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Quantum Merlin-Arthur with an internally separable proof

Roozbeh Bassirian, Bill Fefferman, Itai Leigh, Kunal Marwaha, Pei Wu

Abstract

We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP $\ne$ NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.

Quantum Merlin-Arthur with an internally separable proof

Abstract

We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.

Paper Structure

This paper contains 32 sections, 28 theorems, 69 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Other implications
  4. A complexity-theoretic framework to study multipartite entanglement.
  5. A unifying perspective: $\QMA(2)$ and purity testing.
  6. Techniques
  7. Related work
  8. The complexity class $\QMA(2)$
  9. Convex optimization and quantum entanglement
  10. Measuring quantum entanglement
  11. Outline of the paper
  12. Our setup
  13. $\QMA$ with a restricted set of proofs.
  14. Convex optimization.
  15. Optimization over internally separable states: $\QMA_{\mathsf{IS}} \subseteq \EXP$
  16. ...and 17 more sections

Key Result

Theorem 1.1

For any polynomial $p$ and $1 > c > s > 0$ such that $c - s > \frac{1}{2^{p(n)}}$, we have $\QMA_{\mathsf{IS}}^{c,s} \subseteq \EXP$, where $c,s$ are the completeness and soundness parameters, respectively.

Figures (3)

  • Figure 1: Cartoons of a separable state and a state with a separable subsystem. The left side depicts a separable state, where parts $A$ and $B$ share no entanglement. The right side depicts a state where after tracing out part $A$, parts $B$ and $C$ share no entanglement; this state is in general entangled across every bipartition. The lower cartoons each display a graph where every vertex is a part ($A$, $B$, or $C$), and an edge between two parts $X,Y$ allows bipartite entanglement in the reduced state $\rho_{XY}$. On the right side, part A is entangled with both part B and part C.
  • Figure 2: An illustration of the reduction in \ref{['claim:general-hard-problem']}.
  • Figure 3: The verification circuit recast as an instance of $(\alpha,\beta)$$\mathcal{W}$Isometry Output as in \ref{['claim:general-hard-problemNoPad']}. The poisoning circuit $P_n$ is controlled by register $S$, and applied whenever the qubits in $S$ are not all $0$'s.

Theorems & Definitions (93)

  • Theorem 1.1: Upper bound
  • Theorem 1.2: Lower bound
  • Corollary 1.3
  • Claim 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.8
  • Lemma 1.9: informal
  • Definition 2.1: $\QMA_{\mathcal{W}}(m)$
  • Remark 2.2
  • ...and 83 more