Manipulating many-body quantum states via single-photon resonance

For an atomic Bose-Hubbard dimer quantum control via multiphoton processes have been investigated widely. We here explore how to manipulate the many-body quantum states via single-photon resonance by treating the periodic driving as a weak perturbation. The transition probabilities up to second-order approximation are given as functions of the driving parameters, which are considerable only for the single-photon resonance case. Due to some transition matrix elements vanishing, the first-order quantum transition obeys a selection rule. The non-forbidden transitions involve states of different entanglement entropies and all (part) of the forbidden transitions relate to the entropy balances between two states for odd (even) number of particles. The results provide a new route for manipulating many-body quantum states and entanglement entropies, and controlling the atomic tunnelings of the Bose-Hubbard dimer.     

General Bell-CHSH type and entropic inequalities based on quantum tomograms

Review of Bell-CHSH type and entropic inequalities in composite quantum correlated systems in the probability representation of states is presented. The upper bounds for some new Bell-CHSH type inequalities within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-CHSH type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states. Joint tomographic entropies of multiqubit systems are studied. Limitations on inequalities for tomographic entropies are obtained. A negative result of possible connection between the violation of entropic and Bell-CHSH type inequalities in multi-partite states is reported.     

Entropy scaling and simulability by matrix product states

We investigate the relation between the scaling of block entropies and the efficient simulability by matrix product states (MPSs) and clarify the connection both for von Neumann and Rényi entropies. Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPSs. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time-independent Hamiltonian.     

Some General Properties of Unified Entropies

Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes’ sense and stability in Lesche’s sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum Rényi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurements is proved.     

Note on Entropies of Quantum Dynamical Systems

We review some techniques and notions for quantum information theory. It is shown that the dynamical entropies is discussed and some numerical computations of these entropies are carried for several states.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.     

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party states to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy.Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by quantum relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes.A surprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.     

Uncertainty relations for information entropy in wave mechanics

New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality.     

Quantum uncertainty relations of two generalized quantum relative entropies of coherence

In this paper, we consider the quantum uncertainty relations of two generalized relative entropies of coherence based on two measurement bases. First, we give quantum uncertainty relations for pure states in a d-dimensional quantum system by making use of the majorization technique; these uncertainty relations are then generalized to mixed states. We find that the lower bounds are always nonnegative for pure states but may be negative for some mixed states. Second, the quantum uncertainty relations for single qubit states are obtained by the analytical method. We show that the lower bounds obtained by this technique are always positive for single qubit states. Third, the lower bounds obtained by the two methods described above are compared for single qubit states.     

Some Properties of the kth-partial Rényi Entropy

The kth-partial Rényi entropies for both classical and quantum cases are defined and some properties of them are given. Also, we study the stability of kth-partial Rényi entropy for two states which satisfy majorization condition.     

Entropy inequalities

Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence of the limiting mean entropy for translationally invariant states of quantum continuous systems.Work supported by National Science Foundation Grant GP-9414.     

Instruments and channels in quantum information theory

While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the typical inequalities for the quantum and classical relative entropies, many bounds on the classical information extracted in a quantum measurement, of the type of the Holevo bound, are obtained in a unified manner.     

Quantum correlations and tomographic representation

We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Rényi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.     

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.     

Quantum and Classical Ergotropy from Relative Entropies

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.     

Entropic uncertainty relations for nonextensive quantum scattering

In this paper the angle-angular momentum entropic uncertainty relations are obtained for Tsallis-like entropies for nonextensive quantum scattering of spinless particles. The number-phase entropic uncertainty relations are also proved for nonextensive quantum scattering. Numerical results on the experimental tests of these entropic uncertainty relations, for the nonextensive (q≠1) statistics case are obtained by calculations of Tsallis-like scattering entropies from the 48 experimental sets of the pion-nucleus phase shifts.     

Superposition, Entropy and Schmidt Decomposition of States

Superposition and entropy are compared using the language of the logic of quantum mechanics. It is pointed out that a finite value of the relative quantum entropy of states implies a superposition relation between the states themselves. The superposition relation is then studied by comparing the pure state of the compound system with the product of the reduced states and an intermediate Schmidt state. All the corresponding relative quantum entropies are evaluated in terms of the Schmidt coefficients of the global pure state. Some of the results are extended in case the compound system is in a state represented by a general density operator.     

Anonymous voting for multi-dimensional CV quantum system

We investigate the design of anonymous voting protocols,CV-based binary-valued ballot and CV-based multi-valued ballot with continuous variables(CV) in a multi-dimensional quantum cryptosystem to ensure the security of voting procedure and data privacy.The quantum entangled states are employed in the continuous variable quantum system to carry the voting information and assist information transmission,which takes the advantage of the GHZ-like states in terms of improving the utilization of quantum states by decreasing the number of required quantum states.It provides a potential approach to achieve the efficient quantum anonymous voting with high transmission security,especially in large-scale votes.