This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional (problem c̅E̅q̅T̅, whose communication complexity is O (log n)2) in the quantum version of the SMP and Ω̃ (√n) in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication-SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function c̅E̅q̅T̅ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional (problem c̅E̅q̅T̅, whose communication complexity is O (log n)2) in the quantum version of the SMP and Ω̃ (√n) in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication-SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function c̅E̅q̅T̅ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.

arXiv:1705.07211IEEE Transactions on Information TheoryMay 22, 20196 citations

The Communication Complexity of Private Simultaneous Messages, Revisited

Benny Applebaum, Thomas Holenstein, Manoj Mishra +1 more

Journal of CryptologySep 17, 201914 citations

Bounded-Error Quantum State Identification and Exponential Separations in Communication Complexity

Dmitry Gavinsky, Julia Kempe, Oded Regev +1 more

We consider the following problem of bounded-error quantum state identification: Given either state $\alpha_0$ or state $\alpha_1$ , we are required to output “0”, “1”, or “?” (“don't know"), such that conditioned on outputting “0” or “1”, our guess is correct with high probability. The goal is to maximize the probability of not outputting “?”. We prove the following direct product theorem: If we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is $O(ab)$ . Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with $O(\log n)$ classical bits of communication in the presence of shared randomness, but needs $\Omega(n^{1/3})$ communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef, Jayram, and Kerenidis, this shows that the quantum simultaneous message passing (SMP) model is incomparable with the classical shared-randomness SMP model. Second, we describe a relation that can be computed with $O(\log n)$ classical bits of communication in the presence of shared entanglement, but needs $\Omega((n/\log n)^{1/3})$ communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys much more than quantum communication.We consider the following problem of bounded-error quantum state identification: Given either state $\alpha_0$ or state $\alpha_1$, we are required to output “0”, “1”, or “?” (“don't know"), such that conditioned on outputting “0” or “1”, our guess is correct with high probability. The goal is to maximize the probability of not outputting “?”. We prove the following direct product theorem: If we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is $O(ab)$. Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with $O(\log n)$ classical bits of communication in the presence of shared randomness, but needs $\Omega(n^{1/3})$ communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef, Jayram, and Kerenidis, this shows that the quantum simultaneous message passing (SMP) model is incomparable with the classical shared-randomness SMP model. Second, we describe a relation that can be computed with $O(\log n)$ classical bits of communication in the presence of shared entanglement, but needs $\Omega((n/\log n)^{1/3})$ communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys much more than quantum communication.

SIAM Journal on ComputingJan 1, 200914 citations

The Communication Complexity of Private Simultaneous Messages, Revisited

Benny Applebaum, Thomas Holenstein, Manoj Mishra +1 more

Lecture notes in computer scienceJan 1, 201821 citations

Quantum Encodings, Private Messages, and Communication Complexity

Tom Gur, Miryam Mi-Ying Huang, Er-Cheng Tang

We study the complexity of quantum decomposable randomized encodings (QDRE). We establish a bi-directional connection between the QDRE model and the communication complexity model of quantum private simultaneous message protocols (QPSM), where classical communication is free, and quantum communication is the primary complexity measure. We show the following upper and lower bounds. (1) For every quantum channel mapping $n$ to $m$ qubits, we give a QPSM protocol with quantum communication $m$ , using exponential pre-shared entanglement. In particular, constant-output channels have $O(1)$ quantum communication, and classical-output channels require no quantum communication. (2) With $O(n)$ pre-shared entanglement, every one-qubit-output channel has an $O(n)$ -quantum-communication QPSM protocol. Conversely, there exists a channel and a constant $c>0$ for which any protocol with at most $cn$ pre-shared entanglement requires $Ω(n)$ quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with $O(n)$ pre-shared entangled qubits and only $O(1)$ quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.We study the complexity of quantum decomposable randomized encodings (QDRE). We establish a bi-directional connection between the QDRE model and the communication complexity model of quantum private simultaneous message protocols (QPSM), where classical communication is free, and quantum communication is the primary complexity measure. We show the following upper and lower bounds. (1) For every quantum channel mapping $n$ to $m$ qubits, we give a QPSM protocol with quantum communication $m$, using exponential pre-shared entanglement. In particular, constant-output channels have $O(1)$ quantum communication, and classical-output channels require no quantum communication. (2) With $O(n)$ pre-shared entanglement, every one-qubit-output channel has an $O(n)$-quantum-communication QPSM protocol. Conversely, there exists a channel and a constant $c>0$ for which any protocol with at most $cn$ pre-shared entanglement requires $Ω(n)$ quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with $O(n)$ pre-shared entangled qubits and only $O(1)$ quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.

quant-pharXiv:2609.17690Sep 17, 20260 citations

We consider the following problem of bounded-error quantum state identification: Given either state

\alpha_0

or state

\alpha_1

, we are required to output “0”, “1”, or “?” (“don't know"), such that conditioned on outputting “0” or “1”, our guess is correct with high probability. The goal is to maximize the probability of not outputting “?”. We prove the following direct product theorem: If we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is

O(ab)

. Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with

O(\log n)

classical bits of communication in the presence of shared randomness, but needs

\Omega(n^{1/3})

communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef, Jayram, and Kerenidis, this shows that the quantum simultaneous message passing (SMP) model is incomparable with the classical shared-randomness SMP model. Second, we describe a relation that can be computed with

O(\log n)

classical bits of communication in the presence of shared entanglement, but needs

\Omega((n/\log n)^{1/3})

communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys much more than quantum communication.

n

m

qubits, we give a QPSM protocol with quantum communication

m

, using exponential pre-shared entanglement. In particular, constant-output channels have

O(1)

quantum communication, and classical-output channels require no quantum communication. (2) With

O(n)

pre-shared entanglement, every one-qubit-output channel has an

O(n)

-quantum-communication QPSM protocol. Conversely, there exists a channel and a constant

c>0

for which any protocol with at most

cn

pre-shared entanglement requires

Ω(n)

quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with

O(n)

pre-shared entangled qubits and only

O(1)

quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.