An exponential separation between entanglement-assisted and unassisted one-way quantum communication

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The gist

An exponential separation between entanglement-assisted and unassisted one-way quantum communication demonstrates that shared entanglement can provide an exponential reduction in the cost of

In short

The paper proves an exponential separation between entanglement-assisted and unassisted one-way quantum communication for computing total Boolean functions. It shows that while shared entanglement significantly reduces classical communication costs to O(log n) bits, unassisted quantum protocols require a higher complexity of Omega(n^(1/3)) qubits. This resolves a key question about whether entanglement is strictly necessary for certain computational goals in one-way settings.

Key concepts

Entanglement-Assisted Communication
This model allows Alice and Bob to share pre-existing entangled quantum pairs (EPR pairs) before starting the computation. This shared resource helps reduce the amount of classical information they need to exchange, potentially lowering the total communication cost for solving a problem.
Unassisted One-Way Quantum Communication
This model involves Alice sending a quantum state to Bob, and Bob performing local operations. Crucially, there is no pre-shared entanglement between them. The complexity here measures the minimum number of qubits needed to solve the problem using only this one-way transmission.
Total Boolean Function
A total function is a function that must output a definite result for every possible input combination. In this context, it refers to any computational problem where Alice and Bob need to compute a specific output based on their inputs, rather than just checking membership in a set.
Shifted Equality Problem (ShiftEqG,r)
This is the specific mathematical problem used as an example to demonstrate the separation. It involves determining if a shifted product of two elements in a group G equals a target element h. This problem is constructed to exhibit the desired trade-off between entanglement-assisted and unassisted communication complexities.

Terminology used across episodes

This episode discusses

The paper

An exponential separation between entanglement-assisted and unassisted one-way quantum communication · Read on arXiv

Department of Computer Science, University of Texas at Austin · Hon Hai (Foxconn) Research Institute · School of Computer Science, and Institute for Quantum Computing, University of Waterloo

A longstanding question in quantum communication complexity is whether some task can be accomplished with a small amount of communication in the presence of entanglement, yet require much more quantum communication in the absence of entanglement. Separations of this nature were previously known for relational problems and, in the simultaneous message passing model, for partial functions. But it has remained unresolved whether any such separation exists for a total Boolean function. We resolve this question with an exponential separation in the one-way setting: we exhibit a family of total Boolean functions f n 0,1 n times 0,1 n to 0,1 that can be computed with O(n) bits of one-way classical communication given prior entanglement, but that require Ω(n 1/3) qubits of one-way quantum communication without entanglement. Our function is a special case of the subgroup membership problem, first studied in the communication setting by Aaronson, Le Gall, Russell, and Tani.

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "An exponential separation between entanglement-assisted and unassisted one-way quantum communication".

Mira: An exponential separation between entanglement-assisted and unassisted one-way quantum communication demonstrates that shared entanglement can provide an exponential reduction in the cost of classical communication for computing total Boolean functions.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, what we're looking at here is this paper titled "An exponential separation between entanglement-assisted and unassisted one-way quantum communication," Ryan Anselm, Srijita Kundu, Olivier Lalonde, and Ashwin Nayak. The main point they're making is that they found an exponential separation for total Boolean functions in the one-way setting. Basically, they show a family of functions where you need only O(log n) bits if you have entanglement beforehand for the classical communication part.

Mira: That’s what caught my eye; it resolves a long-standing question about whether such a separation exists for total functions specifically. The paper claims an exponential separation between an entanglement-assisted classical protocol and an unassisted one-way quantum protocol, where the former is bounded by O(log n) bits and the latter by Omega(n(one/three)) qubits <ref:2610.02099#pg0>.

Kai: It seems like they tackle a really deep problem in communication complexity by using a special case of the subgroup membership problem called MembG,k. That function takes Alice's input as a subgroup H of order at most k in a group G, and Bob gets an element g, and they have to decide if g is in H.

Mira: And they construct this specific instance called the "shifted equality problem," ShiftEqG,r, which is reducible to MembG,k. This reduction is what lets them use existing results for subgroup membership while building a function that shows the trade-off they're after.

Kai: The entanglement-assisted upper bound they found involves Alice preparing two copies of a mixed state on Bob's side, and Bob performing a Hadamard test on each copy with respect to the unitary Ug, which applies right multiplication by g. They show this leads to an entanglement-assisted classical communication cost of O(log k) bits.

Mira: And when they restrict the subgroup order such that k is polylog(G), that communication cost drops further to O(log log G), which sets them apart from the unassisted quantum setting that requires at most O(log G) qubits.

Kai: It's a pretty specific construction, and I'm curious about what kind of group structures they use for this separation to hold up as an asymptotic result.

Mira: That’s exactly what they explore further in the technical details, but the core claim is that this separation exists for total functions in the one-way setting. This finding really addresses that central question about whether entanglement helps significantly when you're only considering one-way quantum communication.

Conclusion: Kai: So, looking at the title "An exponential separation between entanglement-assisted and unassisted one-way quantum communication," it really summarizes the main contribution here: they proved that there are total functions where entanglement helps you save a lot on classical bits, but without entanglement, you still need a significant amount of quantum resources for one-way communication.

Mira: I think the implication is that we have to be careful about assuming entanglement can always provide an exponential reduction in cost for every single computation when we're limited to one-way communication. The paper shows that for total functions, this isn't strictly necessary to achieve certain computational goals without entanglement being present.

Kai: So, if you distill it down simply, the key finding is that the entanglement-assisted classical communication cost can be very small, O(log n) bits, while the unassisted quantum and randomized complexities are much larger at Omega(n(one/three)) <ref:2610.02099#pg0>. It establishes a clear asymptotic gap between these two models for a functional problem.

Mira: That gap is what matters because it confirms that entanglement isn't always the deciding factor when you're only talking about one-way quantum communication for solving total functions. It shows that while entanglement helps considerably in the presence of communication, it isn't an absolute necessity to reach certain computational targets in this specific setting.

Kai: It’s a result that shows a fine line between what’s possible with and without entanglement when you restrict ourselves to one-way quantum communication models for total functions. I wonder how this separation translates into practical limits for things we build today.

Mira: That's the big question, Kai; it pushes the boundary on how we think about communication resources in quantum computation and communication simultaneously. It really forces us to re-evaluate the assumptions we make when modeling these tasks in a one-way setting without entanglement <ref:2610.02099#pg2>.

Kai: Exactly; it means we need to keep pushing the boundaries on understanding these limits, even when the results are about theoretical functions rather than practical hardware. That separation is a piece of information that could inform future designs for quantum networks and communication protocols.

Mira: Indeed, this paper provides a concrete framework for studying these separations in total function settings, which is what was previously open to research <ref:2610.02099#pg2>. It’s a solid piece of work showing how specific constructions can yield these kinds of results.

Kai: So, to wrap up this discussion on "An exponential separation between entanglement-assisted and unassisted one-way quantum communication," the core message is that for total functions, the trade-off between classical bits with entanglement and quantum qubits without entanglement is quite stark asymptotically.

Mira: Precisely; it shows that for specific problems, the presence of shared EPR pairs provides a substantial advantage in terms of communication cost reduction over what you can achieve with just one-way quantum messages.

Kai: We'll keep an eye on how this theoretical separation guides our experimental efforts to build systems where these models interact in ways we can actually measure and test.

Mira: I agree; it’s a result that solidifies the understanding of resource trade-offs in this area of communication complexity.

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