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arXiv:2602.00255

Lower bounds on non-local computation from controllable correlation

Richard Cleve, Alex May

quant-phSubmitted Jan 30, 2026• Announced Feb 3, 2026• 0 citations• Revised Apr 29, 2026 (v4)
v1 · Jan 30, 2026v2 · Feb 5, 2026v3 · Apr 14, 2026v4 · Apr 29, 2026
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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.

Comments
v4 includes a number of further minor corrections over v3

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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, SWAP\sqrt{\text{SWAP}}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.
This is a book-length treatment of the subject of non-local quantum computation (NLQC). NLQC is a method for implementing quantum operations that interact two systems without directly bringing the systems together. Instead, a single round of communication and shared entanglement is used. NLQC has appeared in the context of quantum cryptography, computational complexity, communication complexity, quantum gravity, and other applications. The understanding of entanglement cost in NLQC is closely tied to questions in all of these areas. We review upper and lower bounds on entanglement cost, as well as some of the applications of NLQC and its connections to other subjects.
Non-local quantum computation (NLQC) studies how two collaborating players can implement channels on distributed systems using a single simultaneous round of quantum communication and shared entanglement. NLQC has applications in diverse areas, ranging from quantum position-verification to quantum gravity. Recently, it has been realized that the relationships among families of NLQC tasks are highly structured: many seemingly distinct tasks are related by reductions, wherein implementations of one task can be used to efficiently implement a second task. This is analogous to the notion of reduction in complexity theory, and reveals the relative hardness of NLQC tasks. In this work we continue the study of reductions among NLQC tasks. We focus on NLQC examples of the greatest interest in quantum position-verification; in particular examples involving large classical inputs and fixed-size quantum inputs, since these constitute the most feasible protocols for position-verification schemes. Within this setting, we find many new relationships among NLQC tasks. For instance, protocols for the simplest example of redirecting a quantum system based on a classical control imply protocols for controlled single qubit measurements in arbitrary bases, the controlled application of any Clifford unitary, and even the controlled application of any unitary of the form U=C1DC0U=C_1DC_0U=C1​DC0​ with DDD an arbitrary diagonal unitary and C0,C1C_0, C_1C0​,C1​ Clifford circuits. This implies that many feasible position-verification schemes have the same asymptotic scaling for their entanglement cost, and hence a similar level of security. Our techniques rely on ideas from gate teleportation and measurement based quantum computation, among other areas, bringing several new strategies into NLQC which may be of independent interest.
A non-local quantum computation (NLQC) replaces an interaction between two quantum systems with a single simultaneous round of communication and shared entanglement. We study two classes of NLQC, fff-routing and fff-BB84, which are of relevance to classical information theoretic cryptography and quantum position-verification. We give the first non-trivial lower bounds on entanglement in both settings, but are restricted to lower bounding protocols with perfect correctness. Within this setting, we give a lower bound on the Schmidt rank of any entangled state that completes these tasks for a given function f(x,y)f(x,y)f(x,y) in terms of the rank of a matrix g(x,y)g(x,y)g(x,y) whose entries are zero when f(x,y)=0f(x,y)=0f(x,y)=0, and strictly positive otherwise. This also leads to a lower bound on the Schmidt rank in terms of the non-deterministic quantum communication complexity of f(x,y)f(x,y)f(x,y). Because of a relationship between fff-routing and the conditional disclosure of secrets (CDS) primitive studied in information theoretic cryptography, we obtain a new technique for lower bounding the randomness complexity of CDS.
Non-local quantum computation (NLQC) consists on the implementation of a bipartite unitary UABU_{AB}UAB​ by two cooperating distant players by means of a two-round protocol with a single intermediate round of simultaneous communication. It is a major open question to know whether there exists a unitary UABU_{AB}UAB​ whose implementation in NLQC requires extra quantum systems with dimensions exp(nΩ(1)){\rm exp}(n^{Ω(1)})exp(nΩ(1)), where nnn is the dimension of systems AAA and BBB. This type of exponential lower bound is essential for security of position-based quantum cryptography, since the capabilities of the adversaries are precisely described by NLQC. It has also been connected to many other problems, such as the optimality of universal quantum simulators, or the expected properties of holographic quantum gravity. Currently the best lower bounds are only sublinear in nnn. In this paper we consider the family of diagonal ±1{\pm 1}±1 valued unitaries UεU_{\varepsilon}Uε​, indexed by an element of the Boolean hypercube ε∈{±1}n2\varepsilon \in \{\pm 1\}^{n^2}ε∈{±1}n2, i.e. Uε∣i⟩A∣j⟩B=εij∣i⟩A∣j⟩BU_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_BUε​∣i⟩A​∣j⟩B​=εij​∣i⟩A​∣j⟩B​. We show that if UεU_{\varepsilon}Uε​ can be implemented in NLQC with constant accuracy for all ε\varepsilonε, and the dependency on ε\varepsilonε in the first round of the associated strategy is a polynomial of degree O(n14−δ)O(n^{\frac{1}{4}-δ})O(n41​−δ), then the implementation of at least one UεU_{\varepsilon}Uε​ in NLQC requires resources scaling as exp⁡(Ω(n2δ(log⁡n)2))\exp\left(Ω\left(\frac{n^{2δ}}{(\log n)^2}\right)\right)exp(Ω((logn)2n2δ​)). The proof is based on geometric properties of particular Banach spaces, namely their type constants, together with random estimates of Boolean functions valued on them.
A non-local quantum computation (NLQC) replaces an interaction between two quantum systems with a single simultaneous round of communication and shared entanglement. We study two classes of NLQC, f-routing and f-BB84, which are of relevance to classical information theoretic cryptography and quantum position-verification. We give the first non-trivial lower bounds on entanglement in both settings, but are restricted to lower bounding protocols with perfect correctness. Within this setting, we give a lower bound on the Schmidt rank of any entangled state that completes these tasks for a given function f(x,y) in terms of the rank of a matrix g(x,y) whose entries are zero when f(x,y) = 0, and strictly positive otherwise. This also leads to a lower bound on the Schmidt rank in terms of the non-deterministic quantum communication complexity of f(x,y). Because of a relationship between f-routing and the conditional disclosure of secrets (CDS) primitive studied in information theoretic cryptography, we obtain a new technique for lower bounding the randomness complexity of CDS.
New lower bounds for CDS and fff-routing
Understanding the entanglement cost of non-local quantum computation (NLQC) is relevant to complexity theory, cryptography, quantum gravity, and other areas. A central special case is fff-routing, motivated in part by quantum position verification. Proving lower bounds on its entanglement cost in the fully robust setting has been a major open problem in NLQC. Motivated by this problem, we establish two related lower bounds. First, we study the shared-randomness cost of robust conditional disclosure of secrets (CDS). The connection between CDS and fff-routing established by Allerstorfer et al. (Quantum 2024) makes understanding the randomness complexity of robust CDS a natural step toward lower bounds for the fully robust routing problem. We show that the shared-randomness cost of robust CDS is lower bounded by the logarithm of deterministic SMP communication complexity, even when communication and private randomness are unrestricted. Our lower bound is tight for the equality function. Second, we consider one-sided-perfect fff-routing, in which the protocol is exact on one input class and has constant error on the other. By exploiting the positivity of the low-rank matrix in the method of Asadi, Culf, and May (ITCS 2025), we derive a lower bound on the entanglement cost in terms of sign rank. In particular, this yields a linear lower bound on the entanglement cost of routing for the inner-product function in both one-sided-perfect settings, matching the known upper bound.