One ebit is necessary and sufficient for the elegant joint measurement
An exact LOCC lower bound for arbitrary finite-dimensional pure resources, closing the higher-Schmidt-rank gap for the tetrahedral two-qubit measurement.
Yuxuan Zhang · 9 November 2026 · Version 1.0
Result. The exact deterministic LOCC entanglement cost of the two-qubit elegant joint measurement is one ebit, even when the supplied pure resource may have arbitrary finite Schmidt rank. The theorem covers the original QIQC Open Problems catalogue question. This release contains a complete proof candidate with internal review; independent expert review is still outstanding.
Full paper (PDF) · Source and exact checks (ZIP) · Version and SHA-256 metadata
The question
Alice and Bob each hold one qubit. They want to perform the tetrahedral elegant joint measurement using local operations, classical communication, and a shared entangled state. The required output is a classical label known to both parties; no quantum postmeasurement state is prescribed.
One shared Bell pair suffices: Alice teleports her qubit to Bob, who measures both qubits and communicates the outcome. The question is whether a more complicated pure resource, with more Schmidt coefficients but less than one bit of entanglement entropy, could do the same job exactly.
The theorem rules this out. If \(\lambda\) is the vector of Schmidt probabilities of any admissible finite-dimensional pure resource, then
\[H(\lambda)\geq 1.\]Equality requires the nonzero probabilities to be \((1/2,1/2)\). All supplied shared entanglement is included in the cost. The lower bound holds for every finite resource rank before taking the infimum over ranks.
What closes the higher-rank gap
The proof combines two constraints. A coherent detector gives \(\lambda_1\leq 2/3\). For the second constraint, write
\[p=\lambda_1,\qquad q=\lambda_2,\qquad \delta=\sum_{a\geq3}\lambda_a.\]Every exact separable implementation of the measurement satisfies
\[(p-q)^2\leq 4\delta(1-\delta).\]This inequality follows from a two-dimensional adjugate identity and completeness of the entire measurement. It controls the imbalance of the two leading probabilities through the total tail, with no factor depending on the resource dimension. Combining it with the detector constraint yields the entropy lower bound. The full paper supplies the analytic argument, including zero-contraction branches and arbitrary finite resource rank.
Relation to prior work
Akibue, Miyazaki and Osaka prove a matched-rank necessity theorem that settles rank-two resources for this measurement. Their Section 5.3 leaves higher-rank resources open. Bandyopadhyay’s 3 November 2026 paper gives a simpler proof of that matched-rank result and retains the higher-rank EJM question in Section 5.2. The contraction and effect-eigenvector reductions used here are inherited from that line of work.
The new step proposed here is the quantitative adjugate/completeness tail bound and its combination with the detector and entropy arguments. The detector framework itself comes from Bandyopadhyay, Brassard, Kimmel and Wootters. The source package includes a theorem-level comparison. Targeted primary-source checks support the difference from the inspected papers; they do not amount to an exhaustive priority survey.
The result is specific to the EJM. Other two-qubit bases can admit higher-rank resources with less than one ebit. Measurement localization without communication during the local measurement stage also uses a different operational model from the LOCC task considered here.
Evidence and review status
The package contains 37 exact detector checks, 90 exact algebraic and physical-protocol checks, and a second implementation of the central polynomial identity in 15 formal variables. A rank-three positive control implements Bell-pair concentration followed by teleportation. Negative controls detect incorrect algebra and an incomplete measurement. The release runner verifies file hashes and reruns the programs in a fresh temporary directory against their complete expected outputs.
These computations support ingredients of the written proof. They are not a numerical extrapolation to arbitrary rank or a formal proof of the theorem. Derivation, source comparison, computation and drafting used AI assistance under the author’s direction. Review to date is sequential and internal, with no independent expert or proof-assistant verification.
The theorem concerns exact single-implementation cost with pure resources. It makes no claim about mixed-resource cost, approximation, asymptotic net consumption, or communication-free localization. This website release is separate from journal review, arXiv submission and catalogue acceptance.