Quantum Incompatibility in Parallel vs Antiparallel Spins

TL;DR

This work analyzes joint measurability of incompatible qubit observables in parallel versus antiparallel two-copy spin-½ configurations. It constructs explicit 2-copy antiparallel POVMs that enable exact joint measurability of the three Pauli observables $X$, $Y$, and $Z$ (with sharpness parameter $\lambda=1$) and extends the construction to symmetric sets such as $\mathrm{SIC}_4$, revealing a symmetry-driven advantage of the antiparallel configuration. A general theorem shows that for CPTP maps this antiparallel advantage vanishes, while certain PnCP maps yield enhanced compatibility, with a GPT interpretation in which minimal tensor-product composites widen joint measurability further. The work also uncovers connections to foundational tasks, notably the mean King retrodiction problem and Bub’s quantum cryptography, and discusses how the antiparallel scheme can improve device estimation and be demonstrated in finite sub-ensembles. Together, these results illuminate a resource-like aspect of antiparallel spin configurations and chart experimental pathways for observing enhanced compatibility in practice.

Abstract

We explore the joint measurability of incompatible qubit observables on ensembles of parallel and antiparallel spin-1/2 pairs. In parallel configuration, both spins are prepared in the same state, whereas in antiparallel case, each spin is paired with its flipped counterpart. We show that the antiparallel configuration uniquely enables the exact simultaneous prediction of three mutually orthogonal spin components-an advantage not achievable with parallel states. Extending beyond three observables, we examine joint measurability for larger sets of spin measurements and further generalize our analysis to state configurations beyond the parallel and antiparallel cases. As we show, our results reveal a deep connection to the 'mean King retrodiction task' proposed by Vaidman, Aharonov, and Albert, and have implications for a cryptographic protocol introduced by Jeffrey Bub. We further demonstrate how the enhanced compatibility in the antiparallel configuration can facilitate efficient estimation of unknown measurement devices. Finally, we discuss prospects for experimentally realizing the enhanced measurement compatibility in antiparallel configuration by analyzing the effect on finite sub-ensembles of states.

Figures (5)

• Figure 1: (Color online) (a) Naive strategy: Observable $X$ is measured on one copy with red and blue bars respectively denoting the probability of outcomes $+1~\&~-1$ on the given state. The other two observables are jointly measured on the remaining copy with sharpness value $\lambda=1/\sqrt{2}$Busch1986, with blue (red) part in the red (blue) bar indicating the 'unsharpness' in the corresponding outcome. (b) Parallel configuration: All three observables can be jointly measured on two-copies with sharpness value $\lambda=\sqrt{3}/2$Carmeli2016 (b) Antiparallel configuration: All three observables can be jointly measured with sharpness value $\lambda=1$ [Theorem \ref{['theo1']}]
• Figure 2: (Color online) On any sub-ensemble $\mathcal{D}_{\text{GC}}$ on a great circle of the Bloch sphere, and on the sub-ensemble $\mathcal{D}_{\text{Tet}}$ both the parallel and antiparallel configuration ensure compatibility of $\{X,Y,Z\}$ for sharpness parameter $\lambda=1$. On the other hand, on the sub-ensemble $\mathcal{D}_{\text{Oct}}$ antiparallel configuration ensures compatibility up-to $\lambda=1$ (a consequence of Theorem \ref{['theo1']}), whereas parallel configuration allows compatibility up-to $\lambda\approx0.86603$.
• Figure 3: (Color online) The vectors $\{\xi_l\}_{l=0}^7$ (middle). Each vertex is orthogonal to the vertices connected through face diagonal, namely the vectors $\{\xi_1,\xi_2,\xi_4,\xi_7\}$ are mutually orthogonal (left) and the vectors $\{\xi_0,\xi_3,\xi_5,\xi_6\}$ are mutually orthogonal (right).
• Figure 4: (Color online) The observables $\left\{\sigma_{\hat{n}_i}\right\}_{i=1}^3$ are symmetrically chosen by fixing the relative angles between any two of them to be identical. Each of them makes an angle $\theta$ with $\hat{t}=\tfrac{1}{\sqrt{3}}(1,1,1)^{\mathrm{T}}$. Inset: view from the top of the $\hat{t}$ vector.
• Figure 5: (Color online) Plot of the normalization factors $N(\theta)$ and $M(\theta)$ with $\theta$. $N(\theta)<0$ for $\theta\in(\cos^{-1}(1/3),\pi/2]$.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Corollary 1