When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.
When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension d and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.
We investigate the geometric quantum complexity of Bose-Einstein condensate (BEC) at finite temperature. Specifically, we use the Bures and Sjöqvist metrics -- generalizations of the Fubini-Study metric for mixed quantum states, as well as the Nielsen geometric complexity approach based on purification of mixed states. Starting from the Bogoliubov Hamiltonian of BEC, which exhibits an SU(1,1) symmetry, we explicitly derive and compare the complexities arising from these three distinct measures. For the Bures and Sjöqvist metrics, analytical and numerical evaluations of the corresponding geodesics are provided, revealing characteristic scaling behaviors with respect to temperature. In the Nielsen complexity approach, we rigorously handle the gauge freedoms associated with mixed state purification and non-uniqueness unitary operations, demonstrating that the resulting complexity aligns precisely with the Bures metric. Our work provides a comparative study of the geometric complexity of finite-temperature Bose-Einstein condensates, revealing its intimate connections to symmetry structures and temperature effects in BEC systems.
The quantum speed limit establishes a bound on the minimal time required for a quantum system to evolve from a given initial state to a final state. When the mean energy variance is fixed this limitation is captured by the Mandelstam--Tamm bound. The fastest quantum evolution saturating this bound follows a geodesic arc connecting the two states. Such geodesics in the manifold of quantum states are explicitly known when the states are pure (Fubini-Study geodesics) and when they are mixed and given by faithful density matrices (Bures geodesics). In this article we obtain the explicit form of the Bures geodesic arcs joining two non-faithful density matrices, which may have different ranks. For pure states one recovers the Fubini-Study geodesics. A necessary and sufficient condition for the uniqueness of the shortest geodesic arc is given. When the condition is not fulfilled there are infinitely many such arcs, all having length equal to the arccos Bures distance between the two states, in analogy with the arcs of great circles connecting the two poles of a sphere. We discuss the implications of our results for the quantum speed limit.
The quantum geometric tensor (QGT) reveals local geometric properties and associated topological information of quantum states. Here a generalization of the QGT to mixed quantum states at finite temperatures based on the Sj\"oqvist distance is developed. The resulting Sj\"oqvist QGT is invariant under gauge transformations of individual spectrum levels of the density matrix. A Pythagorean-like relation connects the distances and gauge transformations, which clarifies the role of the parallel-transport condition. The real part of the QGT naturally decomposes into a sum of the Fisher-Rao metric and Fubini-Study metric, allowing a distinction between different contributions to the quantum distance. The imaginary part of the QGT is proportional to a weighted summation of the Berry curvatures, which leads to a geometric phase for mixed states under certain conditions. We present three examples of different dimensions to illustrate the temperature dependence of the QGT and a discussion on possible implications.
We compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.
We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators ρ1 and ρ2 in the base space M is given by the shortest distance in the (Hilbert Schmidt) bundle space E of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in E of the geodesic between the ρ1 and ρ2 in M. The horizontal lift is that unique curve in E that orthogonally traverses the fibers F⊂E above the curve in M, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states ρ(s) along the base space geodesic parameterized by affine parameter s between the initial ρ1 and final ρ2 states. We emphasize the role played by geometric mean operator M(s)=ρ1−1/2ρ11/2ρ(s)ρ11/2ρ1−1/2, where the Uhlmann root fidelity between ρ1 and ρ(s) is given by F(ρ1,ρ(s))=Tr[M(s)ρ1]=Tr[ρ11/2ρ(s)ρ11/2], and ρ(s)=M(s)ρ1M(s). We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states ρ(p)=(1−p)I/N+p∣Ψ⟩⟨Ψ∣ with ∣Ψ⟩={∣GHZ⟩,∣W⟩} in dimension N=23. For the latter, we compare expressions in the limit p→1 to the infinite number of possible geodesics between the orthogonal pure states ∣GHZ⟩ and ∣W⟩. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension N=2.