We present a characterization of quantum phase transitions in terms of the the overlap function between two ground states obtained for two different values of external parameters. On the examples of the Dicke and XY models, we show that the regions of criticality of a system are marked by the extremal points of the overlap and functions closely related to it. Further, we discuss the connections between this approach and the Anderson orthogonality catastrophe as well as with the dynamical study of the Loschmidt echo for critical systems.

We introduce the concept of a quantum walk with two particles and study it for the case of a discrete time walk on a line. A quantum walk with more than one particle may contain entanglement, thus offering a resource unavailable in the classical scenario and which can present interesting advantages. In this work, we show how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, the probability to find at least one particle in a certain position after N steps of the walk, as well as the average distance between the two particles, can be larger or smaller than the case of two unentangled particles, depending on the initial conditions we choose. This resource can then be tuned according to our needs, in particular to enhance a given application (algorithmic or other) based on a quantum walk. Experimental implementations are briefly discussed. Given the superposition principle of Quantum Mechanics, quantum walks allow for coherent superpositions of classical random walks and, due to interference effects, can exhibit different features and offer advantages when compared to the classical case. In particular, for a quantum walk on a line, the variance after N steps is proportional to N , rather than √ N as in the classical case (see Fig. 1). Recently, several quantum algorithms with optimal efficiency were proposed based on quantum walks [3], and it was even shown that a continuous time quantum walk on a specific graph can be used for exponential algorithmic speed-up [4].All studies on quantum walks so far have, however, been based on a single walker. In this article we study a discrete time quantum walk on a line with two particles. Classically, random walks with K particles are equivalent to K independent single-particle random walks. In the quantum case though, a walk with K particles may contain entanglement, thus offering a resource unavailable in the classical scenario which can present interesting advantages. Moreover, in the case of identical particles we have to take into account the effects of quantum statistics, giving an additional feature to quantum walks that can also be exploited. In this work we explicitly show that a quantum walk with two particles can indeed be tuned to behave very differently from two independent single-particle quantum walks. This paves the way for new quantum algorithms based on richer quantum walks.Let us start by introducing the discrete time quantum walk on a line for a single particle. The relevant degrees of freedom are the particle's position i (with i ∈ Z) on the line, as well as its coin state. The total Hilbert space is given by H ≡ H P ⊗ H C , where H P is spanned by the orthonormal vectors {|i } representing the position of the particle, and H C is the two-dimensional coin space spanned by two orthonormal vectors which we denote as |↑ and |↓ .Each step of the quantum walk is given by two subsequent operations. First, the coin operation, given bŷ U ...

We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its center, and look at the fidelity between partial ͑either one-site or two-site͒ quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding either to the impurity location or its close vicinity. This enables us to identify the on-site magnetization as the order parameter for the phase transition studied. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, while the other lies elsewhere in the lattice. We also determine the Uhlmann mixed state geometric phase, recently introduced in the study of the structural change of the system state eigenvectors in the vicinity of the lines of thermal phase transitions, and find it to be trivial, both for one-and two-site partial states, except when an electron site is at the impurity. This means that the system partial state eigenvectors do not contribute significantly to the enhanced state distinguishability around the point of this quantum phase transition. Finally, we use the fidelity to analyze the total amount of correlations contained within a composite system, showing that, even for the smallest one-site states, it features an abrupt quantitative change in the vicinity of the point of the quantum phase transition.

We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: The Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the nonanalyticity of the fidelity is directly related to the nonanalyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually noncommuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically nontrivial Uhlmann mixed state geometric phase.