Q12: Which subsystem-spin profiles can we reach?

Working mathematical formulation

Given $N$ labeled qubits, a polynomial-size list $F$ of constant-size subsets, and rational distributions $q_S(j)$ , ask whether a density matrix $\rho$ exists with $\max_{S\in F}\left\lVert p_\rho(J_S)-q_S\right\rVert_{\mathrm{TV}}\le\varepsilon.$ Separately ask for an adaptive STP preparation from $I/2^N$ whose output meets this condition after a specified set of measurement outcomes, with total probability at least $p$ and expected cost at most $T$ . Keep $\varepsilon$ , $p$ , and $T$ explicit; use promise gaps when asking for computational complexity.

Scope & assumptions

These are marginal probability distributions for separate measurement choices, not simultaneously assigned values of noncommuting observables. Compatibility, STP reachability, and efficient preparation are three distinct questions. A profile usually does not identify a full density matrix.

Why this is interesting

This separates a possible pattern of correlations from an efficiently reachable one. It also turns the informal idea of moving singletness into a concrete state-space question.

Explicit variants

Rates of creation, transport, and erasure
Start from $I/2^N$ and specify a target collection of singlet probabilities or retained singlet pairs. Bound the number and depth of adaptive measurements needed to create that structure, transfer it between designated pairs, or remove it from retained systems. Keep the target trace-distance error, success probability, supplied ancillas, and discarded systems explicit; compare protocols under the same geometry.

A place to start

Solve the compatibility and reachability problems for four qubits and a few overlapping pairs, including mixed target states.

Related work

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

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