[Paper Review] Complete and Deterministic Bell State Measurement Using Nonlocal Spin Products

Keiichi Edamatsu|arXiv (Cornell University)|Dec 27, 2016

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper proposes a complete and deterministic Bell state measurement protocol using nonlocal spin product measurements assisted by shared entanglement as a resource. By measuring commuting nonlocal observables $S_{zz}$ and $S_{xx}$, the scheme enables deterministic identification of all four Bell states and realizes a complete Bell filter, offering a scalable solution for photonic and other qubit systems where deterministic nonlocal CNOT gates are unavailable.

ABSTRACT

A simple protocol for complete and deterministic Bell state measurement is proposed. It consists of measurements of nonlocal spin product operators with the help of shared entanglement as an ancillary resource. The protocol realizes not only nonlocal Bell state measurement between a pair of distant qubits but also a complete Bell filter that transmits either one of the Bell states indicated by the measurement outcome. These schemes will be useful in quantum technologies where nonlocal Bell state measurement is indispensable.

Motivation & Objective

• To develop a deterministic and complete Bell state measurement protocol for distant qubits where local operations and classical communication (LOCC) alone are insufficient.
• To overcome the limitations of linear-optical implementations, which cannot achieve deterministic and complete Bell measurements due to the inability to implement deterministic CNOT gates.
• To enable both Bell state measurement and Bell filtering using a single protocol, enhancing utility in quantum communication and computation.
• To provide a practical implementation route using linear optics with path-entangled ancillas, applicable to photonic qubits.

Proposed method

• The protocol uses nonlocal spin product operators $S_{ij} = \sigma_i \otimes \sigma_j$ for $i,j \in \{x,y,z\}$, which commute and thus can be measured simultaneously.
• Measurement of $S_{zz}$ is implemented via local $\sigma_z$ measurements on Alice and Bob’s qubits, with outcomes encoded in the path degrees of freedom of photons using polarizing beamsplitters (PBS).
• Measurement of $S_{xx}$ is achieved by applying a Hadamard gate (via half-wave plates) followed by local $\sigma_x$ measurements on the path qubits.
• The joint outcomes of $S_{zz}$ and $S_{xx}$ uniquely identify the input Bell state, enabling complete and deterministic discrimination.
• A shared maximally entangled Bell state in the path degrees of freedom acts as an ancillary resource to enable the nonlocal measurement.
• The protocol can be extended to implement a complete Bell filter, where the output state is projected onto a specific Bell state based on measurement outcomes.

Experimental results

Research questions

• RQ1 Can a complete and deterministic Bell state measurement be achieved without nonlocal CNOT gates?
• RQ2 Can nonlocal spin product observables $S_{zz}$ and $S_{xx}$ be measured simultaneously using only local operations and classical communication?
• RQ3 Is it possible to realize a deterministic Bell filter using shared entanglement and nonlocal measurements?
• RQ4 Can this protocol be implemented in linear optics with photonic qubits, despite the known limitations of linear optics in implementing deterministic CNOTs?
• RQ5 Can the measurement strength of nonlocal spin products be tuned to explore generalized measurements and uncertainty relations?

Key findings

• The protocol achieves complete and deterministic Bell state measurement by measuring commuting nonlocal spin products $S_{zz}$ and $S_{xx}$, which form a complete set of observables for the Bell basis.
• The scheme enables a complete Bell filter, where the output state is projected onto a specific Bell state corresponding to the measurement outcome, using two ebits of entanglement.
• The method is realizable with current linear optical technology using path-entangled ancillas and polarizing beamsplitters as CNOT gates.
• The protocol overcomes the fundamental limitation of linear optics in implementing deterministic Bell measurements by using ancillary entanglement to simulate nonlocal operations.
• The measurement protocol can be generalized to weak and intermediate-strength measurements, enabling exploration of nonlocal measurement uncertainty relations.
• The scheme is applicable to any qubit system where nonlocal CNOTs are unavailable, including photonic qubits, making it broadly useful in quantum information technologies.