Some Bipartite States do not Arise from Channels

It is well-known that the action of a quantum channel on a state can be represented, using an auxiliary space, as the partial trace of an associated bipartite state. Recently, it was observed that for the bipartite state associated with the optimal average input of the channel, the entanglement of formation is simply the entropy of the reduced density matrix minus the Holevo capacity. It is natural to ask if every bipartite state can be associated with some channel in this way. We show that the answer is negative.Comment: 7 pages; minor typos corrected. To appear in special issue of the IBM Journal of Research and Development for Charles Bennett's 60th birthda

A One-Dimensional Model for Many-Electron Atoms in Extremely Strong Magnetic Fields: Maximum Negative Ionization

We consider a one-dimensional model for many-electron atoms in strong magnetic fields in which the Coulomb potential and interactions are replaced by one-dimensional regularizations associated with the lowest Landau level. For this model we show that the maximum number of electrons is bounded above by 2Z+1 + c sqrt{B}. We follow Lieb's strategy in which convexity plays a critical role. For the case of two electrons and fractional nuclear charge, we also discuss the critical value at which the nuclear charge becomes too weak to bind two electrons.Comment: 23 pages, 5 figures. J. Phys. A: Math and General (in press) 199

Fundamental properties of Tsallis relative entropy

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov inequality is also proven

Bounds for the adiabatic approximation with applications to quantum computation

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.Comment: 15 pages, one figure. Two comments added in Secs. 2 and

Constructive counterexamples to additivity of minimum output R\'enyi entropy of quantum channels for all p>2

We present a constructive example of violation of additivity of minimum output R\'enyi entropy for each p>2. The example is provided by antisymmetric subspace of a suitable dimension. We discuss possibility of extension of the result to go beyond p>2 and obtain additivity for p=0 for a class of entanglement breaking channels.Comment: 4 pages; a reference adde

On 1-qubit channels

The entropy H_T(rho) of a state rho with respect to a channel T and the Holevo capacity of the channel require the solution of difficult variational problems. For a class of 1-qubit channels, which contains all the extremal ones, the problem can be significantly simplified by associating an Hermitian antilinear operator theta to every channel of the considered class. The concurrence of the channel can be expressed by theta and turns out to be a flat roof. This allows to write down an explicit expression for H_T. Its maximum would give the Holevo (1-shot) capacity.Comment: 12 pages, several printing or latex errors correcte

The χ2\chi^2 - divergence and Mixing times of quantum Markov processes

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $\chi^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes

A Random Matrix Model of Adiabatic Quantum Computing

We present an analysis of the quantum adiabatic algorithm for solving hard instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory (RMT). We determine the global regularity of the spectral fluctuations of the instantaneous Hamiltonians encountered during the interpolation between the starting Hamiltonians and the ones whose ground states encode the solutions to the computational problems of interest. At each interpolation point, we quantify the degree of regularity of the average spectral distribution via its Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor spacings. We find that for hard problem instances, i.e., those having a critical ratio of clauses to variables, the spectral fluctuations typically become irregular across a contiguous region of the interpolation parameter, while the spectrum is regular for easy instances. Within the hard region, RMT may be applied to obtain a mathematical model of the probability of avoided level crossings and concomitant failure rate of the adiabatic algorithm due to non-adiabatic Landau-Zener type transitions. Our model predicts that if the interpolation is performed at a uniform rate, the average failure rate of the quantum adiabatic algorithm, when averaged over hard problem instances, scales exponentially with increasing problem size.Comment: 9 pages, 7 figure