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arXiv:2312.12537 · Europhysics Letters (EPL)

Quantum steering ellipsoids and quantum obesity in critical systems

Pedro Rosário, Alan C. Santos

Published Nov 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Quantum obesity (QO) is a novel function introduced to quantify quantum correlations that go beyond traditional measures like entanglement, while also functioning as an entanglement witness. One of the key strengths of QO lies in its analyticity for arbitrary states of bipartite systems, making it a more accessible and versatile tool compared to other measures of quantum correlations, such as quantum discord. In this work, we highlight the importance of QO as a fundamental quantity for identifying signatures of quantum phase transitions, which are critical changes in the ground state of quantum systems driven by quantum fluctuations. We introduce a mechanism based on local filtering operations designed to enhance the critical behavior of QO near phase transition points, providing a deeper understanding of these phenomena. Furthermore, we present a theorem that characterizes how QO transforms under local quantum operations and classical communications (LOCC), which broadens its applicability to a wider range of quantum systems. This opens new avenues for exploring quantum criticality and other novel quantum phenomena by leveraging the analytically computable, pairwise QO, thus offering both theoretical insights and practical applications in quantum information science.

DOI
10.1209/0295-5075/ad8f5e

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Abstract Quantum obesity (QO) is a novel function introduced to quantify quantum correlations that go beyond traditional measures like entanglement, while also functioning as an entanglement witness. One of the key strengths of QO lies in its analyticity for arbitrary states of bipartite systems, making it a more accessible and versatile tool compared to other measures of quantum correlations, such as quantum discord. In this work, we highlight the importance of QO as a fundamental quantity for identifying signatures of quantum phase transitions, which are critical changes in the ground state of quantum systems driven by quantum fluctuations. We introduce a mechanism based on local filtering operations designed to enhance the critical behavior of QO near phase transition points, providing a deeper understanding of these phenomena. Furthermore, we present a theorem that characterizes how QO transforms under local quantum operations and classical communications (LOCC), which broadens its applicability to a wider range of quantum systems. This opens new avenues for exploring quantum criticality and other novel quantum phenomena by leveraging the analytically computable, pairwise QO, thus offering both theoretical insights and practical applications in quantum information science.
This paper investigates quantum obesity (QO), quantum discord (QD), and the quantum steering ellipsoid (QSE) for bipartite Gisin states subjected to Garfinkle-Horowitz-Strominger (GHS) dilation of spacetime on the second qubit. These three quantifiers are introduced to characterize quantum correlations beyond entanglement and can also function as entanglement witnesses. Our results demonstrate a monotonic decrease in the physical accessibility of both QD and QO as the dilation parameter increases within the region-I of the second qubit. Conversely, in the anti-particle region, the accessibility of QD and QO stabilizes at finite values of the dilation parameter owing to the influence of the Pauli exclusion principle and Fermi-Dirac statistics, subsequently increasing gradually. Notably, the QSE in the region-I expands as the Dirac field frequency rises and the dilation parameter diminishes, while the opposite trend is observed in the anti-particle region.
Quantum crossover phenomena play a central role in few-body open quantum systems, yet their identification often relies on global or model-dependent indicators. In this work, we demonstrate that crossovers can be robustly characterized through purely local measurements, establishing a direct connection between local quantum Fisher information and the onset of crossover behavior. We further demonstrate that quantum obesity does not, in general, generalize the quantum steering ellipsoid volume as a universal indicator of crossover. Instead, we identify regimes in which the ellipsoid volume remains insensitive to the transition, while the relevant signatures are encoded in the behavior of the local Bloch vector. These results reveal a geometric distinction between local and global indicators of crossovers.
Abstract This paper investigates quantum obesity, quantum discord, and the quantum steering ellipsoid (QSE) for bipartite Gisin states that are subjected to Garfinkle–Horowitz–Strominger (GHS) dilation of spacetime on the second qubit. These three quantifiers are introduced to characterize quantum correlations beyond entanglement and can also function as entanglement witnesses. A monotonic decrease in the physical accessibility of both quantum discord as demonstrate in the results and quantum obesity as the dilation parameter increases within the region‐I of the second qubit. Conversely, in the anti‐particle region, the accessibility of quantum discord and quantum obesity stabilizes at finite values of the dilation parameter owing to the influence of the Pauli exclusion principle and Fermi‐Dirac statistics, subsequently increasing gradually. Notably, the QSE in the region‐I expand as the Dirac field frequency rises and the dilation parameter diminishes, while the opposite trend is observed in the anti‐particle region.
Graphical representations are invaluable for visualising physical systems and processes. In quantum information theory, the Bloch vector representation of a single qubit is ubiquitous, but visualising higher-dimensional quantum systems is far less straightforward. The quantum steering ellipsoid provides a method for geometrically representing the state of two qubits, the most fundamental system for studying quantum correlations. This thesis constitutes a significant development of the steering ellipsoid formalism. As well as offering new insight into the study of two-qubit states, we extend this powerful geometric approach to explore scenarios beyond two qubits. We find necessary and sufficient conditions for when an ellipsoid inside the Bloch ball describes a valid (i.e. positive semidefinite) two-qubit state. Combined with the notion of ellipsoid chirality, this enables a geometric characterisation of entanglement. We find a family of "maximally obese" two-qubit states whose ellipsoids have maximal volume. These states have optimal correlation properties within the set of all two-qubit states with a single maximally mixed marginal. We study a three-qubit scenario and discover that ellipsoid volume obeys an elegant monogamy of steering relationship. From this we can derive the Coffman-Kundu-Wootters (CKW) inequality for concurrence monogamy, providing an intuitive geometric derivation of this classic result. Remarkably, we find that steering ellipsoids offer a fresh perspective on questions beyond quantum state space. Entanglement witnesses are also very naturally represented and classified using the formalism. This gives a physical interpretation to any ellipsoid inside the Bloch ball as a block positive two-qubit operator, which we may then classify further. We can also use steering ellipsoids to derive some highly nontrivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.
Abstract The swapping of various quantum correlation measures is addressed including, Bell‐nonlocality, Einstein–Podolsky–Rosen steering, usefulness for teleportation, and entanglement, as well as quantum obesity as a quantum correlation more general than entanglement. The effects that local filtering operations have on the swapping of such correlations are particularly focused. Based on a generalization of entanglement swapping with arbitrary two‐qubit states and arbitrary measurements, analytical formulae for the swapping of two‐qudit as well as multi‐qubit quantum states are derived and the exact amount of quantum obesity that is swapped is provided. Then, in order to enhance the final amount of correlations in the post‐swapping two‐qubit state, the action of local filters before as well as after the swapping protocol is explored. In this regard, it is conjectured that applying local filters first is more efficient. This is supported by an analytical proof on the quantum obesity of almost Bell‐diagonal states, as well as numerical evidence for all the considered quantum correlations of arbitrary X‐form states.
The past two decades have witnessed a surge of interest in borrowing tools from quantum information theory to investigate quantum phase transitions (QPTs). The best-known examples are entanglement measures whose nonanalyticities at critical points were tied to QPTs in a plethora of physical models. Here, focusing on the XXZXXZXXZ model, we show how QPTs can be revealed through the quantum steering ellipsoid (QSE), which is a geometric tool capable of characterizing both the strength and type of quantum correlations between two subsystems of a compound system. We find that the QSE associated with the XXZXXZXXZ model changes in shape with the QPTs; that is, it is a needle in the ferromagnetic phase, an oblate spheroid in the gapless phase, and a prolate spheroid in the antiferromagnetic phase. This finding offers an example demonstrating the intriguing possibility of unveiling QPTs in a geometrically visible fashion. Some connections between our results and previous ones are discussed.