Nicolas Cerf

The security of practical quantum key distribution

Quantum key distribution (QKD) is the first quantum information task to reach the level of mature technology, already fit for commercialization. It aims at the creation of a secret key between authorized partners connected by a quantum channel and a classical authenticated channel. The security of the key can in principle be guaranteed without putting any restriction on an eavesdroppers power. This article provides a concise up-to-date review of QKD, biased toward the practical side. Essential theoretical tools that have been developed to assess the security of the main experimental platforms are presented (discrete-variable, continuous-variable, and distributed-phase-reference protocols).

Gaussian quantum information

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography, and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

Security of quantum key distribution using d-level systems.

We consider two quantum cryptographic schemes relying on encoding the key into qudits, i.e., quantum states in a d-dimensional Hilbert space. The first cryptosystem uses two mutually unbiased bases (thereby extending the BB84 scheme), while the second exploits all d+1 available such bases (extending the six-state protocol for qubits). We derive the information gained by a potential eavesdropper applying a cloning-based individual attack, along with an upper bound on the error rate that ensures unconditional security against coherent attacks.

Quantum key distribution using gaussian-modulated coherent states

Quantum continuous variables are being explored as an alternative means to implement quantum key distribution, which is usually based on single photon counting. The former approach is potentially advantageous because it should enable higher key distribution rates. Here we propose and experimentally demonstrate a quantum key distribution protocol based on the transmission of gaussian-modulated coherent states (consisting of laser pulses containing a few hundred photons) and shot-noise-limited homodyne detection; squeezed or entangled beams are not required. Complete secret key extraction is achieved using a reverse reconciliation technique followed by privacy amplification. The reverse reconciliation technique is in principle secure for any value of the line transmission, against gaussian individual attacks based on entanglement and quantum memories. Our table-top experiment yields a net key transmission rate of about 1.7 megabits per second for a loss-free line, and 75 kilobits per second for a line with losses of 3.1 dB. We anticipate that the scheme should remain effective for lines with higher losses, particularly because the present limitations are essentially technical, so that significant margin for improvement is available on both the hardware and software.

Negative entropy and information in quantum mechanics

A framework for a quantum mechanical information theory is introduced that is based entirely on density operators, and gives rise to a unified description of classical correlation and quantum entanglement. Unlike in classical (Shannon) information theory, quantum (von Neumann) conditional entropies can be negative when considering quantum entangled systems, a fact related to quantum nonseparability. The possibility that negative (virtual) information can be carried by entangled particles suggests a consistent interpretation of quantum informational processes.

Quantum search by local adiabatic evolution

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover’s problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order AN, where N is the number of items in the database. We thus recover the advantage of Grover’s standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. ~e-print quant-ph/0001106!, where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.

Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution

A fully general approach to the security analysis of continuous-variable quantum key distribution (CV-QKD) is presented. Provided that the quantum channel is estimated via the covariance matrix of the quadratures, Gaussian attacks are shown to be optimal against all collective eavesdropping strategies. The proof is made strikingly simple by combining a physical model of measurement, an entanglement-based description of CV-QKD, and a recent powerful result on the extremality of Gaussian states [M. M. Wolf, Phys. Rev. Lett. 96, 080502 (2006)10.1103/PhysRevLett.96.080502].

Quantum key distribution over 25 km with an all-fiber continuous-variable system

We report on the implementation of a reverse-reconciliated coherent-state continuous-variable quantum key distribution system, with which we generated secret keys at a rate of more than 2 kb/s over 25 km of optical fiber. Time multiplexing is used to transmit both the signal and phase reference in the same optical fiber. Our system includes all experimental aspects required for a field implementation of a quantum key distribution setup. Real-time reverse reconciliation is achieved by using fast and efficient low-density parity check error correcting codes.

Optical simulation of quantum logic

A constructive method for simulating small-scale quantum circuits by use of linear optical devices is presented. It relies on the representation of several quantum bits by a single photon, and on the implementation of universal quantum gates using simple optical components (beam splitters, phase shifters, etc.). This suggests that the optical realization of small quantum networks with present-day quantum optics technology is a reasonable goal. This technique could be useful for demonstrating basic concepts of simple quantum algorithms or error-correction schemes. The optical analog of a nontrivial three-bit quantum circuit is presented as an illustration.

Optimality of the genetic code with respect to protein stability and amino-acid frequencies

BackgroundThe genetic code is known to be efficient in limiting the effect of mistranslation errors. A misread codon often codes for the same amino acid or one with similar biochemical properties, so the structure and function of the coded protein remain relatively unaltered. Previous studies have attempted to address this question quantitatively, by estimating the fraction of randomly generated codes that do better than the genetic code in respect of overall robustness. We extended these results by investigating the role of amino-acid frequencies in the optimality of the genetic code.ResultsWe found that taking the amino-acid frequency into account decreases the fraction of random codes that beat the natural code. This effect is particularly pronounced when more refined measures of the amino-acid substitution cost are used than hydrophobicity. To show this, we devised a new cost function by evaluating in silico the change in folding free energy caused by all possible point mutations in a set of protein structures. With this function, which measures protein stability while being unrelated to the codes structure, we estimated that around two random codes in a billion (109) are fitter than the natural code. When alternative codes are restricted to those that interchange biosynthetically related amino acids, the genetic code appears even more optimal.ConclusionsThese results lead us to discuss the role of amino-acid frequencies and other parameters in the genetic codes evolution, in an attempt to propose a tentative picture of primitive life.