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De Gruyter eBooks

5 Quantum information and communication

Rainer Müller, Franziska Greinert

Published Oct 23, 2023• 22 citations
Publisher
DOI
10.1515/9783110717457-005

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Quantum information complexity (QIC), introduced by Touchette [Touchette, STOC 2015], has been shown to be one of the most powerful methods for proving quantum communication complexity and has also been shown to be equal to amortized quantum communication complexity. Unfortunately, QIC is generally hard to analyze because it is a sum of quantum conditional mutual information terms, which are difficult to estimate. In this work, we introduce a new variational approach, the information potential, for analyzing QIC and proving quantum communication lower bounds inspired by the resolvent representation for quantum relative entropy. This approach connects the QIC of individual messages to a quadratic form, making it more amenable and thus enables us to bound the cumulative positive increments of the potential throughout an interactive quantum protocol by its QIC. Lower bounds on the growth of the potential therefore translate into lower bounds on both QIC and quantum communication complexity. As an application, we give an optimal $Ω(1/r)$ lower bound on the QIC of the two-bit $\mathsf{AND}$ function as well as an optimal $Ω(n/r)$ lower bound on the quantum communication complexity of $r$-round Set-Disjointness, answering an open problem in~[Braverman, Garg, Ko, Mao, Touchette FOCS 2015]. Moreover, we further prove a direct-sum theorem for bounded-round quantum communication complexity of Set Disjointness. With the tight bound on the QIC of $\mathsf{AND}$ function, we further establish a nearly tight tradeoff for the asymmetric quantum communication complexity of $\mathsf{Set} \mathsf{Disjointness}$: $(q_A+1)(q_B+1)=Ω(n)$, where $q_A$ and $q_B$ denote the total numbers of qubits sent by Alice and Bob, respectively.

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The International Conference on Quantum Information Processing and Communication (QIPC) 2013 was organized within the QUIE2T Coordination Action (Quantum Information and Entanglement Enabled Technologies), an FP7 Coordination Action to support QIPC in Europe. It covered a broad range of topics on quantum information and quantum communication, physical realizations of quantum systems for information technologies and topical subjects such as quantum enhanced measurements, foundations of quantum information, quantum simulations and many-body systems.

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Quantum communication and information networks offer unprecedented processing efficiency and security for data transfers. Technologies like quantum key distribution (QKD), quantum repeaters, quantum memory, and quantum entanglement sources are enabling technologies that ensure secure communication. QKD methods use quantum physics concepts to establish cryptographic keys, while quantum repeaters compensate for signal loss in fiber optic cables. Quantum memory enables storage and retrieval of quantum information, while quantum entanglement generators generate entangled photon pairs. Artificial intelligence and machine learning have significantly improved the efficacy and security of quantum communication. These methods can facilitate ultra-secure, reliable, large-scale communication and even a future quantum internet by analyzing quantum protocols and mitigating noise-induced errors. QKD protocols ensure secure communication channels, while quantum repeaters establish secure long-distance communication lines by mitigating signal loss. Traditional Internet technology is susceptible to surveillance due to Shannon's information theory and mathematics. The quantum internet (QI) uses satellite-based quantum communication and quantum cryptographic protocols for physical QI. Quantum entanglement sources fortify secure communication protocols, improving resistance to attacks. Quantum information networks (QINs) enhance computing, sensing, and security capabilities over extended distances. The development of reliable, robust communication systems depends on these technologies. Quantum-secured networks enhance urban security, smart management, and infrastructure protection by providing data privacy and facilitating secure data transmission, thereby facilitating robust infrastructure and technological development.
Quantum information complexity (QIC), introduced by Touchette [Touchette, STOC 2015], has been shown to be one of the most powerful methods for proving quantum communication complexity and has also been shown to be equal to amortized quantum communication complexity. Unfortunately, QIC is generally hard to analyze because it is a sum of quantum conditional mutual information terms, which are difficult to estimate. In this work, we introduce a new variational approach, the information potential, for analyzing QIC and proving quantum communication lower bounds inspired by the resolvent representation for quantum relative entropy. This approach connects the QIC of individual messages to a quadratic form, making it more amenable and thus enables us to bound the cumulative positive increments of the potential throughout an interactive quantum protocol by its QIC. Lower bounds on the growth of the potential therefore translate into lower bounds on both QIC and quantum communication complexity. As an application, we give an optimal Ω(1/r)Ω(1/r)Ω(1/r) lower bound on the QIC of the two-bit AND\mathsf{AND}AND function as well as an optimal Ω(n/r)Ω(n/r)Ω(n/r) lower bound on the quantum communication complexity of rrr-round Set-Disjointness, answering an open problem in~[Braverman, Garg, Ko, Mao, Touchette FOCS 2015]. Moreover, we further prove a direct-sum theorem for bounded-round quantum communication complexity of Set Disjointness. With the tight bound on the QIC of AND\mathsf{AND}AND function, we further establish a nearly tight tradeoff for the asymmetric quantum communication complexity of SetDisjointness\mathsf{Set} \mathsf{Disjointness}SetDisjointness: (qA+1)(qB+1)=Ω(n)(q_A+1)(q_B+1)=Ω(n)(qA​+1)(qB​+1)=Ω(n), where qAq_AqA​ and qBq_BqB​ denote the total numbers of qubits sent by Alice and Bob, respectively.
The International Conference on Quantum Information Processing and Communication (QIPC) 2013 was organized within the QUIE2T Coordination Action (Quantum Information and Entanglement Enabled Technologies), an FP7 Coordination Action to support QIPC in Europe. It covered a broad range of topics on quantum information and quantum communication, physical realizations of quantum systems for information technologies and topical subjects such as quantum enhanced measurements, foundations of quantum information, quantum simulations and many-body systems.