Optimum unambiguous discrimination of two mixed quantum states

We investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank d that jointly span a 2d-dimensional Hilbert space and are related in a special way. We also present an application of the results to the problem of unambiguous quantum state comparison, generalizing the optimum strategy for arbitrary prior probabilities of the states.Comment: final versio

Optimal unambiguous filtering of a quantum state: An instance in mixed state discrimination

Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e. allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the $N$ given non-orthogonal states, $\{|\psi_{1} >,..., |\psi_{N} > \}$, are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (POVM), to distinguish $|\psi_1>$ from the set $\{|\psi_2 >,...,|\psi_N > \}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [\prl {\bf 90}, 257901 (2003)] as the emphasis there was on appplication of the results to novel probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.Comment: 8 page

Optimum measurement for unambiguously discriminating two mixed states: General considerations and special cases

Based on our previous publication [U. Herzog and J. A. Bergou, Phys.Rev. A 71, 050301(R) (2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.Comment: Submitted to Journal of Physics:Conference Series (Proceedings of the 12th Central European Workshop on Quantum Optics, Ankara, June 2005

Minimal optimal generalized quantum measurements

Optimal and finite positive operator valued measurements on a finite number $N$ of identically prepared systems have been presented recently. With physical realization in mind we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to N=7 and verify that they are minimal up to N=5. We finally propose an expression which gives the size of the minimal optimal measurements for arbitrary $N$.Comment: 9 pages, Late

The Effects of Race and Sex Discrimination Laws

The question of the effects of race and sex discrimination laws on relative economic outcomes for blacks and women has been of interest at least since the Civil Rights and Equal Pay Acts passed in the 1960s. We present new evidence on the effects of these laws based on variation induced first by state anti-discrimination statutes passed prior to the federal legislation and then by the extension of anti-discrimination prohibitions to the remaining states with the passage of federal legislation. This evidence improves upon earlier time-series studies of the effects of anti-discrimination legislation. It is complementary to more recent work that revisits this question using data and statistical experiments that provide 'treatment' and 'comparison' groups. We examine the effects of race and sex discrimination laws on employment and earnings, in each case focusing on outcomes for black females, black males, and white females relative to white males. Overall, we interpret the evidence as corroborating the general conclusion that race discrimination laws positively impacted the relative employment and earnings of blacks, although the evidence is less dramatic than that reported in other research, and there are some cases (in particular, earnings effects for black males) and periods for which we find little positive impact. We find some evidence that sex discrimination/equal pay laws boosted the relative earnings of black and white females. Finally, we find that sex discrimination/equal pay laws reduced the relative employment of both black women and white women.

Age Discrimination Laws and Labor Market Efficiency

In Lazear's (1979) model of efficient long-term incentive contracts, employers impose involuntary retirement based on age. This model implies that age discrimination laws, which bar involuntary terminations based on age, discourage the use of such contracts and reduce efficiency. Alternatively, by making it costly for firms to dismiss older workers paid in excess of their marginal product, such laws may serve as precommitment devices that make credible the long-term commitment to workers that firms must make under Lazear contracts. Given that employers remain able to use financial incentives to induce retirement, age discrimination laws may instead strengthen the bonds between workers and firms and encourage efficient Lazear contracts. We assess evidence on these alternative interpretations of age discrimination laws by estimating the effects of such laws on the steepness of age-earnings profiles. If long-term incentive contracts are strengthened or become more prevalent, average age-earnings profiles should steepen for workers who enter the labor" market after age discrimination laws are passed, and vice versa. The empirical analysis uses decennial Censuses of Population and state-level variation in age discrimination laws induced by state and federal legislation. The evidence indicates that age discrimination laws lead to steeper age-earnings profiles for cohorts entering the labor market, suggesting that these laws encourage the use of Lazear contracts, and increase efficiency.

Optical realization of optimal unambiguous discrimination for pure and mixed quantum states

Quantum mechanics forbids deterministic discrimination among non-orthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal mixed state.Comment: Some minor revision

On Interferometric Duality in Multibeam Experiments

We critically analyze the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality, do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and viceversa.Comment: 26 pages, 1 figure, substantial changes in the text, new material has been added in Section 3. Version to appear in J.Phys.

A two-qubit Bell inequality for which POVM measurements are relevant

A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin.Comment: 7 pages, 1 figur

Programmable quantum state discriminators with simple programs

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, the second performs optimal unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to n copies. In comparison to other works we find that providing n > 1 copies of the data state yields higher success probabilities than providing n > 1 copies of the program states.Comment: 17 pages, 5 figure