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arXiv:2503.13979 · Physical Review A

Complexity of Bose-Einstein condensates at finite temperature

Chang-Yan Wang

Published Feb 17, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/8cv1-2xgh

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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)SU(1,1)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.
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.
It is known that there are infinitely many distinguishability metrics for mixed quantum states. This freedom, in turn, leads to metric-dependent interpretations of physically meaningful geometric quantities such as complexity and volume of quantum states. In this paper, we first present an explicit and unabridged mathematical discussion on the relation between the Sj\"oqvist metric and the Bures metric for arbitrary nondegenerate mixed quantum states, using the notion of decompositions of density operators by means of ensembles of pure quantum states. Then, to enhance our comprehension of the difference between these two metrics from a physics standpoint, we compare the formal expressions of these two metrics for arbitrary thermal quantum states specifying quantum systems in equilibrium with a reservoir at nonzero temperature. For illustrative purposes, we show the difference between these two metrics in the case of a simple physical system characterized by a spin-qubit in an arbitrarily oriented uniform and stationary external magnetic field in thermal equilibrium with a finite-temperature bath. Finally, we compare the Bures and Sj\"oqvist metrics in terms of their monotonicity property.
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 investigate the Uhlmann phase, a generalization of the celebrated Berry phase, for Bose-Einstein condensates (BECs) at finite temperature. The Uhlmann phase characterizes topological properties of mixed states, in contrast to the Berry phase, which is defined for pure states at zero temperature. Using the SU(1,1)\text{SU}(1,1)SU(1,1) symmetry of the Bogoliubov Hamiltonian, we derive a general formula for the Uhlmann phase of BECs. Numerical calculations reveal that the Uhlmann phase can differ from the Berry phase in the zero-temperature limit, contrary to previous studies. As the temperature increases, the Uhlmann phase exhibits a winding behavior, and we relate the total winding degree to the Berry phase. This winding indicates that the Uhlmann phase takes values on a Riemann surface. Furthermore, we propose an experimental scheme to measure the Uhlmann phase of BECs by purifying the density matrix using an atomic interferometer.
It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., incomplete information) about a quantum system, while pure quantum states are states of perfect knowledge (i.e., complete information) with vanishing von Neumann entropy. In this paper, we propose an information geometric theoretical construct to describe and, to a certain extent, understand the complex behavior of evolutions of quantum systems in pure and mixed states. The comparative analysis is probabilistic in nature, it uses a complexity measure that relies on a temporal averaging procedure along with a long-time limit, and is limited to analyzing expected geodesic evolutions on the underlying manifolds. More specifically, we study the complexity of geodesic paths on the manifolds of single-qubit pure and mixed quantum states equipped with the Fubini-Study metric and the Sj\"oqvist metric, respectively. We analytically show that the evolution of mixed quantum states in the Bloch ball is more complex than the evolution of pure states on the Bloch sphere. We also verify that the ranking based on our proposed measure of complexity, a quantity that represents the asymptotic temporal behavior of an averaged volume of the region explored on the manifold during the evolution of the systems, agrees with the geodesic length-based ranking. Finally, focusing on geodesic lengths and curvature properties in manifolds of mixed quantum states, we observed a softening of the complexity on the Bures manifold compared to the Sj\"oqvist manifold.
The geometry and topology of quantum states are intimately related at zero temperature through exact bounds that constrain geometric quantities from below by topological invariants. At finite-temperature, however, the analogous relations remain unclear. Here we establish rigorous geometric lower bounds for one-dimensional (1D) chiral-symmetric systems at finite temperature within the Uhlmann's framework for mixed states. We show that the Bures length is bounded by a continuous geometric phase angle. We further derive a temperature-dependent bound that interpolates between the zero-temperature limit and a trivial high-temperature regime. Our results are verified analytically and numerically using the Su-Schrieffer-Heeger (SSH) model and the spinless Kitaev chain model. Finally, we discuss potential ways to detect the geometry of the density matrix in quantum circuits, with the quantum imaginary time evolution (QITE) method.