Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.
Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.
Given an unknown quantum state described by one of two possible density operators, the Helstrom bound provides the minimum discrimination error probability (DEP) by optimizing over all possible quantum measurements. However, it is unrealistic to implement arbitrary measurements in practice due to physical limitations of measurement apparatuses. This paper considers a quantum state discrimination scenario where a fixed measurement apparatus is available. In this setting, we advocate the use of quantum pre-processing (QPP) to realize effectively different measurements from that of the fixed apparatus. Applying optimal QPP prior to measurement with the fixed apparatus allows one to minimize the DEP. This paper derives the minimum DEP, determines the QPP required to achieve it, and provides necessary and sufficient conditions for this minimum DEP with optimal QPP to coincide with the Helstrom bound.
A set of pairwise non-orthogonal quantum states cannot be perfectly discriminated, and this remains true even when one bit of classical partial information is available prior to the measurement. Contrary to the usual intuition that earlier information is at least as valuable as later information, we show that the same bit can be more useful when it arrives after the measurement. We present a framework for state discrimination in which the sender provides classical information in response to a request from the receiver, which we refer to as a query. In some cases, pairwise non-orthogonal states can be perfectly discriminated when the query is allowed to depend on the measurement outcome. We give a general method for finding the optimal strategy in this setting, which turns out to be the standard minimum-error discrimination problem for an auxiliary ensemble.
Abstract The discrimination of quantum states is a central problem in quantum information science and technology. Meanwhile, partial post-selection has emerged as a valuable tool for quantum state engineering. In this work, we bring these two areas together and ask whether partial measurements can enhance the discrimination performance between two unknown and non-orthogonal pure states. Our framework is general: the two unknown states interact with the same environment—set in a pure state—via an arbitrary unitary transformation. A measurement is then performed on one of the output modes (i.e. a partial measurement), modeled by an arbitrary positive operator-valued measure (POVM). We then allow classical communication to inform the unmeasured mode of the outcome of the partial measurement, which is subsequently measured by a POVM that is optimal in the sense that the discrimination probability of error is minimized. The two POVMs act locally and classical information is exchanged between the two modes, representing a single-round (feed-forward) form of local operations with classical communication. Under these considerations, we first show that, as expected, the minimum error probability, averaged over all possible conditional states, cannot be reduced below the minimum error probability of discriminating the original input states. Then, we devise a generic setup produces specific examples where the conditional discrimination can achieve strictly lower error probabilities than the original optimal measurement, illustrating that while post-selection does not improve the average performance, it can enable better discrimination in certain post-selected ensembles.
In quantum state discrimination, typically the design of measurement operators or probe state is formulated assuming the set of possible states is perfectly known, but this may yield designs which are sensitive to deviations in the realized set of states. For example, the channel through which a transmitted state is sent may not be deterministic, but instead characterized by a classical distribution over quantum channels. In this paper, we consider the design of measurement schemes and probe states for quantum detection over an uncertain quantum channel. We present stochastic gradient-based algorithms to maximize the expected performance over the channel distribution under two design scenarios, joint design and two-stage design. We consider various design objectives, including the detection probability and mutual information, with the latter leading to a hybrid scheme consisting of von Neumann measurement and classical hypothesis test. Furthermore, we introduce a channel discrimination scheme which leverages the isometric extension of a quantum channel which increases the channel distinguishability, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. In addition, we apply amortized optimization to train a recurrent neural network in order to improve the convergence speed of the proposed algorithms. Finally, we apply the proposed algorithms to multi-copy channel discrimination as well as a novel joint channel-state discrimination scenario.
In quantum information technology, crucial information is regularly encoded in different quantum states. To extract information, the identification of one state from the others is inevitable. However, if the states are non-orthogonal and unknown, this task will become awesomely tricky, especially when our resources are also limited. Here, we introduce the quantum stochastic neural network (QSNN), and show its capability to accomplish the binary discrimination of quantum states. After a handful of optimizing iterations, the QSNN achieves a success probability close to the theoretical optimum, no matter whether the states are pure or mixed. Other than binary discrimination, the QSNN is also applied to classify an unknown set of states into two types: entangled ones and separable ones. After training with four samples, it can classify a number of states with acceptable accuracy. Our results suggest that the QSNN has the great potential to process unknown quantum states in quantum information.
We study the problem of calibrating a quantum receiver for optical coherent states when transmitted on a quantum optical channel with variable transmissivity, a common model for long-distance optical-fiber and free/deep-space optical communication [1]–[7]. We optimize the error probability of legacy adaptive receivers, such as Kennedy’s and Dolinar’s [8], [9], on average with respect to the channel transmissivity distribution. We then compare our results with the ultimate error probability attainable by a general quantum device, computing the Helstrom bound for mixtures of coherent-state hypotheses, for the first time to our knowledge, and with homodyne measurements. With these tools, we first analyze the simplest case of two different transmissivity values; we find that the strategies adopted by adaptive receivers exhibit strikingly new features as the difference between the two transmissivities increases. Finally, we employ a recently introduced library of shallow reinforcement learning methods [10], demonstrating that an intelligent agent can learn the optimal receiver setup from scratch by training on repeated communication episodes on the channel with variable transmissivity and receiving rewards if the coherent-state message is correctly identified.