensor product of a two-dimensional ancilla space and the $N$-dimensional system space.

Agreeing to disagree in a countable space of equiprobable states of nature

An example is given in which agents agree to disagree in a countable space of equiprobable states of nature. Even in this unorthodox setting, if the sets of the information partitions are intervals, an agreement theorem holds. A result that describes the margin for disagreement is also obtained.