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arXiv:2112.01605 · Physical Review A

Optimal unambiguous discrimination of Bell-like states with linear optics

Dov Fields, János A. Bergou, Mark S. Hillery, Siddhartha Santra, Vladimir S. Malinovsky

Published Aug 8, 2022• 4 citations
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Abstract

Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillae, can facilitate their potential implementation. Here, we consider the task of unambiguously discriminating between Bell-like states using linear optics and without the use of ancillary photons. This is a basic problem relevant in diverse settings, for example, in the measurement of the output of an entangling quantum circuit or for entanglement swapping at a quantum repeater station. While it is known that exact Bell states of two qubits can be discriminated with an optimal success probability of 50%, we find, surprisingly, that for Bell-like states the optimal probability can be only 25%. We analyze a set of Bell-like states in terms of their distinguishability, entanglement as measured by concurrence, and parameters of the beam-splitter network used for unambiguous discrimination. Further, we provide the linear optical configuration comprising single-photon detectors and beam splitters with input-state-dependent parameters that achieves optimal discrimination in the Bell-like case.

DOI
10.1103/physreva.106.023706

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Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillae, can facilitate their potential implementation. Here, we consider the task of unambiguously discriminating between Bell-like states using linear optics and without the use of ancillary photons. This is a basic problem relevant in diverse settings, for example, in the measurement of the output of an entangling quantum circuit or for entanglement swapping at a quantum repeater station. While it is known that exact Bell states of two qubits can be discriminated with an optimal success probability of 50%, we find, surprisingly, that for Bell-like states the optimal probability can be only 25%. We analyze a set of Bell-like states in terms of their distinguishability, entanglement as measured by concurrence, and parameters of the beam-splitter network used for unambiguous discrimination. Further, we provide the linear optical configuration comprising single-photon detectors and beam splitters with input-state-dependent parameters that achieves optimal discrimination in the Bell-like case.
A linear optics-based scheme to implement various quantum information processing tasks is of paramount importance due to ease of implementation and low noise. Many information-theoretic tasks depend on the successful discrimination of Bell states. A no-go theorem has been proved in literature which tells that it is not possible to perfectly discriminate among the four Bell states by restricting measurement apparatus to linear optical elements. The success probability is only 50%50\%50%. Through using extra resources such as hyper entanglement, ancillary entanglement, and even a minimum amount of non-linearity complete Bell-state discrimination can be achieved. The success probability for Bell-like state discrimination is only 25%25\%25%. We find that this can be boosted up to 50%50\%50% using hyperentanglement in polarization, momentum, or OAM degrees of freedom of the photons which is in contrast to the Bell-state discrimination scenario where 100%100\%100% can be achieved. Furthermore, we find that by using correlation in time of the photons all four Bell states can be distinguished with 100%100\%100% success probability while for the Bell-like state discrimination, it strictly lies between 25%25\%25% and 50%50\%50% depending on the state parameter with only three Bell-like states being distinguishable. We also observe a similar contrast when we use ancillary entangled photons. While the success probability for all four Bell-state discrimination increases as 1−12N1-\frac{1}{2^N}1−2N1​ where N is the number of ancillary photons for Bell-like states it depends again on the state parameters and can be less than 25%25\%25% in some cases. Also adding further ancillary photons decreases the success probability. We then show that using non-linear gadgets namely SFG 100%100\%100% success probability can be achieved even for Bell-like state discrimination.
One major challenge in utilizing linear optical setups for quantum computing, is that only a limited set of operations can be preformed. Achieving universal quantum computing using only linear optical components comes at the cost of either requiring an untenable number of ancillary photons or the acceptance of some probability of failure. Understanding the constraints and optimizing general linear optical operations under these constraints is both an important and challenging task. Here we consider the problem of perfectly discriminating between Bell-like states without the use of ancillary photons. For the discrimination between Bell states, the optimal success probability cannot exceed 50%. Here we derive the optimal approach for Bell-like states and show that the success probability without ancillary photons can only reach 25%.
We propose a framework to analyze linear optical circuits based on an analogy with stabilizer formalism in quantum circuits, which provides efficiently computable formulas related to state discriminations. Hence, we analyze a Bell-state discrimination scheme with linear optics and ancillary single photons. With an increasing number of ancilla photons, the success probability of Bell-state discrimination has a maximum of 403512≃0.787\frac{403}{512} \simeq 0.787512403​≃0.787 at 282828 ancilla photons. By contrast, the corresponding two-qubit measurement asymptotically approaches a maximally entangling measurement.
Retrieving classical information encoded in optical modes is at the heart of many quantum information processing tasks, especially in the field of quantum communication and sensing. Yet, despite its importance, the fundamental limits of optical mode discrimination have been studied only in a few specific examples. Here we present a toolbox to find the optimal discrimination of any set of optical modes. The toolbox uses linear and semidefinite programming techniques, which provide rigorous (not heuristic) bounds, and which can be efficiently solved on standard computers. We study both probabilistic and unambiguous single-shot discrimination in two scenarios: the channel-discrimination scenario, typical of metrology, in which the verifier holds the light source and can set up a reference frame for the phase; and the source-discrimination scenario, more frequent in cryptography, in which the verifier only sees states that are diagonal in the photon-number basis. Our techniques are illustrated with several examples. Among the results, we find that, for many sets of modes, the optimal state for mode discrimination is a superposition or mixture of at most two number states; but this is not general, and we also exhibit counterexamples.
Bell measurements, entailing the projection onto one of the Bell states, play a key role in quantum information and communication, where the outcome of a variety of protocols crucially depends on the success probability of such measurements. Although in the case of qubit systems, Bell measurements can be implemented using only linear optical components, the same result is no longer true for qudits, where at least the use of ancillary photons is required. In order to circumvent this limitation, one possibility is to introduce nonlinear effects. In this work, we adopt the latter approach and propose a scalable Bell measurement scheme for high-dimensional states, exploiting multiple squeezer devices applied to a linear optical circuit for discriminating the different Bell states. Our approach does not require ancillary photons, is not limited by the dimension of the quantum states, and is experimentally scalable, thus paving the way toward the realization of an effective high-dimensional Bell measurement.
Photonic quantum gates are fundamental building blocks for a wide range of optical quantum information processing tasks. We propose an efficient linear-optical scheme for implementing a post-selected three-qubit fan-out gate (also known as a controlled-NOT-NOT gate) using only two ancillary single photons and linear optical elements. The scheme can be generalized to an nnn-qubit fan-out gate, requiring (n−1)(n-1)(n−1) ancillary single photons and (3n−3)(3n-3)(3n−3) polarizing beam splitters (PBSs), with a success probability of (14)n−1\left(\frac{1}{4}\right)^{n-1}(41​)n−1. Compared to the standard gate decomposition approach, which requires (2n−2)(2n-2)(2n−2) ancillary single photons, (5n−5)(5n-5)(5n−5) PBSs, and yields a success probability of (18)n−1\left(\frac{1}{8}\right)^{n-1}(81​)n−1. Our scheme significantly reduces the resource overhead and improves the success probability. We further evaluate the gate performance and demonstrate improved robustness compared to gate decomposition-based methods.