Exponential separation of quantum communication and classical information

Anshu, Anurag; Touchette, Dave; Yao, Penghui; Yu, Nengkun

doi:10.1145/3055399.3055401

Cited by 9 publications

(14 citation statements)

References 64 publications

(36 reference statements)

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“…As a consequence, we infer that the quantum information cost of a protocol may be arbitrarily smaller than the communication cost of any protocol without shared entanglement for compressing its messages. Anshu, Touchette, Yao, and Yu [3] had previously proven a similar separation when the compression protocol is allowed to use shared entanglement. However, their separation is exponential: they exhibited an interactive protocol for a Boolean function with quantum information cost that is exponentially smaller than the communication cost of any interactive quantum protocol that computes the function.…”

Section: Implications and Related Workmentioning

confidence: 83%

On the Entanglement Cost of One-Shot Compression

Hadiashar

Nayak

2020

Quantum

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We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.

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Section: Implications and Related Workmentioning

confidence: 83%

On the Entanglement Cost of One-Shot Compression

Hadiashar

Nayak

2020

Quantum

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“…Another property of our construction is that the number of bits of input given to Alice is ≈ 7 log d. Thus, the lower bound on the expected communication cost is of the order of the input size. This may be contrasted with the well known exponential separations between information and communication [25], [26] and their recent quantum counterpart [27], where the lower bound on the communication cost is doubly exponentially smaller than the input size.…”

Section: Our Resultsmentioning

confidence: 98%

Expected Communication Cost of Distributed Quantum Tasks

Anshu¹,

Garg²,

Harrow³

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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A central question in classical information theory is that of source compression, which is the task where Alice receives a sample from a known probability distribution and needs to transmit it to the receiver Bob with small error. This problem has a one-shot solution due to Huffman, in which the messages are of variable length and the expected length of the messages matches the asymptotic and i.i.d. compression rate of the Shannon entropy of the source. In this work, we consider a quantum extension of above task, where Alice receives a sample from a known probability distribution and needs to transmit a part of a pure quantum state (that is associated to the sample) to Bob. We allow entanglement assistance in the protocol, so that the communication is possible through classical messages, for example using quantum teleportation. The classical messages can have a variable length and the goal is to minimize their expected length. We provide a characterization of the expected communication cost of this task, by giving a lower bound that is near optimal up to some additive factors. A special case of above task, and the quantum analogue of the source compression problem, is when Alice needs to transmit the whole of her pure quantum state. Here we show that there is no one-shot interactive scheme which matches the asymptotic and i.i.d. compression rate of the von Neumann entropy of the average quantum state. This is a relatively rare case in quantum information theory where the cost of a quantum task is significantly different from its classical

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“…The following Shearer-type inequality for quantum information was shown in Ref. [ATYY17]. Classical variants appeared in [GKR15,RS15].…”

Section: Mutual Informationmentioning

confidence: 99%

Quantum Log-Approximate-Rank Conjecture is Also False

Anshu¹,

Boddu²,

Touchette³

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In a recent breakthrough result, Chattopadhyay, Mande and Sherif showed an exponential separation between the log approximate rank and randomized communication complexity of a total function f , hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of f . Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

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Exponential separation of quantum communication and classical information

Cited by 9 publications

References 64 publications

On the Entanglement Cost of One-Shot Compression

On the Entanglement Cost of One-Shot Compression

Expected Communication Cost of Distributed Quantum Tasks

Quantum Log-Approximate-Rank Conjecture is Also False

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