Q16: How are product tests expressed in subsystem spin?

Working mathematical formulation

Let Merlin send two unentangled $m$ -qubit registers with joint state $\rho\otimes\sigma$ ; neither register is assumed to be a product over its $m$ qubits. The product-test verifier measures singlet/triplet on every corresponding qubit pair across the registers and accepts if every outcome is triplet. Its accepting projector is $\Pi=\bigotimes_{i=1}^m\frac{I+\operatorname{SWAP}_i}{2}.$ Analyze its completeness and soundness as a subroutine of a stated two-witness verification protocol, retaining the unentanglement promise.

Scope & assumptions

For honest identical pure copies, acceptance is $2^{-m}\sum_{S\subseteq[m]}\operatorname{Tr}(\rho_S^2)$ . This identity is not by itself a full $\mathsf{QMA}(2)$ soundness proof: dishonest registers can be different, though unentangled with each other. Larger logical blocks require block-SWAP instruments rather than separate qubit projections.

Why this is interesting

The unentanglement promise in $\mathsf{QMA}(2)$ is a real resource assumption. The spin description should clarify that promise rather than suggest a SWAP test can impose it on arbitrary witnesses.

Explicit variants

Spin-based verifier classes
Give a uniform verifier classical input $x$ , one arbitrary polynomial-size quantum witness (or two unentangled witnesses), maximally mixed ancillary qubits, polynomially many adaptive STP measurements, and polynomial-time classical control. Acceptance is a classical bit with a $2/3$ versus $1/3$ gap. Characterize the resulting classes and the cost of simulations of standard $\mathsf{QMA}$ or $\mathsf{QMA}(2)$ verifiers. Any logical encoding must retain soundness against physical witnesses outside its intended code space.
Low-spin cross-copy conditions
Design a specified invariant test on two unentangled registers that uses low-spin outcomes, and prove its exact acceptance formula and a soundness bound for the intended promise problem. The standard product test accepts symmetric outcomes, so a proposed low-spin mechanism needs its own argument.
Short unentangled proofs and de Finetti
For a chosen $\mathsf{NP}$ promise problem, can an STP-restricted verifier use $O(\sqrt{n}\,\operatorname{polylog}n)$ -qubit unentangled witnesses with an explicit completeness–soundness gap? State the number of witnesses, readout and controller restrictions. Any de Finetti step needs exchangeability across copies; collective $\mathrm{SU}(2)$ invariance alone is insufficient.
Pairwise versus higher-arity witness verification
Define the spin-only verifier class with at most $k$ -qubit total-spin instruments, two unentangled witnesses, mixed ancillas, and a uniform polynomial-time classical controller. Compare fixed $k=2$ , larger fixed $k$ , and efficiently described growing arity. Determine under which physical-witness and encoding conventions the class equals $\mathsf{QMA}(2)$ .

A place to start

Work through the Harrow–Montanaro test on two copies of a bipartite pure state and relate its rejection probability to local purities.

Related work

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

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Why this is interesting

Explicit variants

Spin-based verifier classes

Low-spin cross-copy conditions

Short unentangled proofs and de Finetti

Pairwise versus higher-arity witness verification

A place to start

Related work

Related questions

Discuss this question