Symmetric joint measurement as a complement to the elegant joint measurement

TL;DR

The work addresses the limitation of existing symmetric joint measurements by introducing a two-qubit symmetric joint measurement with concurrence $C=\frac{1}{2}|\sin\theta|$ that spans $[0,\tfrac{1}{2}]$, complementing the elegant joint measurement (EJM) which lies in $[\tfrac{1}{2},1]$. It constructs basis states $|\Phi_k\rangle$ from parameterized single-qubit components, analyzes their rotational symmetry in single-qubit reductions, and demonstrates that the basis reduces to the original EJM at $\theta=\frac{\pi}{2}$, while interpolating to the product basis at $\theta=0$; a quantum circuit is provided to implement the measurement. The authors apply this construction to a triangular network to reveal permutation-symmetric output statistics and show quantum nonlocality beyond trilocal bounds for suitable $\theta$, with $p(a=b=c)=\frac{4+21\sin^2\theta}{64}$ in that scenario. Furthermore, the approach is generalized to even-numbered multiqubit systems, laying a foundation for scalable symmetric joint measurements. Overall, the paper broadens the family of symmetry-preserving joint measurements, offering a bridge between PBM and EJM and enabling new tests of network nonlocality and multi-qubit joint measurements.

Abstract

Traditional Bell state measurement (BSM) and product basis measurements (PBM) have been integral to nearly the entire development of quantum computing. Unlike the BSM and the PBM, a recently proposed two-qubit joint measurement called the elegant joint measurement (EJM) exhibits novel tetrahedral symmetry in its single-qubit reduced states. In [Phys.Rev.Lett.126:220401], a parameterized two-qubit iso-entangled basis was proposed, with concurrence between 1/2 and 1, perfectly spanning the original EJM and conventional BSM. We present a two-qubit symmetric joint measurement having concurrence from 0 to 1/2, which is complementary to [Phys.Rev.Lett.126:220401] and contains the PBM and the original EJM. We investigate the symmetry of the current structure and its application in triangular networks. The results indicate that the reduction vectors of the current basis states exhibit rotational symmetry, rather than the aforementioned mirror symmetry; moreover, the output probability distributions of three parties in the network explicitly demonstrate the expected permutation symmetry. Furthermore, we generalize the two-qubit symmetric joint measurement to the multiqubit systems with an even number of qubits.

Figures (4)

• Figure 1: Comparison of concurrence for the current states (green solid line) and previous parameterized EJMs (orange dash-dot line). Both vary with the parameter $\theta$, distributed below and above $C=0.5$ (red dashed line, corresponding to the original EJM without parameters), respectively.
• Figure 2: Rotational symmetry of the vectors corresponding to the reduced states $\langle \Phi_{k}|\vec{\sigma} \otimes I|\Phi_{k}\rangle$ and $\langle \Phi_{k}|I \otimes \vec{\sigma}|\Phi_{k}\rangle$. One vector can be transformed into the other by a 180-degree rotation about the axis $(\cos \varphi_{k}, \sin \varphi_{k}, 0)$.
• Figure 3: Quantum circuit for detecting the generalized three-parameter EJM.
• Figure 4: A triangular network of three parties connected by three independent sources. Each source creates a pair of Bell states $|\psi_{+}\rangle = (|01\rangle + |10\rangle)/\sqrt{2}$, which are distributed to neighboring parties. All parties perform the current symmetric joint measurements; the outputs for Alice, Bob, and Charlie are denoted by $a$, $b$, and $c$, respectively.