General relation is derived which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time-scale proportional to delta^(-2) where delta is the strength of perturbation, whereas faster, typically gaussian decay on shorter time scale proportional to delta^(-1) is predicted for integrable, or generally non-ergodic dynamics. This surprising result is demonstrated in quantum Ising spin-1/2 chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of non-ergodic and non-integrable motion in thermodynamic limit.
We establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics. The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices. The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases,…
After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our definitions are consistent with usual understanding of ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings are introduced. They are computed for right triangular billiards and the results indicate a very close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue that, besides ergodicity and mixi…
This paper is a physicist's review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here we present a unified, graph-based view of all archetypical models of such universality (billiards, particles in random media, interacting spin or fermion systems). We find phenomenological relations between the onset of ergodicity (Gaussian-random delocalization of eigenstates) and the structure of the appropriate graphs, and we construct a heuristic picture of summing trajectories on graphs that describes why a generic interacting system should be ergodic.…
Recent discovery of persistent revivals in quantum simulators based on Rydberg atoms have pointed to the existence of a new type of dynamical behavior that challenged the conventional paradigms of integrability and thermalization. This novel collective effect has been named quantum many-body scars by analogy with weak ergodicity breaking of a single particle inside a stadium billiard. In this overview, we provide a pedagogical introduction to quantum many-body scars and highlight the newly emerged connections with the semiclassical quantization of many-body systems. We discuss the relation bet…
For a real number $r>0$, let $F(r)$ be the family of all stationary ergodic quantum sources with von Neumann entropy rates less than $r$. We prove that, for any $r>0$, there exists a blind, source-independent block compression scheme which compresses every source from $F(r)$ to $r n$ qubits per input block length $n$ with arbitrarily high fidelity for all large $n$. As our second result,we show that the stationarity and the ergodicity of a quantum source $\{ρ_m \}_{m=1}^{\infty}$ are preserved by any trace-preserving completely positive linear map of the tensor product form ${\cal E}^{\otimes …
Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence …
We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the ener…
We study the ergodic -- non-ergodic transition in a generalized Dicke model with independent co- and counter rotating light-matter coupling terms. By studying level statistics, the average ratio of consecutive level spacings, and the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we show that the ergodic -- non-ergodic transition in the Dicke model is a consequence of the proximity to the integrable limit of the model when one of the couplings is set to zero. This can be interpreted as a hint for the existence of a quantum analogue of the classical Kolmogorov-Arnold-M…
A geometric interpretation for the A-fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the A-fidelity are studied by numerical method. An alternative generalization of the A-fidelity, which has the same geometric picture, to a $N$-state quantum system is also discussed.
It was shown in [T. Morimae, Phys. Rev. A {\bf81}, 060307(R) (2010)] that the gate fidelity of an inaccurate one-way quantum computation is upper bounded by a decreasing function of the amount of entanglement in the register. This means that a strong entanglement causes the low gate fidelity in the one-way quantum computation with inaccurate measurements. In this paper, we derive similar entanglement-fidelity relations for the inaccurate ancilla-driven quantum computation. These relations again imply that a strong entanglement in the register causes the low gate fidelity in the ancilla-driven …
We derive fidelity decay and parametric energy correlations for random matrix ensembles where time--reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities can be established.
Recently, some quantum algorithms have been implemented by quantum adiabatic evolutions. In this paper, we discuss the accurate relation between the running time and the distance of the initial state and the final state of a kind of quantum adiabatic evolutions. We show that this relation can be generalized to the case of mixed states.
We study the Bures measure of entanglement and the geometric measure of entanglement as special cases of entanglement measures based on fidelity, and find their tighter monogamy inequalities over tri-qubit systems as well as multi-qubit systems. Furthermore, we derive the monogamy inequality of concurrence for qudit quantum systems by projecting higher-dimensional states to qubit substates.
We investigate the dynamics of information in isolated multi-qubit systems. It is shown that information is in not only local form but also nonlocal form. We apply a measure of local information based on fidelity, and demonstrate that nonlocal information can be directly related to some appropriate well defined entanglement measures. Under general unitary transformations, local and nonlocal information will exhibit unambiguous complementary behavior with the total information conserved.
Pointer states are long-lasting high-fidelity states in open quantum systems. We show how any pure state in a non-Markovian open quantum system can be made to behave as a pointer state by suitably engineering the coupling to the environment via open-loop periodic control. Engineered pointer states are constructed as approximate fixed points of the controlled open-system dynamics, in such a way that they are guaranteed to survive over a long time with a fidelity determined by the relative precision with which the dynamics is engineered. We provide quantitative minimum-fidelity bounds by identif…
The bounds of concurrence in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98 (2007) 140505] and [C. Zhang \textit{et. al.}, Phys. Rev. A 78 (2008) 042308] are proved by using two properties of the fidelity. In two-qubit systems, for a given value of concurrence, the states achieving the maximal upper bound, the minimal lower bound or the maximal difference upper-lower bound are determined analytically.
Recent results obtained in quantum measurements indicate that the fundamental relations between three physical properties of a system can be represented by complex conditional probabilities. Here, it is shown that these relations provide a fully deterministic and universally valid framework on which all of quantum mechanics can be based. Specifically, quantum mechanics can be derived by combining the rules of Bayesian probability theory with only a single additional law that explains the phases of complex probabilities. This law, which I introduce here as the law of quantum ergodicity, is base…
Motivated by recent development in quantum fidelity and fidelity susceptibility, we study relations among Lie algebra, fidelity susceptibility and quantum phase transition for a two-state system and the Lipkin-Meshkov-Glick model. We get the fidelity susceptibility for SU(2) and SU(1,1) algebraic structure models. From this relation, the validity of the fidelity susceptibility to signal for the quantum phase transition is also verified in these two systems. At the same time, we obtain the geometric phase in these two systems in the process of calculating the fidelity susceptibility. In additio…
Based on the assumption that the receiver Bob can apply any unitary transformation, Horodecki {\it et al.} [Phys. Lett. A {\bf 222}, 21 (1996)] proved that any mixed two spin-1/2 state which violates the Bell-CHSH inequality is useful for teleportation. Here, we further show that any X state which violates the Bell-CHSH inequality can also be used for nonclassical teleportation even if Bob can only perform the identity or the Pauli rotation operations. Moreover, we showed that the maximal difference between the two average fidelities achievable via Bob's arbitrary transformations and via the s…
We investigate the connection between sub-Planck structure in the Wigner function and the output fidelity of continuous-variable teleportation protocols. When the teleporting parties share a two-mode squeezed state as an entangled resource, high fidelity in the output state requires a squeezing large enough that the smallest sub-Planck structures in an input pure state are teleported faithfully. We formulate this relationship, which leads to an explicit relation between the fine-scale structure in the Wigner function and large-scale extent of the Wigner function, and we treat specific examples…
A quantum system can be described and characterized by at least two different concepts, namely, its physical and informational properties. Here, we explicitly connect these two concepts, by equating the time-energy cost which is the product of the largest energy of a Hamiltonian of quantum dynamics and the evolution time, and the entanglement fidelity which is the informational difference between an input state and the corresponding output state produced by a quantum channel characterized by the Hamiltonian. Specifically, the worst-case entanglement fidelity between the input and output states…
We analyze uncertainty relations on finite dimensional Hilbert spaces expressed in terms of classical fidelity, which are stronger then metric uncertainty relations introduced by Fawzi, Hayden and Sen. We establish validity of fidelity uncertainty relations for random unitary matrices with optimal parameters (up to universal constants) which improves upon known results for the weaker notion of metric uncertainty. This result is then applied to locking classical information in quantum states and allows to obtain optimal locking in Hellinger distance, improving upon previous results on locking i…
Motivated by the growing importance of fidelity in quantum critical phenomena, we establish a general relation between fidelity and structure factor of the driving term in a Hamiltonian through a newly introduced concept: fidelity susceptibility. Our discovery, as shown by some examples, facilitates the evaluation of fidelity in terms of susceptibility using well developed techniques such as density matrix renormalization group for the ground state, or Monte Carlo simulations for the states in thermal equilibrium.
Purity and coherence of a quantum state are recognized as useful resources for various information processing tasks. In this article, we propose a fidelity based valid measure of purity and coherence monotone and establish a relationship between them. This formulation of coherence is extended to quantum correlation relative to measurement. We have also studied the role of weak measurement on purity.
A conceptually simple and experimentally prevalent class of entanglement witnesses, known as fidelity witnesses, detect entanglement via a state's fidelity with a pure reference state. While existence proofs guarantee that a suitable witness can be constructed for every entangled state, such assurances do not apply to fidelity witnesses. Recent results have found that entangled states that cannot be detected by a fidelity witness, known as unfaithful states, are exceedingly common among bipartite states. In this paper, we show that even among two-qubit states, the simplest of all entangled sta…
Quantum teleportation (QT) is a fundamentally remarkable communication protocol that also finds many important applications for quantum informatics. Given a quantum entangled resource, it is crucial to know to what extent one can accomplish the QT. This is usually assessed in terms of output fidelity, which can also be regarded as an operational measure of entanglement. In the case of multipartite communication when each communicator possesses a part of $N$-partite entangled state, not all pairs of communicators can achieve a high fidelity due to monogamy property of quantum entanglement. We h…
We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provided a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power.
We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behavior. The irreversible work, rescaled by square of the quench's a…
High fidelity quantum state transfer is an essential part of quantum information processing. In this regard, we address the problem of maximizing the fidelity in a quantum state transformation process satisfying the Liouville-von Neumann equation. By introducing fidelity as the performance index, we aim at maximizing the similarity of the final state density operator with the one of the desired target state. Optimality conditions in the form of a Maximum Principle of Pontryagin are given for the matrix-valued dynamic control systems propagating the probability density function. These provide a…
We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy $D$ on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of…
We study bipartite entangled states in arbitrary dimensions and obtain different bounds for the entanglement measures in terms of teleportation fidelity. We find that there is a simple relation between negativity and teleportation fidelity for pure states but for mixed states, an upper bound is obtained for negativity in terms of teleportation fidelity using convex-roof extension negativity (CREN). However, with this it is not clear how to distinguish betweeen states useful for teleportation and positive partial transpose (PPT) entangled states. Further, there exists a strong conjecture in the…
In the present work, we investigate the intrinsic relation between quantum fidelity susceptibility (QFS) and the dynamical structure factor. We give a concise proof of the QFS beyond the perturbation theory. With the QFS in the Lehmann representation, we point out that the QFS actually the negative-two-power moment of dynamical structure factor, and illuminate the inherent relation between physical quantities in the linear response theory. Moreover, we discuss the generalized fidelity susceptibility (GFS) of any quantum relevant operator, that may not be coupled to the driving parameter, prese…
The fidelity and entanglement entropy in an antiferromagnetic-ferromagnetic alternating Heisenberg chain are investigated by using the method of density-matrix renormalization-group. The effect of anisotropy on fidelity and entanglement entropy are investigated. The relations between fidelity, entanglement entropy and quantum phase transition are analyzed. It is found that the quantum phase transition point can be well characterized by both the ground-state entropy and fidelity for large system.
We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotinicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to t…
Various fidelity measures can be defined between two quantum processes especially when at least one of them is non-unitary. In this paper we consider two such measures of state averaged process fidelity, put forward an efficient procedure to compute them and investigate their mathematical properties as well as their relationship.
A general relation between quantum phase transitions and the second derivative of the fidelity (or the "fidelity susceptibility") is proposed. The validity and the limitation of the fidelity susceptibility in characterizing quantum phase transitions is thus established. Moreover, based on the bosonization method, general formulas of the fidelity and the fidelity susceptibility are obtained for a class of one-dimensional gapless systems known as the Tomonaga-Luttinger liquid. Applying these formulas to the one-dimensional spin-1/2 $XXZ$ model, we find that quantum phase transitions, even of the…
We analyze the average fidelity (say, F) and the fidelity deviation (say, D) in noisy-channel quantum teleportation. Here, F represents how well teleportation is performed on average and D quantifies whether the teleportation is performed impartially on the given inputs, that is, the condition of universality. Our analysis results prove that the achievable maximum average fidelity ensures zero fidelity deviation, that is, perfect universality. This structural trait of teleportation is distinct from those of other limited-fidelity probabilistic quantum operations, for instance, universal-NOT or…
Taking into account the density matrices with non-full ranks, we show that the fidelity susceptibility is determined by the support of the density matrix. Combining with the corresponding expression of the quantum Fisher information, we rigorously prove that the fidelity susceptibility is proportional to the quantum Fisher information. As this proof can be naturally extended to the full rank case, this proportional relation is generally established for density matrices with arbitrary ranks. Furthermore, we give an analytical expression of the quantum Fisher information matrix, and show that th…
This paper introduces a new formalism for quantum cellular automata (QCAs), based on evolving tensor products of qubits using local unitary operators. It subsequently uses this formalism to analyze and validate several conjectures, stemming from a formal analogy between quantum computational complexity theory and classical thermodynamics, that have arisen recently in the context of black hole physics. In particular, the apparent resonance and thermalization effects present within such QCAs are investigated, and it is demonstrated that the expected exponential relationships between the quantum …
