Q24: What changes with higher-arity total-spin measurements?

Working mathematical formulation

For a fixed $k\ge 3$ , compare pairwise STP with the instrument $M_S(\rho)=\sum_j \lvert j\rangle\langle j\rvert\otimes\Pi_{S,j}\rho\Pi_{S,j}$ on any $k$ selected qubits $S$ . Can the complete instrument, including its retained state and correlations with a reference, be approximated within diamond-norm error $\varepsilon$ by pairwise STP with polynomial resources? State ancillary input and failure probabilities. Treat growing $k$ as a separate regime.

Scope & assumptions

Measuring an entire coupling tree refines the eigenspaces of total spin. It is not automatically the same quantum instrument.

Why this is interesting

Larger measurements may reduce overhead or expose a cleaner primitive even when they do not change unrestricted computational power.

Explicit variants

Power of k-STP
Define $k$ -STP using maximally mixed inputs, uniform polynomial-time classical control, and projective total-spin measurements on subsets of size at most $k$ . Each measurement preserves coherence within the reported total-spin sector. For fixed $k\ge2$ , the unrestricted bounded-error decision class is already $\mathsf{BQP}$ . Determine what changes in efficient sampling protocols and physical overhead, and under depth, geometry, or adaptivity restrictions. Treat growing $k$ and its implementation cost separately.
Depth and adaptivity with three or four qubits
Do three- or four-qubit total-spin measurements reduce the measurement depth or adaptive rounds needed for universality or for a specified task? Compare with pairwise STP at the same error and ancillary budget. Count disjoint-subset measurement layers separately from batches whose schedules are fixed in advance.

A place to start

Construct the three- and four-qubit instruments, then determine which can be simulated efficiently by pairwise STP without unintended refinement of their eigenspaces.

Related work

These works provide context and established results. They need not be the origin of this exact formulation.

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