We propose protocols for determining the distances in Hilbert space between pure and mixed quantum states prepared on a quantum computer. In the case of pure quantum states, the protocol is based on measuring the square of modulus of scalar product between certain states. Determination of the distance between mixed quantum states is reduced to measuring the squares of modules of scalar products between all pure states included in the mixed states. In addition, we develop a protocol that allows one to determine the speed of evolution of the spin system simulated by a quantum computer. These protocols we apply to measure distances and speeds of evolution of different quantum systems implemented on the ibmq-santiago quantum computer.
We propose protocols for determining the distances in Hilbert space between pure and mixed quantum states prepared on a quantum computer. In the case of pure quantum states, the protocol is based on measuring the square of modulus of scalar product between certain states. Determination of the distance between mixed quantum states is reduced to measuring the squares of modules of scalar products between all pure states included in the mixed states. In addition, we develop a protocol that allows one to determine the speed of evolution of the spin system simulated by a quantum computer. These protocols we apply to measure distances and speeds of evolution of different quantum systems implemented on the ibmq-santiago quantum computer.
Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories and quantum computers, quantum communications, and quantum phase-space simulations.
The measures of distances between points in a Hilbert space are one of the basic theoretical concepts used to characterize properties of a quantum system with respect to some etalon state. These are not only used in studying fidelity of signal transmission and basic quantum phenomena but also applied in measuring quantum correlations, and also in quantum machine learning. The values of quantum distance measures are very difficult to determine without completely reconstructing the state. Here we demonstrate an interferometric approach to measuring distances between quantum states that in some cases can outperform quantum state tomography. We propose a direct experimental method to estimate such distance measures between two unknown two-qubit mixed states such as Uhlmann-Jozsa fidelity (or the Bures distance), the Hilbert-Schmidt distance, and the trace distance. The fidelity is estimated via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method is based on the multiparticle interactions (i.e., interference) between copies of the unknown pairs of qubits.
The counter-intuitive features of Quantum Mechanics make it possible to solve problems and perform tasks that are beyond the abilities of classical computers and classical communication devices. The area of quantum information processing studies how representing information by quantum states can help achieving such improvements.In this research, we use basic notions of quantum information (mainly entanglement, Bloch sphere, and geometrical distances between quantum states) for analyzing relations of quantum states to each other and quantum cryptographic protocols.Entanglement is an important feature of quantum states. Intuitively (and, partly, inaccurately), entanglement represents quantum (non-classical) correlations between several different quantum systems. Entanglement is one of the most important quantum phenomena, and it has many uses in quantum information, quantum communication, and quantum computing.Some quantum states can be geometrically represented by the Bloch sphere: the unit sphere in the three-dimensional Euclidean space. The "standard" quantum states, to which the laws of Quantum Mechanics directly apply, are called pure states. Other states are the mixed states: probability distributions ("mixtures") of several pure states. The points on the Bloch sphere are the pure states, and those inside the Bloch sphere are the mixed states. This geometrical representation is useful and intuitive for many purposes.We provide a geometrical analysis of entanglement for all the quantum mixed states of rank 2 (all the mixtures of exactly two pure states): for any such state (in any dimension), we define a generalized Bloch sphere by using the two pure states, and we analyze this state and its neighbor states inside this Bloch sphere. We look at the set of non-entangled states ("separable states") in the Bloch sphere and characterize it into exactly five possible classes. We give examples for each class and prove that there are no other classes. In addition, we suggest possible definitions of "entanglement measures" by using the "trace distance" from the nearest separable state.Many types of distances between quantum states can be defined. One of the most useful distances is the trace distance, which bounds the "distinguishability" between the states. The trace distance is very useful in quantum information and quantum cryptography, and it also has a simple geometrical interpretation: it is half of the Euclidean distance between the states in the Bloch sphere.Quantum key distribution (QKD) protocols make it possible for two participators to achieve the classically-impossible task of generating a secret random shared key even if their adversary is computationally unlimited. Several important QKD protocols, including the first protocol of Bennett and Brassard (BB84), have their unconditional security proved against adversaries performing the most general attacks in a theoretical (idealized) setting. We discuss a slightly different protocol, named "BB84-INFO-z", and prove it secure against a broad class of attacks (the collective attacks). Moreover, we make use of the "trace distance" for making our security proof more "composable" than similar security proofs for BB84: namely, for making a step towards proving that the secret key remains secret even when the two participators actually use it for cryptographic purposes.
The geometric measure of entanglement of mixed states is investigated based on its relation to spin correlations. We derive an analytical expression for the entanglement of rank-2 mixed states composed of arbitrary multiqubit pure states within a two-dimensional Hilbert subspace. The dependence of the entanglement of rank-2 mixed multiqubit states on the parameters characterizing arbitrary pure states in the mixture is studied both analytically and using quantum computing. For a particular class of mixed states, namely, equal-probability mixtures of an arbitrary multiqubit state and the Schrödinger cat state, we construct quantum protocols for detecting the geometric measure of entanglement based on its relation to spin correlations. We quantify the entanglement using both the AerSimulator and the ibm_marrakesh quantum computer. The effects of quantum-computing errors are also analyzed. The simulation results reproduce the analytical dependencies, while measurements on the quantum device demonstrate good agreement with theoretical predictions within estimated error bounds.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, p.675 (1995) and H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, p.6787 (2001)], is explored for bipartite and multipartite pure and mixed states. It is determined for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states.