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Entanglement cost in non-local quantum computation

This book-length review comprehensively examines non-local quantum computation (NLQC), detailing its methodology of using shared entanglement and a single communication round to interact distant systems, while analyzing entanglement cost bounds and exploring its applications across quantum cryptography, complexity theory, and quantum gravity.

Original authors: Alex May

Published 2026-09-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex May

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, information is not just a string of ones and zeros; it is a physical state that can be shared between distant people in a way that defies everyday logic. When two particles are "entangled," they share a deep connection where measuring one instantly influences the other, no matter how far apart they are. This phenomenon is not just a curiosity; it is a resource, much like electricity or fuel, that scientists can use to perform tasks that would otherwise be impossible. One of the most intriguing challenges in this field is figuring out how to perform a quantum calculation when the two parts of the system are separated by space and cannot touch. Normally, to make two quantum systems interact, you must bring them together. But what if you cannot? What if the laws of physics or the layout of a secure facility prevent them from ever meeting? This is the central puzzle of non-local quantum computation: how to make two separated systems act as if they are together, using only a single round of communication and a shared supply of entanglement.

The answer to this puzzle is not just a theoretical exercise. It touches on the security of future communication systems, the limits of how fast computers can solve problems, and even the nature of space and time itself. If we can understand exactly how much entanglement is needed to perform these distant interactions, we can build better codes to protect secrets, design more efficient quantum computers, and perhaps even understand how the universe stitches together the fabric of reality. The question is simple to state but incredibly difficult to answer: given a specific quantum interaction, how much shared entanglement is required to recreate it without the systems ever meeting?

A new book-length study by Alex Maya, a researcher at the Perimeter Institute for Theoretical Physics, takes a comprehensive look at this problem. The work does not just offer a single solution; it maps out the entire landscape of what is known, what is possible, and where the biggest mysteries remain. The author treats entanglement as a currency, asking how much of it must be spent to buy a specific quantum interaction. The findings reveal a complex picture where some tasks are cheap to perform, while others might require an astronomical amount of resources, though the exact cost for many of these expensive tasks remains a subject of intense debate.

The study begins by establishing that it is always possible to perform any quantum interaction this way, provided you have enough entanglement. The researchers describe a general method that works for any situation, but it comes with a steep price tag: the amount of entanglement needed grows exponentially with the size of the system. This means that for a large system, the resource cost becomes so huge that it is practically impossible to implement. However, the book shows that for many specific types of interactions, the cost is much lower. If the interaction is simple, or if it follows a specific structure, the entanglement required can be kept manageable. The author explores these efficient methods, showing how certain patterns in the calculation can be exploited to save resources.

One of the most fascinating discoveries in the work is the unexpected link between quantum entanglement and the complexity of classical computer programs. The book demonstrates that the difficulty of performing a non-local quantum task is often tied to how much memory a classical computer would need to solve a related problem. For example, if a task can be solved by a classical computer using a very small amount of memory, then the quantum version of that task can be performed with a relatively small amount of entanglement. This connection is surprising because it bridges two very different fields: the study of how quantum systems share information and the study of how classical computers process data. It suggests that the limits of quantum resources are deeply rooted in the fundamental structure of computation itself.

The research also delves into the practical side of these ideas, particularly in the realm of cryptography. There is a method called quantum position verification, which is designed to prove that a person is physically located at a specific spot. To attempt to bypass this system, a dishonest party would need to perform a non-local quantum computation to fake their location. The book explains that the security of these systems depends entirely on how much entanglement is required to pull off the attempt. If the cost is too high, the attempt is impossible, and the system is secure. The author shows that for many proposed schemes, the cost is indeed high enough to prevent circumvention, but for others, the cost might be low enough to be a vulnerability. This makes the study of entanglement costs a critical tool for designing secure communication networks.

Perhaps the most profound implication of the work lies in its connection to gravity and the structure of the universe. The book draws a parallel between these quantum tasks and the way gravity works in theories that describe our universe as a hologram. In these theories, the three-dimensional world we experience is a projection of information stored on a two-dimensional surface. The author argues that the way quantum systems interact across space without touching is exactly the mechanism that allows the holographic universe to function. The entanglement between different regions of space is what holds the geometry of the universe together. If the entanglement is too weak, the connection breaks, and the space itself might fall apart. This suggests that the rules of quantum information are not just abstract math but are the very building blocks of spacetime.

Despite these advances, the book makes it clear that we do not have the full picture yet. While we know how to perform these tasks efficiently for some specific cases, we still do not know the exact cost for the most difficult interactions. The author points out that proving a strict lower limit on the entanglement required for certain complex tasks remains one of the biggest open problems in the field. If we could prove that some tasks require a massive amount of entanglement, it would not only secure our cryptographic systems but also provide new insights into the limits of computation and the nature of the universe. Until then, the relationship between the cost of entanglement and the complexity of the task remains a rich and evolving story, waiting for the next breakthrough to reveal its final chapters.

Technical Summary: Entanglement Cost in Non-Local Quantum Computation

1. Problem Statement

The central problem addressed in this work is the quantification of entanglement cost in Non-Local Quantum Computation (NLQC). An NLQC is a protocol where two spatially separated parties, Alice and Bob, implement a quantum channel $\mathcal{N}_{AB \to A'B'}$ that acts jointly on their input systems without bringing the systems together. Instead, they utilize a shared entangled resource state and a single simultaneous round of quantum communication.

The core question is: Given a target quantum channel $\mathcal{N}$ , what is the minimal amount of shared entanglement (measured in EPR pairs or qubits) required to implement $\mathcal{N}$ as an NLQC? This problem is framed as a resource-theoretic inquiry, investigating the trade-off between entanglement and local interactions. The paper notes that while general upper bounds exist, they are often exponential, and known lower bounds are typically linear, leaving a significant gap in understanding the true cost for most channels.

2. Methodology

The paper employs a multi-faceted approach combining quantum information theory, classical complexity theory, and holographic principles:

Protocol Construction (Upper Bounds): The author develops specific protocols to implement arbitrary channels and specific classes of unitaries. Key techniques include:
- Port-Teleportation: A generalized teleportation scheme where a state is teleported to one of $N$ "ports" (EPR pairs), allowing the correction of Pauli errors to be deferred or handled via tracing out unused ports.
- Garden-Hose Protocols: A model where entanglement is used to route a quantum state based on classical inputs, analogous to water flowing through pipes. This is linked to the complexity class L (logspace).
- Code-Routing: Using quantum error-correcting codes (specifically secret sharing schemes) to split quantum information and route shares based on classical inputs, linking NLQC to formula size complexity.
- Pauli-Adapted Teleportation: A hybrid of Bell-basis and port-teleportation that exploits the stabilizer structure of unitaries with low T-count (number of non-Clifford T-gates).
Lower Bound Techniques: The author utilizes information-theoretic and cryptographic tools to establish limits on entanglement:
- Entropy and Mutual Information: Deriving entanglement lower bounds from entropy lower bounds and controllable correlation measures.
- Monogamy-of-Entanglement (MoE) Games: Reducing measurement tasks to MoE games (e.g., the BB84 game) to bound the success probability of protocols without entanglement, thereby establishing a necessity for correlation.
- Parallel Repetition: Analyzing the scaling of success probabilities when tasks are repeated in parallel to strengthen lower bounds.
Relational Approach: The paper introduces a framework of reductions between different NLQC tasks (e.g., $f$ -routing, $f$ -measure, $f$ -Bell). By showing that one task reduces to another (often with constant overhead), the author transfers upper and lower bounds across different classes of problems without re-proving them from scratch.
Holographic Connections: The work connects NLQC to the AdS/CFT correspondence and the holographic principle, specifically analyzing how bulk scattering regions (where interactions occur) must be reproduced by boundary entanglement in the absence of a boundary scattering region.

3. Key Contributions and Results

A. Upper Bounds on Entanglement Cost

General Channels: Using port-teleportation, the paper confirms that any channel on $n$ qubits can be implemented as an NLQC with an entanglement cost of $O(2^n)$ (specifically $E \approx 2n \cdot 2^{2n}$ for constant diamond norm error). This establishes that exponential entanglement is sufficient for universality.
T-Depth and T-Count: For unitaries with low complexity in terms of non-Clifford gates:
- T-Depth: The entanglement cost is bounded by $O((Kn)^d)$ , where $d$ is the T-depth and $K$ is a constant. This links NLQC cost to the depth of the circuit.
- T-Count: Using Pauli-adapted teleportation, the cost for a unitary with $t$ T-gates is shown to be roughly $O(2^t)$ , improving upon generic bounds for circuits with few non-Clifford gates.
f-Routing: For the specific task of routing a quantum state based on a Boolean function $f(x,y)$ $f (x, y)$ :
- Garden-Hose: Efficient protocols exist for functions in the complexity class L (logspace), with cost scaling polynomially in the garden-hose complexity.
- Sub-Exponential Bounds: Through connections to Conditional Disclosure of Secrets (CDS), the paper highlights that all functions can be $f$ -routed with sub-exponential entanglement ( $2^{O(\sqrt{n} \log n)}$ ), a result that surpasses simple garden-hose or formula-size bounds.

B. Lower Bounds

Linear Bounds: The paper reviews and refines linear lower bounds for specific tasks. For instance, routing an $n$ -qubit system requires at least $\Omega(n)$ entanglement.
Measurement Tasks: Using MoE games, it is shown that implementing the "measuring task" (determining a bit in a basis chosen by a distant party) requires entanglement; without it, the success probability is bounded by $\cos^2(\pi/8) \approx 0.85$ .
Parallel Repetition: The success probability of parallel repeated measurement tasks decays exponentially, implying that the entanglement cost must grow linearly with the number of repetitions.

C. Connections to Other Fields

Quantum Cryptography (QPV): NLQC is identified as the mechanism for bypassing security in Quantum Position Verification (QPV). The difficulty of implementing an NLQC (entanglement cost) directly correlates to the security of QPV schemes.
Classical Cryptography (CDS & PSM): A major contribution is the rigorous equivalence between $f$ -routing and Quantum Conditional Disclosure of Secrets (CDQS). This implies that upper bounds on classical CDS randomness cost translate to entanglement cost in NLQC, and vice versa.
Quantum Gravity: The paper applies NLQC to the Connected Wedge Theorem. It argues that if a bulk scattering region exists but the boundary scattering region is empty, the boundary CFT must possess sufficient entanglement (mutual information $\Omega(1/G_N)$ ) to support the interaction via NLQC. This provides a quantum information-theoretic derivation of geometric constraints in AdS/CFT.

4. Significance and Claims

The paper positions NLQC as a unifying framework that bridges quantum information, complexity theory, and quantum gravity. Its significance lies in:

Resource Theory: It provides a concrete setting to study the interplay between entanglement and local operations, revealing that complexity measures (like T-depth or formula size) naturally emerge as upper bounds on entanglement cost.
Cryptography: It establishes that the security of QPV schemes relies on the hardness of NLQC. The equivalence between $f$ -routing and CDS suggests that breakthroughs in classical cryptographic lower bounds could immediately yield quantum entanglement lower bounds, and vice versa.
Holography: It offers a novel perspective on the emergence of spacetime geometry from entanglement. The argument that bulk interactions are simulated by boundary NLQC provides a mechanism for how $d$ -dimensional physics can be encoded in $d-1$ dimensions without local interaction in the boundary.
Open Problems: The paper modestly concludes that while exponential upper bounds are known, super-linear lower bounds (e.g., polynomial or exponential) for general channels remain unproven. Proving such bounds is highlighted as a critical open problem, as it would simultaneously improve QPV security, establish new complexity lower bounds, and constrain computations in quantum gravity.

The work does not claim to have solved the central problem of determining the exact entanglement cost for all channels but rather maps the landscape of known bounds, establishes deep connections between disparate fields, and identifies the specific gaps (particularly the lack of strong lower bounds) that define the current frontier of research.