Koenraad M. R. Audenaert

A high-resolution sensor based on tri-aural perception

The authors present a high-resolution sensor composed of three ultrasonic sensors, one transmitter/receiver and two extra receivers, which allows a significant improvement in the information-extraction process. With this sensor the position (distance and bearing) of all isolated objects in an approximately 25 degrees field of view can be determined using information contained in one single snapshot of a moderately complex scene. Within limits, the sensor system can also discriminate between different types of reflectors, in particular, walls and edges, based on their radius of curvature. These results are all based on the determination of the arrival times of the echoes present at the three receivers. A noise model that accounts for the measured variations of the arrival times is used to derive limits on the resolution of the results provided by the sensor. Based on this model it is shown that, to a large extent, the sensor results are impervious to measurement variations common to all three receivers. Results obtained in a realistic environment are compared with those obtained from a conventional time-of-flight sensor. >

Discriminating States: The Quantum Chernoff Bound

We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because it does not seem to share some of the undesirable features of other distance measures.

Entanglement cost under positive-partial-transpose-preserving operations

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT-operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this easily computable entanglement measure. As examples we discuss general Werner states and arbitrary bi-partite Gaussian states. Equipped with this result we then prove that for the anti-symmetric Werner state PPT-cost and PPT-entanglement of distillation coincide giving the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible.

Asymptotic Error Rates in Quantum Hypothesis Testing

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing.The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support.The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.

Entanglement properties of the harmonic chain

We study the entanglement properties of a closed chain of harmonic oscillators that are coupled via a translationally invariant Hamiltonian, where the coupling acts only on the position operators. We consider the ground state and thermal states of this system, which are Gaussian states. The entanglement properties of these states can be completely characterized analytically when one uses the logarithmic negativity as a measure of entanglement.

Evenly distributed unitaries: on the structure of unitary designs

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

A sharp continuity estimate for the von Neumann entropy

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

Accurate ranging of multiple objects using ultrasonic sensors

The authors propose a measurement setup consisting of a number of ultrasonic sensors used in parallel to perform triangulation measurements. The sensor system is based on two ideas. The first idea was to use signal processing techniques borrowed from existing radar and sonar systems. This allows the accurate determination of the position of multiple objects. Processing data in real time demands a fairly powerful processing system. The second idea was to assign a microprocessor to each transducer. To support the use of multiple sensors in the final measurement setup, transputers were used as processing elements as they offer easy scalability because of their serial links. This sensor measured the distance to multiple objects very accurately and with a resolution of 2 cm. It is shown that these techniques could be implemented in a cost-effective manner.<<ETX>>

Maximally entangled mixed states of two qubits

In this paper we investigate how much entanglement in a mixed two-qubit system can be created by global unitary transformations. The class of states for which no more entanglement can be created by global unitary operations is clearly a generalization of the class of Bell states to mixed states, and gives strict bounds on how the degree of mixing of a state limits its entanglement. This question is of considerable interest as entanglement is the magic ingredient of quantum information theory and experiments always deal with mixed states. Recently, Ishizaka and Hiroshima @1# independently considered the same question. They proposed a class of states and conjectured that the entanglement of formation @2# and the negativity @3# of these states could not be increased by any global unitary operation. Here we rigorously prove their conjecture and furthermore prove that the states they proposed are the only ones having the property of maximal entanglement. Closely related to the issue of generalized Bell states is the question of characterizing the set of separable density matrices @5#, as the entangled states closest to the maximally mixed state necessarily have to belong to the proposed class of maximal entangled mixed states. We can thus give a complete characterization of all nearly entangled states lying on the boundary of the sphere of separable states surrounding the maximally mixed state. As a by-product this gives an alternative derivation of the well-known result of Zyczkowski et al. @3# that all states for which the inequality Tr(r 2 )< 1

A comparison of the entanglement measures negativity and concurrence

In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger than √[(1 − C)2 + C2] − (1 − C), where C is the concurrence of the state. Furthermore, we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.