Abstract The task of state discrimination for a set of mutually orthogonal pure states is trivial if one has access to the corresponding sharp (projection-valued) measurement, but what if we are restricted to an unsharp measurement? Given that any realistic measurement device will be subject to some noise, such a problem is worth considering. In this paper, we consider minimum error state discrimination for mutually orthogonal states with a noisy measurement. We show that by considering repetitions of commutative Lüders measurements on the same system we are able to increase the probability of successfully distinguishing states. In the case of binary Lüders measurements, we provide a full characterisation of the success probabilities for any number of repetitions. This leads us to identify a ‘rule of three’, where no change in probability is obtained from a second measurement but there is noticeable improvement after a third. We also provide partial results for N-valued commutative measurements where the rule of three remains, but the general pattern present in binary measurements is no longer satisfied.
Abstract The task of state discrimination for a set of mutually orthogonal pure states is trivial if one has access to the corresponding sharp (projection-valued) measurement, but what if we are restricted to an unsharp measurement? Given that any realistic measurement device will be subject to some noise, such a problem is worth considering. In this paper, we consider minimum error state discrimination for mutually orthogonal states with a noisy measurement. We show that by considering repetitions of commutative Lüders measurements on the same system we are able to increase the probability of successfully distinguishing states. In the case of binary Lüders measurements, we provide a full characterisation of the success probabilities for any number of repetitions. This leads us to identify a ‘rule of three’, where no change in probability is obtained from a second measurement but there is noticeable improvement after a third. We also provide partial results for N-valued commutative measurements where the rule of three remains, but the general pattern present in binary measurements is no longer satisfied.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
A measurement strategy is developed for a different kind of hypothesis testing. It assigns, with minimum probability of error, the state of a quantum system to one or the other of two complementary subsets of a set of N given nonorthogonal quantum states occurring with given a priori probabilities. A general analytical solution is obtained for N states that are restricted to a two-dimensional subspace of the Hilbert space of the system. The result for the special case of three arbitrary but linearly dependent states is applied to a variety of sets of three states that are symmetric and equally probable. It is found that, in this case, the minimum-error probability for distinguishing one of the states from the other two is only about half as large as the minimum-error probability for distinguishing all three states individually.
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 nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ 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 (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.
The discrimination of two mixed quantum states is a fundamental task in quantum state estimation and quantum information theory. In quantum state discrimination a quantum system is assumed to be in one of two possible – in general mixed – non-orthogonal quantum states. The discrimination then consists of a measurement strategy that allows to decide in which state the system was before the measurement. In unambiguous state discrimination the aim is to make this decision without errors, but it is allowed to give an inconclusive answer.
Especially interesting are measurement strategies that minimize the probability of an inconclusive answer. A starting point for the analysis of this optimization problem was a result by Eldar et al. [Phys. Rev. A 69, 062318 (2004)], which provides non-operational necessary and sufficient conditions for a given measurement strategy to be optimal. These conditions are reconsidered and simplified in such a way that they become operational. The simplified conditions are the basis for further central results: It is shown that the optimal measurement strategy is unique, a statement that is e.g. of importance for the complexity analysis of optimal measurement devices. The optimal measurement strategy is derived for the case, where one of the possible input states has at most rank two, which was an open problem for many years. Furthermore, using the optimality criterion it is shown that there always exists a threshold probability for each state, such that below this probability it is optimal to exclude this state from the discrimination strategy.
If the two states subject to discrimination can be brought to a diagonal structure with (2×2)-dimensional blocks, then the unambiguous discrimination of these states can be reduced to the unambiguous discrimination of pure states.
A criterion is presented that allows to identify the presence of such a structure for two self-adjoint operators. This criterion consists of the evaluation of three commutators and allows an explicit construction of the (2×2)-dimensional blocks.
As an important application of unambiguous state discrimination, unambiguous
state comparison, i.e., the question whether two states are identical or not, is generalized and optimal measurements for this problem are constructed.
If for a certain family of states, a physical device maps the input state to an output state, such that a second device can be built that yields back the original input state, such a map is called reversible on this family. With respect to state discrimination, such reversible maps are particularly interesting, if the output states are pure. A complete characterization of all families that allow such a reversible and purifying map is provided. If the states are mapped to pure states, but the map itself is not reversible, upper and lower bounds are analyzed for the “deviation from perfect faithfulness”, a quantity which measures the deviation from a reversible mapping.