Induced measures in the space of mixed quantum states

We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x K mixed states, if the reduction takes place with respect to the first subsystem). For K=N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of $N \times N$ pure states behaves as lnN-1/2.Comment: 12 latex pages, 2 figures in epsf, submited to J. Phys. A. ver.3, some improvements and a few references adde

Mutually Unbiased Bases and Complementary Spin 1 Observables

The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction z, the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.Comment: 7 pages, 3 figures, comments welcom

A characterization of positive linear maps and criteria of entanglement for quantum states

Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $\rho$ on $H\otimes K$ is separable if and only if $(\Phi\otimes I)\rho\geq 0$ for all positive finite rank elementary operators $\Phi$. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.Comment: 20 page

Notes on entropic characteristics of quantum channels

One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange are derived. We also give upper bounds on the map $(q,s)$-entropies in terms of the output entropy, corresponding to the completely mixed input.Comment: 10 pages, no figures. The statement of Proposition 1 is explicitly illustrated with the depolarizing channel. The bibliography is extended and updated. More explanations. To be published in Cent. Eur. J. Phy

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat. Phy

Many body physics from a quantum information perspective

The quantum information approach to many body physics has been very successful in giving new insight and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.Comment: Lectures from the Les Houches School on "Modern theories of correlated electron systems". Improved version new references adde