No Absolute Hierarchy of Quantum Complementarity

Abstract

Bohr's principle of complementarity, prohibiting simultaneous access to certain physical properties within a single experimental arrangement, is considered to be a defining feature of quantum mechanics. It is commonly viewed as inducing an intrinsic hierarchy among incompatible observables: some sets of quantum properties are fundamentally more incompatible than others, as quantified by the maximal sharpness permitting their joint measurement. We show that this hierarchy ceases to be absolute in the multi-copy regime. Analyzing qubit spin observables, we prove a No-Comparison Theorem establishing that no global ordering of incompatible observable sets is preserved across all finite-copy configurations. In particular, two sets of observables can exhibit reversed complementarity ordering depending solely on whether the available resources are arranged as identical copies or as parallel-antiparallel pairs. Thus, the degree of quantum incompatibility is not an intrinsic property of observables alone but depends on the global configuration of the prepared quantum probes. Our results uncover a configuration-dependent structure of complementarity, reveal a subtle role of entanglement in shaping the structure of measurement limitations, and call for a reassessment of quantum information protocols under finite resources.

Figures (4)

• Figure 1: Multi-copy joint measurability: A device $\mathcal{G}^{[k]}$ accesses $k$-copy of identical qubit states in each experimental run, and yields the outcome statistics $\{p({\bf a})\}_{{\bf a}}$, with $\mathbf{a}:=a_1\cdots a_N \in \{+1,-1\}^N$. The device jointly measures a set $\mathcal{O}$ of $N$ distinct observables up to an optimal sharpness value $\lambda~(=\lambda^{[k]}_{\mathcal{O}})$ if the outcome statistics $\{p({\bf a})\}_{\bf a}$ can be processed to reproduce measurement statistics of each observable in $\mathcal{O}$ with that effective sharpness. The device $\mathcal{G}^{[k_1|k_2]}$, on the other hand, accesses $k_1$ copies of a qubit state along with $k_2$ copies of its spin-flipped version and ensures joint measurability up to a sharpness value $\lambda~(=\lambda^{[k_1|k_2]}_{\mathcal{O}})$.
• Figure 2: Configuration-Dependent Complementarity: The triangular set $\mathtt{SyTri}$ is perfectly joint measurable on three identical qubit, while the tetrahedral set $\mathtt{SyTet}$ is not. Conversely, on anti-parallel configuration $\mathtt{SyTet}$ is perfectly jointly measurable but $\mathtt{SyTri}$ is not. For the $\mathtt{MUB}$ observables $\lambda^{_{[2]}}_{\mathtt{MUB}}=\sqrt{3}/2$Carmeli2016, whereas $\lambda^{_{[1|1]}}_{\mathtt{MUB}}=1$Patra2026. On the other hand, for the observable set $\mathtt{SyTri}$ we have $\lambda^{_{[2]}}_{\mathtt{SyTri}}=\lambda^{_{[1|1]}}_{\mathtt{SyTri}}=2\sqrt{2}/3$, thereby supporting the thesis of No-Comparison theorem.
• Figure 3: Three symmetrically arranged spin observables: Symmetric configuration of three spin observables $\mathcal{O}_\theta \equiv \{\sigma_{\hat{n}_i}\}_{i=1}^3$. Each observable's direction $\hat{n}_i$ makes an angle $\theta \in (0, \pi)$ with the reference axis $\hat{t} = (1,1,1)^{\text{T}}/\sqrt{3}$. The angle between any two distinct observables is $\varphi = \cos^{-1}\left((3\cos^2\theta - 1)/2\right)$.
• Figure 4: For any triple of spin observables $\mathcal{O}_{\tilde{\theta}}$ with $\tilde{\theta}\in \tilde{\mathcal{R}}$ and any triple of spin observables $\mathcal{O}_\theta$ with $\theta \in \mathcal{R}$, we observe that $\lambda^{_{[2]}}_{\mathcal{O}_{\tilde{\theta}}} > \lambda^{_{[2]}}_{\mathcal{O}_\theta}$ while $\lambda^{_{[1|1]}}_{\mathcal{O}_{\tilde{\theta}}} < \lambda^{_{[1|1]}}_{\mathcal{O}_\theta}$, thereby re-establishing the core thesis of the No-Comparison theorem. As the region $\tilde{\mathcal{R}}$ decreases, the region $\mathcal{R}$ increases, a fact evident from the two sub-figures. The red and blue curves coincide at $\theta = 0$, $\pi/2$, and $\pi$. At $\theta = 0$ and $\theta = \pi$, the set $\mathcal{O}_\theta$ contains only a single measurement. The case $\theta = \pi/2$ is particularly interesting, as the three distinct observables lie in the great plane orthogonal to the $\hat{t}:=\tfrac{1}{\sqrt{3}}(1,1,1)$ axis. The relationship between the sharpness thresholds for parallel and anti-parallel configurations, in this case, is consistent with the claim of Proposition \ref{['prop2']}

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Proposition 1
• Proposition 2
• Definition 4
• Lemma 1
• proof
• proof