Quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting instantaneously two impenetrable barriers dividing the ring into two chambers. In the process, the candidate wave functions, as the insertion points become nodes, get entangled with the barriers and can, if judiciously chosen, be distinguished with smaller error probability. As a consequence, the Helstrom bound under idealised conditions can be violated.
The problem of quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting either adiabatically or instantaneously an impenetrable barrier. The insertion point, independent of the shape of the initial wave function, becomes a node. The resulting modified wave functions can be, if the initial functions are judiciously chosen, distinguished with a smaller error probability, and as a consequence the Helstrom bound can be violated under idealised conditions.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting an impenetrable barrier in the middle of the square well, which is either a nodal or non-nodal point of the wave function. The energy required to insert the barrier is dependent on the initial state. This enables the experimenter to gain additional information beyond the standard probing of the state envisaged by Helstrom and to improve the distinguishability of the states. It is shown that under some conditions the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary
m quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimum-error probability for ambiguous discrimination among arbitrary
m mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if
m=2, the optimal inconclusive probability of unambiguous discrimination
QU and the minimum-error probability of ambiguous discrimination
QE between arbitrary given
m mixed quantum states have the relationship
QU≥2QE. In this paper, we show that, however, if
m>2, the relationship
QU≥2QE may not hold again in general, and there may be no supremum of
QU/QE for more than two states, which may also reflect an essential difference between discrimination for two-states and multi-states. (4) A number of examples are constructed.
For the optimal success probability under minimum-error discrimination between $r\ensuremath{\ge}2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the known general bounds, lower and upper. We also present the example where the values of the new general lower and upper bounds on the optimal success probability are tighter than the values of most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to
r>2 the form of the Helstrom bound. For
r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
The minimum error probability for distinguishing between two quantum states is bounded by the Helstrom limit, derived under the assumption that measurement strategies are restricted to positive operator-valued measurements. We explore scenarios in which the error probability for discriminating two quantum states can be reduced below the Helstrom bound under some constrained access of resources, indicating the use of measurement operations that go beyond the standard positive operator-valued measurements framework. We refer to such measurements as non-positive operator-valued measurements. While existing literature often associates these measurements with initial entanglement between the system and an auxiliary, followed by joint projective measurement and discarding the auxiliary, we demonstrate that initial entanglement between system and auxiliary is not necessary for the emergence of such measurements in the context of state discrimination. Interestingly, even initial product states can give rise to effective non-positive measurements on the subsystem, and achieve sub-Helstrom discrimination error when discriminating quantum states of the subsystem.