research-article

Exponential separation of quantum communication and classical information

Authors:

Dave Touchette,

Nengkun YuAuthors Info & Claims

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

Pages 277 - 288

https://doi.org/10.1145/3055399.3055401

Published: 19 June 2017 Publication History

Abstract

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable.

As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold.

The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha.

As another application of the techniques that we develop, a simple proof for an optimal trade-off between Alice's and Bob's communication is given, even when allowing pre-shared entanglement, while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/2^{O (b)} bits to Bob. We also present a classical protocol achieving this bound.

Supplementary Material

MP4 File (d2_sa_t1.mp4)

Download
156.18 MB

References

[1]

Anurag Anshu, Dave Touchette, Penghui Yao, Nengkun Yu. Exponential Separation of Quantum Communication and Classical Information. arXiv:1611.08946, 2016. https://arxiv.org/abs/1611.08946

[2]

Huzihiro Araki and Elliott H. Lieb. Entropy inequalities. Comm. Math. Phys., 18(2):160–170, 1970.

[3]

Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 70:1895–1899, Mar 1993.

[4]

Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM Journal on Computing, 42(3), pages 1327– 1363.

[5]

Mark Braverman, Ankit Garg, Young Kun Ko, Jieming Mao, and Dave Touchette. Near-optimal bounds on bounded-round quantum communication complexity of disjointness. In Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS’15), pages 773–791, Washington, DC, USA, 2015. IEEE Computer Society.

Digital Library

[6]

Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In J. Comput. Syst. Sci., pages 702–732.

Digital Library

[7]

Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Trans. Inf. Theory, 60(10), 6058–6069, 2014.

[8]

Mark Braverman. Interactive information complexity. SIAM Journal on Computing, 44(6), pages 1698–1739, 2015.

[9]

Mark Braverman. A hard-to-compress interactive task? In Proceedings of the 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton’13), pages 8–12, Oct 2013.

[10]

Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct product via round-preserving compression. In Proceedings of the 40th International Conference on Automata, Languages, and Programming (ICALP’13) - Volume Part I, pages 232–243, Berlin, Heidelberg, 2013. Springer-Verlag.

Digital Library

[11]

Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In Proceedings of the 54th IEEE Symposium on Foundations of Computer Science (FOCS’13), pages 746–755, Washington, DC, USA, 2013. IEEE Computer Society.

Digital Library

[12]

Mark Braverman and Omri Weinstein. A discrepancy lower bound for information complexity. In Algorithmica, vol 76, no. 3, pages 846–864. Springer-Verlag New York, Inc., 2016.

Digital Library

[13]

Mark Braverman and Omri Weinstein. An interactive information odometer and applications. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pages 341–350, New York, NY, USA, 2015. ACM.

Digital Library

[14]

Richard Cleve and Harry Buhrman. Substituting quantum entanglement for communication. Phys. Rev. A, 56:1201–1204, Aug 1997.

[15]

F.R.K Chung, R.L Graham, P Frankl, and J.B Shearer. Some intersection theorems for ordered sets and graphs. Journal of Combinatorial Theory, Series A, 43(1):23 – 37, 1986.

Digital Library

[16]

Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew C. Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS’01), page 270–278, Washington, DC, USA, 2001. IEEE Computer Society.

Digital Library

[17]

STOC’17, June 2017, Montreal, Canada Anurag Anshu, Dave Touchette, Penghui Yao, and Nengkun Yu

[18]

Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. Wiley Series in Telecommunications. John Wiley & Sons, New York, NY, USA, 1991.

Digital Library

[19]

D. Dieks. Communication by EPR devices. Physics Letters A, 92(6):271 – 272, 1982.

[20]

Igor Devetak and Jon Yard. Exact cost of redistributing multipartite quantum states. Phys. Rev. Lett., 100:230501, Jun 2008.

[21]

Lila Fontes, Rahul Jain, Iordanis Kerenidis, Mathieu Laurière, and Jérémie Roland. Relative Discrepancy Does not Separate Information and Communication Complexity. ACM Trans. Comput. Theory, vol 9, no. 1, Article 4 (December 2016), 15 pages.

Digital Library

[22]

Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM Journal on Computing, 23(5):1001–1018, 1994.

Digital Library

[23]

Christopher A Fuchs and Jeroen Van De Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory, 45(4):1216– 1227, 1999.

Digital Library

[24]

Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. In Proceedings of the 55th IEEE Symposium on Foundations of Computer Science (FOCS’14) pages 176–185, Washington, DC, USA, Oct 2014. IEEE Computer Society.

Digital Library

[25]

Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication for boolean functions. J. ACM, 63, 5, Article 46 (November 2016), 31 pages.

Digital Library

[26]

Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of communication and external information. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC’16), pages 977–986, New York, NY, USA, 2016. ACM.

Digital Library

[27]

Rahul Jain. New strong direct product results in communication complexity. J. ACM, 62, 3, Article 20 (June 2015), 27 pages.

Digital Library

[28]

Rahul Jain and Ashwin Nayak. Short proofs of the quantum substate theorem. IEEE Trans. Inf. Theory, 58(6):3664–3669, 2012.

Digital Library

[29]

Rahul Jain and Ashwin Nayak. The space complexity of recognizing wellparenthesized expressions in the streaming model: The index function revisited. IEEE Trans. Inf. Theory, 60(10):6646–6668, 2014.

[30]

Rahul Jain, Attila Pereszlényi, and Penghui Yao. A direct product theorem for the two-party bounded-round public-coin communication complexity. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS’12), pages 167–176, Washington, DC, USA, 2012. IEEE Computer Society.

Digital Library

[31]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS’02), pages 429–438, Washington, DC, USA, 2002. IEEE Computer Society.

Digital Library

[32]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In Proceedings of the 40th International Conference on Automata, Languages, and Programming (ICALP’03), pages 300–315, Berlin, Heidelberg, 2003. Springer-Verlag.

Digital Library

[33]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A lower bound for the bounded round quantum communication complexity of set disjointness. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science (FOCS’03), pages 220–229, Washington, DC, USA, 2003.

Digital Library

[34]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. Prior entanglement, message compression and privacy in quantum communication. In Proceedings of the 20th Annual IEEE Conference on Computational Complexity (CCC’05), pages 285–296, Washington, DC, USA, 2005. IEEE Computer Society.

Digital Library

[35]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. Optimal direct sum and privacy trade-off results for quantum and classical communication complexity. CoRR, abs/0807.1267, 2008. https://arxiv.org/abs/0807.1267

[36]

Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A new informationtheoretic property about quantum states with an application to privacy in quantum communication. J. ACM, 56, 6, Article 33 (September 2009), 32 pages.

Digital Library

[37]

Rahul Jain and Penghui Yao. A strong direct product theorem in terms of the smooth rectangle bound. CoRR, abs/1209.0263, 2012.

[38]

https://arxiv.org/abs/1209.0263

[39]

Hartmut Klauck. Lower bounds for quantum communication complexity. SIAM Journal on Computing, 37(1):20–46, April 2007.

Digital Library

[40]

Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, and David Xiao. Lower bounds on information complexity via zero-communication protocols and applications. SIAM Journal on Computing, 44(5):1550–1572, 2015.

Digital Library

[41]

Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1996.

Digital Library

[42]

Hartmut Klauck, Ashwin Nayak, Amnon Ta-Shma, and David Zuckerman. Interaction in quantum communication and the complexity of set disjointness. In Proceedings of the 33rd annual ACM symposium on Theory of computing (STOC ’01), pages 124–133, New York, NY, USA, 2001. ACM.

Digital Library

[43]

Gillat Kol. Interactive compression for product distributions. In In Proceedings of the 48th annual ACM symposium on Theory of Computing (STOC ’16), pages 987–998, New York, NY, USA, 2016. ACM.

Digital Library

[44]

Elliott H Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Advances in Mathematics, 11(3):267 – 288, 1973.

[45]

Sophie Laplante, Virginie Lerays, and Jérémie Roland. Classical and quantum partition bound and detector inefficiency. In Proceedings of the 39th International Colloquium Conference on Automata, Languages, and Programming (ICALP’12) - Volume Part I, pages 617–628, Berlin, Heidelberg, 2012. Springer-Verlag.

Digital Library

[46]

Elliott H. Lieb and Mary Beth Ruskai. Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys., 14(12):1938–1941, 1973.

[47]

Troy Lee and Adi Shraibman. Lower bounds in communication complexity. Foundations and Trends in Theoretical Computer Science, 3(4):263–399, 2007.

Digital Library

[48]

Mathieu Laurière and Dave Touchette. The flow of information in interactive quantum protocols :the cost of forgetting. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS ’17), 2017.

[49]

Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. J. Comput. Syst. Sci., 57(1):37–49, August 1998.

Digital Library

[50]

Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, UK, 2000.

Digital Library

[51]

Noam Nisan. The communication complexity of threshold gates. In Proceedings of Combinatorics, Paul Erdös is Eighty, pages 301–315, 1994.

[52]

Ashwin Nayak and Dave Touchette. Augmented index and quantum streaming for DYCK(2). Technical Report arXiv:1610.04937, 2016.

[53]

https://arxiv.org/abs/1610.04937

[54]

Jaikumar Radhakrishnan. Entropy and counting. Computational Mathematics, Modelling and Algorithms (Ed. J.C.Misra), pages 146–168, 2003.

[55]

Oded Regev and Bo’az Klartag. Quantum one-way communication can be exponentially stronger than classical communication. In Proceedings of the 43rd annual ACM symposium on Theory of computing (STOC ’11), pages 31–40, New York, NY, USA, 2011. ACM.

Digital Library

[56]

Sivaramakrishnan N. Ramamoorthy and Makrand Sinha. On the communication complexity of greater-than. In Proceedings of the 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton’15), pages 442–444, Sept 2015.

[57]

Anup Rao and Makrand Sinha. Simplified separation of information and communication. ECCC, 15:057, 2015. https://eccc.weizmann.ac.il/report/2015/057/

[58]

Alexander A. Sherstov. The pattern matrix method for lower bounds on quantum communication. In Proceedings of the 40th annual ACM symposium on Theory of computing (STOC ’08), pages 85–94, New York, NY, USA, 2008. ACM.

Digital Library

[59]

Alexander Sherstov. Compressing interactive communication under product distributions. In Proceedings of the 57th IEEE Symposium on Foundations of Computer Science (FOCS’16), pages 535–544, Oct 2016.

[60]

Dave Touchette. Quantum information complexity. In Proceedings of the 47th annual ACM symposium on Theory of computing (STOC ’15), pages 317–326, New York, NY, USA, 2015. ACM.

Digital Library

[61]

Emanuele Viola. The communication complexity of addition. In Proceedings of the 24th annual ACM-SIAM symposium on Discrete algorithms (SODA ’13), Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, pages 632–651. 2013.

Digital Library

[62]

John Watrous. Theory of Quantum Information, lecture notes, https://cs.uwaterloo.ca/ watrous/LectureNotes.html

[63]

Mark M. Wilde. Quantum Information Theory. Cambridge University Press, New York, 2013.

Digital Library

[64]

W. K. Wotters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299:802, October 1982.

[65]

Andrew C. Yao. Some complexity questions related to distributive computing (preliminary report). In Proceedings of the 11th Annual ACM Symposium on Theory of Computing (STOC ’79), pages 209–213, New York, NY, USA, 1979. ACM.

Digital Library

[66]

Andrew C. Yao. Quantum circuit complexity. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science (FOCS’93), pages 352–361, Nov 1993.

Digital Library

[67]

Jon Yard and Igor Devetak. Optimal quantum source coding with quantum side information at the encoder and decoder. IEEE Trans. Inf. Theory, 55(11):5339– 5351, Nov 2009.

Digital Library

Cited By

Fang KZhao JLi XLi YDuan R(2023)Quantum NETwork: from theory to practiceScience China Information Sciences10.1007/s11432-023-3773-466:8Online publication date: 5-Jul-2023
https://doi.org/10.1007/s11432-023-3773-4
Berera ACalderón-Figueroa J(2022)Viability of quantum communication across interstellar distancesPhysical Review D10.1103/PhysRevD.105.123033105:12Online publication date: 28-Jun-2022
https://doi.org/10.1103/PhysRevD.105.123033
Bab Hadiashar SNayak A(2020)On the Entanglement Cost of One-Shot CompressionQuantum10.22331/q-2020-06-25-2864(286)Online publication date: 25-Jun-2020
https://doi.org/10.22331/q-2020-06-25-286
Show More Cited By

Index Terms

Exponential separation of quantum communication and classical information
1. Theory of computation
  1. Computational complexity and cryptography
    1. Communication complexity
    2. Quantum complexity theory

Recommendations

Quantum Information Complexity
STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum tasks. These are the fully quantum generalizations of the analogous quantities for bipartite classical ...
Exponential Separation of Information and Communication for Boolean Functions
STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2^k. This ...
Exponential Separation of Information and Communication for Boolean Functions

We show an exponential gap between communication complexity and information complexity by giving an explicit example of a partial boolean function with information complexity ≤ O(k), and distributional communication complexity ≥ 2^k. This shows that a ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

June 2017

1268 pages

ISBN:9781450345286

DOI:10.1145/3055399

General Chairs:
Hamed Hatami
McGill University
,
Pierre McKenzie
Université de Montréal
,
Program Chair:
Valerie King
University of Victoria

Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 June 2017

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

STOC '17

Sponsor:

SIGACT

STOC '17: Symposium on Theory of Computing

June 19 - 23, 2017

Montreal, Canada

Acceptance Rates

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
109
Total Downloads

Downloads (Last 12 months)16
Downloads (Last 6 weeks)3

Reflects downloads up to 09 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Fang KZhao JLi XLi YDuan R(2023)Quantum NETwork: from theory to practiceScience China Information Sciences10.1007/s11432-023-3773-466:8Online publication date: 5-Jul-2023
https://doi.org/10.1007/s11432-023-3773-4
Berera ACalderón-Figueroa J(2022)Viability of quantum communication across interstellar distancesPhysical Review D10.1103/PhysRevD.105.123033105:12Online publication date: 28-Jun-2022
https://doi.org/10.1103/PhysRevD.105.123033
Bab Hadiashar SNayak A(2020)On the Entanglement Cost of One-Shot CompressionQuantum10.22331/q-2020-06-25-2864(286)Online publication date: 25-Jun-2020
https://doi.org/10.22331/q-2020-06-25-286
Anshu ABoddu NTouchette D(2019)Quantum Log-Approximate-Rank Conjecture is Also False2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2019.00063(982-994)Online publication date: Nov-2019
https://doi.org/10.1109/FOCS.2019.00063
Sinha Mde Wolf R(2019)Exponential Separation between Quantum Communication and Logarithm of Approximate Rank2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2019.00062(966-981)Online publication date: Nov-2019
https://doi.org/10.1109/FOCS.2019.00062
Anshu AGarg AHarrow AYao P(2018)Expected Communication Cost of Distributed Quantum Tasks2018 IEEE International Symposium on Information Theory (ISIT)10.1109/ISIT.2018.8437550(806-810)Online publication date: Jun-2018
https://doi.org/10.1109/ISIT.2018.8437550

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents