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%. 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%. We find that this can be boosted up to
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% can be achieved. Furthermore, we find that by using correlation in time of the photons all four Bell states can be distinguished with
100% success probability while for the Bell-like state discrimination, it strictly lies between
25% and
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
1−2N1 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% in some cases. Also adding further ancillary photons decreases the success probability. We then show that using non-linear gadgets namely SFG
100% success probability can be achieved even for Bell-like state discrimination.
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.
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%.
In quantum communication and fusion-based quantum computation, photonic Bell measurements are fundamentally limited when only passive linear optics is employed. While for qubits, some Bell states can be unambiguously identified with static beam splitters and no extra photons or entanglement, additional auxiliary photons or at least additional auxiliary degrees of freedom with a certain level of additional entanglement are needed to approach or attain a complete, deterministic Bell measurement. Here, we prove an exact resource threshold when the same two photons carry system qudits of dimension
d and a fixed auxiliary entangled state
Φ, possibly distributed over several additional degrees of freedom, with total Schmidt rank
rΦ. We show that a single conclusive Bell-label functional can occur for
rΦ⩾⌈d/2⌉, but deterministic discrimination of all
d2 Bell-state labels requires
rΦ⩾d. A maximally entangled rank-
d auxiliary state achieves the bound by local Bell-basis sorting between each photon's system and auxiliary degrees of freedom. Thus, the auxiliary Schmidt rank is a certified resource for ancilla-photon-free, embedded photonic Bell measurements.
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
512403≃0.787 at
28 ancilla photons. By contrast, the corresponding two-qubit measurement asymptotically approaches a maximally entangling measurement.
The use of higher-dimensional photonic encodings (qudits) instead of two-dimensional encodings (qubits) can improve the loss tolerance and reduce the computational resources of photonic-based quantum information processing. To harness this potential, efficient schemes for entangling operations such as the high-dimensional generalization of a linear optics Bell measurement will be required. We show how an efficient high-dimensional entangled state analyzer can be implemented with a linear optics interferometer and auxiliary photonic states. The degree of entanglement of the auxiliary state is much less than in previous protocols as quantified by an exponentially smaller Schmidt rank. In addition, the auxiliary state only occupies a single spatial mode, allowing it to be generated deterministically from a single quantum emitter coupled to a small qubit register. The reduced complexity of the auxiliary states results in a high robustness to imperfections and we show that auxiliary states with fidelities above 0.9 for qudit dimensions 4 can be generated in the presence of qubit error rates on the order of 10%. This paves the way for experimental demonstrations with current hardware.