Quantum capacity under adversarial quantum noise: arbitrarily varying quantum channels

We investigate entanglement transmission over an unknown channel in the presence of a third party (called the adversary), which is enabled to choose the channel from a given set of memoryless but non-stationary channels without informing the legitimate sender and receiver about the particular choice that he made. This channel model is called arbitrarily varying quantum channel (AVQC). We derive a quantum version of Ahlswede's dichotomy for classical arbitrarily varying channels. This includes a regularized formula for the common randomness-assisted capacity for entanglement transmission of an AVQC. Quite surprisingly and in contrast to the classical analog of the problem involving the maximal and average error probability, we find that the capacity for entanglement transmission of an AVQC always equals its strong subspace transmission capacity. These results are accompanied by different notions of symmetrizability (zero-capacity conditions) as well as by conditions for an AVQC to have a capacity described by a single-letter formula. In he final part of the paper the capacity of the erasure-AVQC is computed and some light shed on the connection between AVQCs and zero-error capacities. Additionally, we show by entirely elementary and operational arguments motivated by the theory of AVQCs that the quantum, classical, and entanglement-assisted zero-error capacities of quantum channels are generically zero and are discontinuous at every positivity point.Comment: 49 pages, no figures, final version of our papers arXiv:1010.0418v2 and arXiv:1010.0418. Published "Online First" in Communications in Mathematical Physics, 201

Incompressible strips in dissipative Hall bars as origin of quantized Hall plateaus

We study the current and charge distribution in a two dimensional electron system, under the conditions of the integer quantized Hall effect, on the basis of a quasi-local transport model, that includes non-linear screening effects on the conductivity via the self-consistently calculated density profile. The existence of ``incompressible strips'' with integer Landau level filling factor is investigated within a Hartree-type approximation, and non-local effects on the conductivity along those strips are simulated by a suitable averaging procedure. This allows us to calculate the Hall and the longitudinal resistance as continuous functions of the magnetic field B, with plateaus of finite widths and the well-known, exactly quantized values. We emphasize the close relation between these plateaus and the existence of incompressible strips, and we show that for B values within these plateaus the potential variation across the Hall bar is very different from that for B values between adjacent plateaus, in agreement with recent experiments.Comment: 13 pages, 11 figures, All color onlin

Trading quantum for classical resources in quantum data compression

We study the visible compression of a source E of pure quantum signal states, or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy S(E) qubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal is the optimal quantum rate for a given classical rate of R bits per signal. Our main result is an explicit characterization of this trade--off function by a simple formula in terms of only single signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parameterized states. We also show that our result leads to a number of corollaries characterizing the trade--off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so--called remote state preparation problem. Finally, we develop a probability--free version of our main result which may be interpreted as an answer to the question: ``How many classical bits does a qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.Comment: 51 pages, 7 figure

Realistic modelling of quantum point contacts subject to high magnetic fields and with current bias at out of linear response regime

The electron and current density distributions in the close proximity of quantum point contacts (QPCs) are investigated. A three dimensional Poisson equation is solved self-consistently to obtain the electron density and potential profile in the absence of an external magnetic field for gate and etching defined devices. We observe the surface charges and their apparent effect on the confinement potential, when considering the (deeply) etched QPCs. In the presence of an external magnetic field, we investigate the formation of the incompressible strips and their influence on the current distribution both in the linear response and out of linear response regime. A spatial asymmetry of the current carrying incompressible strips, induced by the large source drain voltages, is reported for such devices in the non-linear regime.Comment: 16 Pages, 9 Figures, submitted to PR

Number theoretic example of scale-free topology inducing self-organized criticality

In this work we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology.Comment: 4 pages, 3 figures. Physical Review Letters, in pres

Self-consistent local-equilibrium model for density profile and distribution of dissipative currents in a Hall bar under strong magnetic fields

Recent spatially resolved measurements of the electrostatic-potential variation across a Hall bar in strong magnetic fields, which revealed a clear correlation between current-carrying strips and incompressible strips expected near the edges of the Hall bar, cannot be understood on the basis of existing equilibrium theories. To explain these experiments, we generalize the Thomas-Fermi--Poisson approach for the self-consistent calculation of electrostatic potential and electron density in {\em total} thermal equilibrium to a {\em local equilibrium} theory that allows to treat finite gradients of the electrochemical potential as driving forces of currents in the presence of dissipation. A conventional conductivity model with small values of the longitudinal conductivity for integer values of the (local) Landau-level filling factor shows that, in apparent agreement with experiment, the current density is localized near incompressible strips, whose location and width in turn depend on the applied current.Comment: 9 pages, 7 figure

Quantum Network Coding

Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether or not quantum network coding is possible. Its classical counterpart is another good example to show that digital information flow can be done much more efficiently than conventional (say, liquid) flow. Our answer to the question is similar to the case of cloning, namely, it is shown that quantum network coding is possible if approximation is allowed, by using a simple network model called Butterfly. In this network, there are two flow paths, s_1 to t_1 and s_2 to t_2, which shares a single bottleneck channel of capacity one. In the classical case, we can send two bits simultaneously, one for each path, in spite of the bottleneck. Our results for quantum network coding include: (i) We can send any quantum state |psi_1> from s_1 to t_1 and |psi_2> from s_2 to t_2 simultaneously with a fidelity strictly greater than 1/2. (ii) If one of |psi_1> and |psi_2> is classical, then the fidelity can be improved to 2/3. (iii) Similar improvement is also possible if |psi_1> and |psi_2> are restricted to only a finite number of (previously known) states. (iv) Several impossibility results including the general upper bound of the fidelity are also given.Comment: 27pages, 11figures. The 12page version will appear in 24th International Symposium on Theoretical Aspects of Computer Science (STACS 2007

Self-consistent Coulomb picture of an electron-electron bilayer system

In this work we implement the self-consistent Thomas-Fermi approach and a local conductivity model to an electron-electron bilayer system. The presence of an incompressible strip, originating from screening calculations at the top (or bottom) layer is considered as a source of an external potential fluctuation to the bottom (or top) layer. This essentially yields modifications to both screening properties and the magneto-transport quantities. The effect of the temperature, inter-layer distance and density mismatch on the density and the potential fluctuations are investigated. It is observed that the existence of the incompressible strips plays an important role simply due to their poor screening properties on both screening and the magneto-resistance (MR) properties. Here we also report and interpret the observed MR Hysteresis within our model.Comment: 12 pages, 12 figures, submitted to PR

Compression of quantum measurement operations

We generalize recent work of Massar and Popescu dealing with the amount of classical data that is produced by a quantum measurement on a quantum state ensemble. In the previous work it was shown how spurious randomness generally contained in the outcomes can be eliminated without decreasing the amount of knowledge, to achieve an amount of data equal to the von Neumann entropy of the ensemble. Here we extend this result by giving a more refined description of what constitute equivalent measurements (that is measurements which provide the same knowledge about the quantum state) and also by considering incomplete measurements. In particular we show that one can always associate to a POVM with elements a_j, an equivalent POVM acting on many independent copies of the system which produces an amount of data asymptotically equal to the entropy defect of an ensemble canonically associated to the ensemble average state and the initial measurement (a_j). In the case where the measurement is not maximally refined this amount of data is strictly less than the von Neumann entropy, as obtained in the previous work. We also show that this is the best achievable, i.e. it is impossible to devise a measurement equivalent to the initial measurement (a_j) that produces less data. We discuss the interpretation of these results. In particular we show how they can be used to provide a precise and model independent measure of the amount of knowledge that is obtained about a quantum state by a quantum measurement. We also discuss in detail the relation between our results and Holevo's bound, at the same time providing a new proof of this fundamental inequality.Comment: RevTeX, 13 page

Monogamy of entanglement and other correlations

It has been observed by numerous authors that a quantum system being entangled with another one limits its possible entanglement with a third system: this has been dubbed the "monogamous nature of entanglement". In this paper we present a simple identity which captures the trade-off between entanglement and classical correlation, which can be used to derive rigorous monogamy relations. We also prove various other trade-offs of a monogamy nature for other entanglement measures and secret and total correlation measures.Comment: 7 pages, revtex