Pure State Systems Research Articles

Extensions of the Mandelstam–Tamm quantum speed limit to systems in mixed states

The Mandelstam–Tamm quantum speed limit (QSL) puts a bound on how fast a closed system in a pure state can evolve. In this paper, we derive several extensions of this QSL to closed systems in mixed states. We also compare the strengths of these extensions and examine their tightness. The most widely used extension of the Mandelstam–Tamm QSL originates in Uhlmann’s energy dispersion estimate. We carefully analyze the underlying geometry of this estimate, an analysis that makes apparent that the Bures metric, or equivalently the quantum Fisher information, will rarely give rise to tight extensions. This observation leads us to address whether there is a tightest general extension of the Mandelstam–Tamm QSL. Using a geometric construction similar to that developed by Uhlmann, we prove that this is indeed the case. In addition, we show that tight evolutions of mixed states are typically generated by time-varying Hamiltonians, which contrasts with the case for systems in pure states.

Wave-Particle-Entanglement-Ignorance Complementarity for General Bipartite Systems.

Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave–particle–entanglement–ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems.

Entanglement classification via neural network quantum states

The task of classifying the entanglement properties of a multipartite quantum state poses a remarkable challenge due to the exponentially increasing number of ways in which quantum systems can share quantum correlations. Tackling such challenge requires a combination of sophisticated theoretical and computational techniques. In this paper we combine machine-learning tools and the theory of quantum entanglement to perform entanglement classification for multipartite qubit systems in pure states. We use a parameterisation of quantum systems using artificial neural networks in a restricted Boltzmann machine architecture, known as Neural Network Quantum States, whose entanglement properties can be deduced via a constrained, reinforcement learning procedure. In this way, Separable Neural Network States can be used to build entanglement witnesses for any target state.

Symmetric logarithmic derivative for general n-level systems and the quantum Fisher information tensor for three-level systems

Within a geometrical context, we derive an explicit formula for the computation of the symmetric logarithmic derivative for arbitrarily mixed quantum systems, provided that the structure constants of the associated unitary Lie algebra are known. To give examples of this procedure, we first recover the known formulae for two-level mixed and three-level pure state systems and then apply it to the novel case of U(3), that is for arbitrarily mixed three-level systems (q-trits). Exploiting the latter result, we finally calculate an expression for the Fisher tensor for a q-trit considering also all possible degenerate subcases.

Quantum Zeno Effect with Mixed Initial States

Quantum Zeno effect with mixed initial state is studied here. Frequent projective measurements performed on a bipartite joint pure state system will result in the quantum Zeno effect on the subsystem of interest. This shows the existence of Quantum Zeno effect in the system with mixed initial states.

AN EFFICIENT SEPARABILITY CRITERION FOR n-PARTITE ARBITRARILY DIMENSIONAL QUANTUM STATES

In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.

Entanglement dynamics of spin systems in pure states

We investigate numerically the appearance and evolution of entanglement in spin systems prepared initially in a pure state. We consider the dipolar coupling spin systems of different molecular structures: benzene ${\text{C}}_{6}{\text{H}}_{6}$, cyclopentane ${\text{C}}_{5}{\text{H}}_{10}$, sodium butyrate ${\text{CH}}_{3}{({\text{CH}}_{2})}_{2}{\text{CO}}_{2}\text{Na}$, and calcium hydroxyapatite ${\text{Ca}}_{5}(\text{OH}){({\text{PO}}_{4})}_{3}$. Numerical simulations show that the close relationship exists between the intensity of second order (2Q) coherences and concurrences of nearest spins in a cyclopentane molecule.

Classification and quantification of qubit and qutrit multipartite pure state entanglement

Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.

CLASSIFICATION AND QUANTIFICATION OF ENTANGLED BIPARTITE QUTRIT PURE STATES

A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.

Extremal entanglement for triqubit pure states

A complete analysis of entangled triqubit pure states is carried out based on a new simple entanglement measure. An analysis of all possible extremally entangled pure triqubit states with up to eight terms is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis presented are most helpful for finding different entanglement types in multipartite pure state systems.

Maximum Fisher information in mixed state quantum systems

We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.

Entanglement and Properties of Composite Quantum Systems: A Conceptual and Mathematical Analysis

Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of the paper deals with composite systems in pure states. After a detailed discussion and a precise formal analysis of the case of systems of distinguishable particles, the problems of entanglement and the one of the properties of subsystems of systems of identical particles are thoroughly discussed. This part is the most interesting and new and it focuses in all details various subtle questions which have never been adequately discussed in the literature. Some inappropriate assertions which appeared in recent papers are analyzed. The relations of the main subject of the paper with the nonlocal aspects of quantum mechanics, as well as with the possibility of deriving Bell's inequality are also considered.

Misure massimizzanti l'informazione di Fisher nei sistemi quantistici

Braunstein and Caves (1994) showed that Helstrom's quantum information maximises expected Fisher information over all possible measurements on a given quantum system. For one-dimensional parameter pure state system, we derived necessary and/or sufficient conditions for measurement in order to be a maximising one, according to some assumptions either on the dimension of the physical space of the system or on the kind of measurement to be applied.

Education finance reform: A dynamic perspective

We use a dynamic Tiebout model to analyze the consequences of moving from a pure local system of education finance to a pure state system of finance in which each student receives the same resources. While much of the education finance literature focuses on the static or immediate effects of such a change, our analysis also examines the dynamic effects. Numerical simulations for a calibrated version of our model indicate that these dynamic effects are very important. Comparing steady states, we find that aggregate welfare increases on the order of 10 percent following the switch to a state system. The key to this welfare gain is that a local system yields inefficiently low investment in human capital of children from low-income families.

Education finance reform: A dynamic perspective

We use a dynamic Tiebout model to analyze the consequences of moving from a pure local system of education finance to a pure state system of finance in which each student receives the same resources. While much of the education finance literature focuses on the static or immediate effects of such a change, our analysis also examines the dynamic effects. Numerical simulations for a calibrated version of our model indicate that these dynamic effects are very important. Comparing steady states, we find that aggregate welfare increases on the order of 10 percent following the switch to a state system. The key to this welfare gain is that a local system yields inefficiently low investment in human capital of children from low-income families.