Lower bounds on non-local computation from controllable correlation

TL;DR

This work addresses the entanglement cost of non-local quantum computation by introducing two general lower-bound techniques: controllable correlation and controllable entanglement. Each method yields a bound on the entanglement of formation required for NLQC of a given unitary, with explicit forms such as $E_f(L:R)_\Psi \ge \frac{\lambda_1-\lambda_2}{2}-\Delta(\epsilon,n_A)$ for controllable correlation and $E_f(L:R)_\Psi \ge \lambda_1-2\lambda_2^{1/4}-2\gamma^{1/8}$ for controllable entanglement (small approximation parameters). The techniques apply broadly, with non-trivial bounds for Haar-random two-qubit unitaries and many common two-qubit gates; notably, the CNOT gate attains a tight bound via the controllable entanglement approach, and parallel repetition holds in several regimes, including noisy settings. The paper also provides numerical methods and results for evaluating these bounds on concrete gates, discusses the dimension and entropy implications, and outlines open questions such as extension to quantum channels beyond unitaries. Overall, the work significantly advances understanding of NLQC entanglement costs and offers practical tools for certifying lower bounds across a wide range of gates.

Abstract

Understanding entanglement cost in non-local quantum computation (NLQC) is relevant to complexity, cryptography, gravity, and other areas. This entanglement cost is largely uncharacterized; previous lower bound techniques apply to narrowly defined cases, and proving lower bounds on most simple unitaries has remained open. Here, we give two new lower bound techniques that can be evaluated for any unitary, based on their controllable correlation and controllable entanglement. For Haar random two qubit unitaries, our techniques typically lead to non-trivial lower bounds. Further, we obtain lower bounds on most of the commonly studied two qubit quantum gates, including CNOT, DCNOT, $\sqrt{\text{SWAP}}$, and the XX interaction, none of which previously had known lower bounds. For the CNOT gate, one of our techniques gives a tight lower bound, fully resolving its entanglement cost. The resulting lower bounds have parallel repetition properties, and apply in the noisy setting.

Figures (4)

• Figure 1: Local and non-local computations. a) A unitary $U_{AB}$ is implemented by directly interacting the input systems. b) A non-local quantum computation. The goal is for the action of this circuit on the $AB$ systems to approximate $U_{AB}$.
• Figure 2: Results of a numerical optimization computing the controllable entanglement and controllable correlation for some simple gates; these values are lower bounds on the entanglement of formation in any resource state that suffices to complete the corresponding gate as an NLQC. The reference state is the choice of state on $QA$ used in deriving the lower bound; see figure \ref{['fig:NLQC_correlation']}. $\rho_{cc}$ refers to the maximally classically correlated pair of qubits, while $\Psi^+$ is a Bell state. Matrix expressions for the listed gates are in appendix \ref{['sec:gates']}.
• Figure 3: A non-local quantum computation implementing a unitary $U_{AB}$. To prove lower bounds on the entanglement cost, we consider placing the input system $A$ in a state $P_{QA}$ correlated with a reference system $Q$. We indicate this with the dashed line. The state $P_{QA}$ need not be pure. We find that if adjusting the input on $B$ changes the amount of correlation between $A$ and $Q$ in the final state, that $L:R$ must be entangled.
• Figure 4: Histogram showing the value of the controllable correlation lower bound computed for $100,000$ two qubit unitaries drawn from the Haar distribution. The average value of the lower bound is $\approx 0.230$.

Theorems & Definitions (21)

• Definition 1
• Definition 2
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• Definition 6
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• Definition 10