Fidelity has always been an important concept in quantum optics. Recently, it was found that fidelity can also play a key role in quantum information and communication theory. Fidelity can be interpreted as the probability that a decoded message possesses the same information content as the message prior to coding and transmission. In this paper, we give a formula of Bures fidelity for displaced squeezed thermal states directly by the displacement and squeezing parameters and briefly discuss how the results can apply to quantum information theory.
Fidelity has always been an important concept in quantum optics. Recently, it was found that fidelity can also play a key role in quantum information and communication theory. Fidelity can be interpreted as the probability that a decoded message possesses the same information content as the message prior to coding and transmission. In this paper, we give a formula of Bures fidelity for displaced squeezed thermal states directly by the displacement and squeezing parameters and briefly discuss how the results can apply to quantum information theory.
In the theory of quantum transmission of information the concept of fidelity plays a fundamental role. An important class of channels, which can be experimentally realized in quantum optics, is that of Gaussian quantum channels. In this work we present a general formula for fidelity in the case of two arbitrary Gaussian states. From this formula one can get a previous result [H. Scutaru, J. Phys. A 31, 3659 (1998)], for the case of a single mode. The concept of fidelity used in this paper is the standard one [D. Bures, Trans. Am. Math. Soc. 135, 199 (1969); A. Uhlmann, Rep. Math. Phys. 9, 273 (1976); R. Josza, J. Mod. Opt. 41, 2314 (1994)]. It can be defined by $F({\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}{)}^{\underset{=}{\mathrm{def}}}{\mathrm{max}}_{|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉}|〈{\ensuremath{\psi}}_{1}|{\ensuremath{\psi}}_{2}〉{|}^{2},$ where $|{\ensuremath{\psi}}_{i}〉,$ i=1,2 are purifications of the density matrices ${\ensuremath{\rho}}_{i}.$
It is known that, by extending the equivalence of the Fisher information matrix to its quantum version, the Bures metric, the quantum Jeffreys prior can be determined from the volume element of the Bures metric. We compute the Bures metric for the displaced squeezed thermal state and analyse the quantum Jeffreys prior and its marginal probability distributions. To normalize the marginal probability density function, it is necessary to provide a range of values of the squeezing parameter or the inverse temperature. We find that if the range of the squeezing parameter is kept narrow, there are significant differences in the marginal probability density functions in terms of the squeezing parameters for the displaced and undisplaced situations. However, these differences disappear as the range increases. Furthermore, marginal probability density functions against temperature are very different in the two cases.
Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two- or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.
Clarke and Barron have recently established that the (classical) Jeffreys' prior yields universal codes - ones which do relatively well (in the sense of relative entropy) no matter what the true state. Twamley has recently computed the Bures metric - proportional to the quantum Fisher information (statistical distinguishability) metric, as shown by Braunstein and Caves - for squeezed thermal states. The volume elements of these metrics - that is, the quantum Jeffreys' prior - are found to be simply the product of a function of the squeeze factor and a function of the inverse temperature, the phase being irrelevant in this regard. A computational strategy to find a universal coding using the quantum Jeffreys' prior - previously implemented for the two-level quantum systems - is then discussed.
One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.
Fidelity plays an important role in quantum information theory. In this paper, two pairs of metrics of quantum states are introduced based on the Uhlmann-Jozsa fidelity and super-fidelity. They are proved to satisfy the axioms of metrics and return to the Sine metric and Bures metric for the qubit case. The CP expansive property and convex property of the metrics are studied. These metrics have deep connections with the known important metrics such as trace metric, Sine metric, Bures metric, and spectral metric, and are useful to quantify entanglement. I. Background, problem, and result Suppose one has two quantum states ρ and σ, then the Uhlmann-Jozsa fidelity [1] between ρ and σ is given by F(ρ, σ) = [Tr ρ 1 2 σρ 1 2] 2. Uhlmann-Jozsa fidelity plays an important role in quantum information theory and quantum computation [1], [2]. We know that for the case of qubit, Uhlmann-Jozsa fidelity has a simple form. From the Bloch sphere representation of quantum states, a qubit is described by a density matrix as([3]): ρ(u) = 1 (I + σ · u), 2 where I is the 2 × 2 unit matrix and σ = (σ1, σ2, σ3) are the Pauli matrices. Assume ρ(u) and ρ(v) are two states of one qubit, then they can represented by two vectors u and v in the Bloch sphere. The Uhlmann-Jozsa fidelity for qubits has an elegant form([4] and [5]): F(ρ(u), ρ(v)) = 1 2 [1 + u · v + √ 1 − |u | 2 √ 1 − |v | 2], where u · v is the inner product of two vectors u and v, and |u | is the magnitude of u. We know that for N × N quantum states, the Uhlmann-Jozsa fidelity has no simple form like the case of qubits. To use the simple form of fidelity, we note that in [6], the authors introduce a new fidelity, called super-fidelity, and proved that Uhlmann-Jozsa fidelity can be bounded above by the super-fidelity G(ρ1, ρ2): F(ρ1, ρ2) ≤ G(ρ1, ρ2)