>> Joseph M. Renes > Publications > Refereed Publications
- Duality of Privacy Amplification Against Quantum Adversaries and Data Compression with Quantum Side Information
Joseph M. Renes
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 18 (2010).
arXiv:1003.0703 [quant-ph]Information processing protocols are typically built out of simpler parts, called primitives, and two of the most important such primitives are privacy amplification (PA) and data compression. The former extracts the truly secret part of some classical data, while the latter squeezes it into the smallest possible form. We show these tasks are dual in the setting of quantum information processing. Specifically, the tasks of PA of classical information against quantum adversaries and classical data compression with quantum side information are dual in the sense that the ability to perform one implies the ability to perform the other. The duality arises because the two protocols are connected by complementarity and the uncertainty principle in the quantum setting. Applications include a new uncertainty principle formulated in terms of smooth min- and max-entropies, which are useful in the study of one-shot protocols, as well as new conditions for approximate quantum error correction.
- Quantum Computational Renormalization in the Haldane Phase
Stephen D. Bartlett, Gavin K. Brennen, Akimasa Miyake, and Joseph M. Renes
Physical Review Letters 105, 110502 (2010).
arXiv:1004.4906 [quant-ph]Single-spin measurements on the ground state of an interacting spin lattice can be used to perform a quantum computation. We show how such measurements can mimic renormalization group transformations and remove the short-ranged variations of the state that can reduce the fidelity of a computation. This suggests that the quantum computational ability of a spin lattice could be a robust property of a quantum phase. We illustrate our idea with the ground state of a rotationally invariant spin-1 chain, which can serve as a quantum computational wire not only at the Affleck-Kennedy-Lieb-Tasaki point, but within the Haldane phase.
- The Uncertainty Principle in the Presence of Quantum Memory
Mario Berta, Matthias Christandl, Roger Colbeck, Joseph M. Renes, and Renato Renner
Nature Physics 6, 659-662 (2010).
arXiv:0909.0950 [quant-ph]The uncertainty principle, originally formulated by Heisenberg, clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future, it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.
- Conjectured Strong Complementary Information Tradeoff
Joseph M. Renes and Jean-Christian Boileau
Physical Review Letters 103, 020402-4 (2009).
arXiv:0806.3984 [quant-ph]We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.
- Optimal State Merging Without Decoupling
Jean-Christian Boileau and Joseph M. Renes
in Proceedings of the Fourth Workshop on Theory of Quantum Computation, Communication, and Cryptography,
edited by A. M. Childs and M. Mosca (Springer Verlag, Berlin, 2009).
Lecture Notes in Computer Science 5906, 76 (2009).
arXiv:0905.1324 [quant-ph]We construct an optimal state merging protocol by adapting a recently-discovered optimal entanglement distillation protcol [Renes and Boileau, Phys. Rev. A . 73, 032335 (2008)]. The proof of optimality relies only on directly establishing sufficient "amplitude" and "phase" correlations between Alice and Bob and not on usual techniques of decoupling Alice from the environment. This strengthens the intuition from quantum error-correction that these two correlations are all that really matter in two-party quantum information processing.
- Symmetric Extension in Two-Way Quantum Key Distribution
Geir Ove Myhr, Joseph M. Renes, Andrew C. Doherty, and Norbert Lütkenhaus
Physical Review A 79, 042329-10 (2009).
arXiv:0812.3607 [quant-ph]We introduce symmetric extensions of bipartite quantum states as a tool for analyzing protocols that distill secret key from quantum correlations. Whether the correlations are coming from a prepare-and-measure quantum key distribution scheme or from an entanglement-based scheme, the protocol has to produce effective states without a symmetric extension in order to succeed. By formulating the symmetric extension problem as a semidefinite program, we solve the problem for Bell-diagonal states. Applying this result to the six-state and Bennett-Brassard 1984 schemes, we show that for the entangled states that cannot be distilled by current key distillation procedures, the failure can be understood in terms of a failure to break a symmetric extension.
- Physical Underpinnings of Privacy
Joseph M. Renes and Jean-Christian Boileau
Physical Review A 78, 032335-12 (2008).
arXiv:0308.3096 [quant-ph]One of the remarkable features of quantum mechanics is the ability to ensure secrecy. Private states embody this effect, as they are precisely those multipartite quantum states from which two parties can produce a shared secret that cannot under any circumstances be correlated with an external system. Naturally, these play an important role in quantum key distribution QKD and quantum information theory. However, a general distillation method has heretofore been missing. Inspired by Koashi’s complementary control scenario [M. Koashi, e-print arXiv:0704.3661 [quant-ph] (2007)], we give a new definition of private states in terms of one party’s potential knowledge of two complementary measurements made on the other and use this to construct a general method of private state distillation using quantum error-correcting codes. The procedure achieves the same key rate as recent, more information-theoretic approaches while demonstrating the physical principles underlying privacy of the key. Additionally, the same approach can be used to establish the hashing inequality for entanglement distillation, as well as the direct quantum coding theorem.
- Improved One-Way Rates for BB84 and 6-State Protocols
Oliver Kern and Joseph M. Renes
Quantum Information and Computation 8, 756-772 (2008).
arXiv:0712.1494 [quant-ph]We study the advantages to be gained in quantum key distribution (QKD) protocols by combining the techniques of local randomization, or noisy preprocessing, and structured (nonrandom) block codes. Extending the results of [Smith, Renes, and Smolin, quant-ph/0607018] pertaining to BB84, we improve the best-known lower bound on the error rate for the 6-state protocol from 14.11% for local randomization alone to at least 14.59%. Additionally, we also study the effects of iterating the combined preprocessing scheme and find further improvements to the BB84 protocol already at small block lengths.
- Structured Codes Improve the Bennett-Brassard-84 Quantum Key Rate
Graeme Smith, Joseph M. Renes, and John A. Smolin
Physical Review Letters 100, 170502-4 (2008).
arXiv:quant-ph/0607018A central goal in information theory and cryptography is finding simple characterizations of optimal communication rates under various restrictions and security requirements. Ideally, the optimal key rate for a quantum key distribution (QKD) protocol would be given by a single-letter formula involving optimization over a single use of an effective channel. We explore the possibility of such a formula for the simplest and most widely used QKD protocol, Bennnett-Brassard-84 with one-way classical post-processing. We show that a conjectured single-letter formula is false, uncovering a deep ignorance about good private codes and exposing unfortunate complications in the theory of QKD. These complications are not without benefit—with added complexity comes better key rates than previously thought possible. The threshold for secure key generation improves from a bit error rate of 0.124 to 0.129.
- Equiangular Tight Frames from Paley Tournaments
Joseph M. Renes
Linear Algebra and Its Applications 426, 497-501 (2007).
arXiv:math/0408287We prove the existence of equiangular tight frames having n = 2d − 1 elements drawn from either C^d or C^(d−1) whenever n is either 2^k − 1 for k ∈ N, or a power of a prime such that n ≡ 3 mod 4. We also find a simple explicit expression for the prime power case by establishing a connection to a 2d-element equiangular tight frame based on quadratic residues.
- Noisy Processing and Distillation of Private Quantum States
Joseph M. Renes and Graeme Smith
Physical Review Letters 98, 020502 (2007).
arXiv:quant-ph/0603262We provide a simple security proof for prepare and measure quantum key distribution protocols employing noisy processing and one-way postprocessing of the key. This is achieved by showing that the security of such a protocol is equivalent to that of an associated key distribution protocol in which, instead of the usual maximally entangled states, a more general private state is distilled. In addition to a more general target state, the usual entanglement distillation tools are employed (in particular, Calderbank-Shor-Steane–like codes), with the crucial difference that noisy processing allows some phase errors to be left uncorrected without compromising the privacy of the key.
- Generalized Decoding, Effective Channels, and Simplified Security Proofs in Quantum Key Distribution
Joseph M. Renes and Markus Grassl
Physical Review A 74, 022317-9 (2006).
arXiv:quant-ph/0505061Prepare and measure quantum key distribution protocols can be decomposed into two basic steps: delivery of the signals over a quantum channel and distillation of a secret key from the signal and measurement records by classical processing and public communication. Here we formalize the distillation process for a general protocol in a purely quantum-mechanical framework and demonstrate that it can be viewed as creating an "effective" quantum channel between the legitimate users Alice and Bob. The process of secret key generation can then be viewed as entanglement distribution using this channel, which enables application of entanglement-based security proofs to essentially any prepare and measure protocol. To ensure secrecy of the key, Alice and Bob must be able to estimate the channel noise from errors in the key, and we further show how symmetries of the distillation process simplify this task. Applying this method, we prove the security of several key distribution protocols based on equiangular spherical codes.
- Unconditional Security of a Three State Quantum Key Distribution Protocol
J.-C. Boileau, K. Tamaki, J. Batuwantudawe, R. Laflamme, and Joseph M. Renes
Physical Review Letters 94, 040503-4 (2005).
arXiv:quant-ph/0408085Quantum key distribution (QKD) protocols are cryptographic techniques with security based only on the laws of quantum mechanics. Two prominent QKD schemes are the Bennett-Brassard 1984 and Bennett 1992 protocols that use four and two quantum states, respectively. In 2000, Phoenix et al. proposed a new family of three-state protocols that offers advantages over the previous schemes. Until now, an error rate threshold for security of the symmetric trine spherical code QKD protocol has been shown only for the trivial intercept-resend eavesdropping strategy. In this Letter, we prove the unconditional security of the trine spherical code QKD protocol, demonstrating its security up to a bit error rate of 9.81%. We also discuss how this proof applies to a version of the trine spherical code QKD protocol where the error rate is evaluated from the number of inconclusive events.
- Equiangular Spherical Codes in Quantum Cryptography
Joseph M. Renes
Quantum Information and Computation 5, 081-092 (2005).
arXiv:quant-ph/0409043Quantum key distribution protocols based on equiangular spherical codes are introduced and their behavior under the intercept/resend attack investigated. Such protocols offer a greater range of secure noise tolerance and speed options than protocols based on their cousins, the mutually-unbiased bases, while also enabling the determination of the channel noise rate without the need to sacrifice key bits. For fixed number of signal states in a given dimension, the spherical code protocols offer Alice and Bob more noise tolerance at the price of slower key generation rates.
- Spherical-Code Key-Distribution Protocols for Qubits
Joseph M. Renes
Physical Review A 70, 052314 (2004).
arXiv:quant-ph/0402135Recently spherical codes were introduced as potentially more capable ensembles for quantum key distribution. Here we develop specific key-creation protocols for the two qubit-based spherical codes, the trine and tetrahedron, and analyze them in the context of a suitably tailored intercept/resend attack, both in standard form, and in a “gentler” version whose back action on the quantum state is weaker. When compared to the standard unbiased basis protocols, Bennett-Brassard 1984 (BB84) and six-state, two distinct advantages are found. First, they offer improved tolerance of eavesdropping, the trine besting its counterpart BB84 and the tetrahedron the six-state protocol. Second, the key error rate may be computed from the sift rate of the protocol itself, removing the need to sacrifice key bits for this purpose. This simplifies the protocol and improves the overall key rate.
- Optimal Protocols and Tradeoffs in Quantum Key Distribution
Joseph M. Renes
in Proceedings of the Seventh International Conference on Quantum Communication, Measurement and Computing,
edited by S. M. Barnett, O. Hirota, P. Ohberg, J. Jeffers, and E. Andersson (American Institute of Physics, Melville, NY, 2004).
AIP Conference Proceedings 734, 327-330 (2004).
Quantum key distribution protocols based on equiangular spherical codes are introduced and a security comparison made to random and unbiased-bases schemes in two and three dimensions. Attention is limited to intercept/resend attacks for simplicity. In each case, a general tradeoff between eavesdropper resistance and key generation speed is observed, with the largest possible spherical code found to be the most robust, inviting the question of their optimality in general.
- Symmetric Informationally Complete Quantum Measurements
Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, and Carlton M. Caves
Journal of Mathematical Physics 45, 2171-2180 (2004).
arXiv:quant-ph/0310075We consider the existence in arbitrary finite dimensions d of a positive operator valued measure POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ‘‘symmetric, informationally complete’’ POVM (SIC–POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
- Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
Carlton M. Caves, Christopher A. Fuchs, Kiran K. Manne, and Joseph M. Renes
Foundations of Physics 34, 193-209 (2004).
arXiv:quant-ph/0306179We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields—settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.
- Quantum Information and Precision Measurement
Andrew M. Childs, John Preskill, and Joseph M. Renes
Journal of Modern Optics 47, 155-176 (2000).
arXiv:quant-ph/9904021We describe some applications of quantum information theory to the analysis of quantum limits on measurement sensitivity. A measurement of a weak force acting on a quantum system is a determination of a classical parameter appearing in the master equation that governs the evolution of the system; limitations on measurement accuracy arise because it is not possible to distinguish perfectly among the different possible values of this parameter. Tools developed in the study of quantum information and computation can be exploited to improve the precision of physics experiments; examples include superdense coding, fast database search, and the quantum Fourier transform.