Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

A geometric representation of entanglement

Shahabeddin Mostafanazhad aslmarand, Warner A. Miller

Published Jun 23, 2021• 0 citations
OpenAlex

Abstract

In this paper, we introduce a novel way of witnessing entanglement in quantum systems. This geometrical approach will enable us to differentiate systems with high quantum correlation from systems with low quantum correlation. We will show using this geometrical approach to quantum entanglement; one can address the entanglement between specific parts of the quantum network without the necessity to calculate all pairwise entanglement between nodes in the network. We will also show that for particular quantum networks, this geometrical approach will be the geometrical realization of squashed entanglement. Our approach is inspired by Schumacher's singlet state triangle inequality, that used an information geometry-based entropic distance, however, unlike Schumacher, who uses classical entropy, we will use von Neumann entropy to reach a new inequality and then will generalize this inequality using entropic areas and volumes and higher dimensional volumes for multipartite quantum systems.

Related papers

Filtering of higher-dimensional entanglement networks using information volumes

Shahabeddin Mostafanazhad aslmarand, Warner A. Miller, Doyeol Doyeol +2 more

We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher's singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher-dimensional volumes for multipartite quantum systems. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enables us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well-motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach, and in particular to a random sample of a thousand density matrices.

arXiv:2106.12508•Jun 23, 2021•0 citations

Quantum information geometry in the space of measurements

Warner A. Miller

We introduce a new approach to evaluating entangled quantum networks using information geometry. Quantum computing is powerful because of the enhanced correlations from quantum entanglement. For example, larger entangled networks can enhance quantum key distribution (QKD). Each network we examine is an n-photon quantum state with a degree of entanglement. We analyze such a state within the space of measured data from repeated experiments made by n observers over a set of identically-prepared quantum states – a quantum state interrogation in the space of measurements. Each observer records a 1 if their detector triggers, otherwise they record a 0. This generates a string of 1’s and 0’s at each detector, and each observer can define a binary random variable from this sequence. We use a well-known information geometry-based measure of distance that applies to these binary strings of measurement outcomes,1–3 and we introduce a generalization of this length to area, volume and higher-dimensional volumes.4 These geometric equations are defined using the familiar Shannon expression for joint and mutual entropy.5 We apply our approach to three distinct tripartite quantum states: the |GHZi state, the |Wi state, and a separable state |Pi. We generalize a well-known information geometry analysis of a bipartite state to a tripartite state. This approach provides a novel way to characterize quantum states, and it may have favorable scaling with increased number of photons.

arXiv:1812.09810•SPIE Quantum Information Science, Sensing, and Computation•May 16, 2018•6 citations

A Geometry of entanglement and entropy

Ramita Sarkar, Soumik Mahanti, Prasanta Kumar Panigrahi

This paper explores the fundamental relationship between the geometry of entanglement and von Neumann entropy, shedding light on the intricate nature of quantum correlations. We provide a comprehensive overview of entanglement, highlighting its crucial role in quantum mechanics. Our focus centers on the connection between entanglement, von Neumann entropy, a measure of the information content within quantum systems and the geometry of composite Hilbert spaces. We discuss various methods for quantifying and characterizing entanglement through a geometric perspective and elucidate how this connection unveils the nature of quantum entanglement, offering valuable insights into the underlying structure of quantum systems. This study underscores the significance of geometry as a key tool for understanding the rich landscape of quantum correlations and their implications across various domains of physics and information theory. An example of entanglement as an indispensable resource for the task of state teleportation is presented at the end.

arXiv:2402.15880•Feb 24, 2024•0 citations

Quantum Entanglement Modeling via Geometric Formalism

Jincheng Zhang

Quantum entanglement, a fundamental phenomenon in quantum mechanics, presents significant challenges in information processing and secure communication. Current modeling approaches often struggle to adequately capture the intricate dynamics of entangled states, particularly when dealing with complex correlations. This paper proposes a novel geometric model for quantum entanglement, aiming to provide a more intuitive and mathematically rigorous framework for analyzing entanglement's evolution and measurement. We develop a model based on the concept of entangled states as geometric objects, utilizing a complex structure to represent and track the correlations between entangled particles. This model offers a potentially more powerful and flexible approach to understanding and manipulating quantum entanglement, with implications for quantum information processing.

•Sep 6, 2026•0 citations

Quantum Reactivity: An Indicator of Quantum Correlation

Shahabeddin Mostafanazhad aslmarand, Warner A. Miller, Verinder S. Rana +1 more

Geometry is often a valuable guide to complex problems in physics. In this paper, we introduce a novel geometric quantity called quantum reactivity (QR) to probe quantum correlations in higher-dimensional quantum systems. Much like quantum discord, QR is not a measure of quantum entanglement but can be useful in quantum information processes where a notion of quantum correlation in higher dimensions is needed. Both quantum discord and QR are extendable to an arbitrarily large number of qubits; however, unlike discord, QR satisfies the invariance under unitary operations. Our approach parallels Schumacher's singlet state triangle inequality, which used an information geometry-based entropic distance. We use a generalization of information distance to area, volume, and higher-dimensional volumes and then use these to define a quantity that we call QR, which is the familiar ratio of surface area to volume. We examine a spectrum of multipartite states (Werner, W, GHZ, randomly generated density matrices, etc.) and demonstrate that QR can provide an ordering of these quantum states as to their degree of quantum correlation.

arXiv:1902.02391•Entropy•Dec 19, 2019•7 citations

Experimental realization of Schumacher's information geometric Bell inequality

Tahereh Rezaei, Shahabeddin Mostafanazhad aslmarand, Robert Snyder +5 more

Quantum mechanics can produce correlations that are stronger than classically allowed. This stronger–than–classical correlation is the “fuel” for quantum computing. In 1991 Schumacher forwarded a beautiful geometric approach, analogous to the well-known result of Bell, to capture non-classicality of this correlation for a singlet state. He used a well-established information distance defined on an ensemble of identically–prepared states. He calculated that for certain detector settings used to measure the entangled state, the resulting geometry violated a triangle inequality — a violation that is not possible classically. This provided a novel information–based geometric Bell inequality in terms of a “covariance distance.” Here we experimentally reproduce his construction and demonstrate a definitive violation for a Bell state of two photons based on the usual spontaneous parametric down-conversion in a paired BBO crystal. The state we produced had a visibility of Vad=0.970. We discuss generalizations to higher dimensional multipartite quantum states.

arXiv:2106.00194•Physics Letters A•May 20, 2021•9 citations
In this paper, we introduce a novel way of witnessing entanglement in quantum systems. This geometrical approach will enable us to differentiate systems with high quantum correlation from systems with low quantum correlation. We will show using this geometrical approach to quantum entanglement; one can address the entanglement between specific parts of the quantum network without the necessity to calculate all pairwise entanglement between nodes in the network. We will also show that for particular quantum networks, this geometrical approach will be the geometrical realization of squashed entanglement. Our approach is inspired by Schumacher's singlet state triangle inequality, that used an information geometry-based entropic distance, however, unlike Schumacher, who uses classical entropy, we will use von Neumann entropy to reach a new inequality and then will generalize this inequality using entropic areas and volumes and higher dimensional volumes for multipartite quantum systems.
We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher's singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher-dimensional volumes for multipartite quantum systems. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enables us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well-motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach, and in particular to a random sample of a thousand density matrices.
We introduce a new approach to evaluating entangled quantum networks using information geometry. Quantum computing is powerful because of the enhanced correlations from quantum entanglement. For example, larger entangled networks can enhance quantum key distribution (QKD). Each network we examine is an n-photon quantum state with a degree of entanglement. We analyze such a state within the space of measured data from repeated experiments made by n observers over a set of identically-prepared quantum states – a quantum state interrogation in the space of measurements. Each observer records a 1 if their detector triggers, otherwise they record a 0. This generates a string of 1’s and 0’s at each detector, and each observer can define a binary random variable from this sequence. We use a well-known information geometry-based measure of distance that applies to these binary strings of measurement outcomes,1–3 and we introduce a generalization of this length to area, volume and higher-dimensional volumes.4 These geometric equations are defined using the familiar Shannon expression for joint and mutual entropy.5 We apply our approach to three distinct tripartite quantum states: the |GHZi state, the |Wi state, and a separable state |Pi. We generalize a well-known information geometry analysis of a bipartite state to a tripartite state. This approach provides a novel way to characterize quantum states, and it may have favorable scaling with increased number of photons.
This paper explores the fundamental relationship between the geometry of entanglement and von Neumann entropy, shedding light on the intricate nature of quantum correlations. We provide a comprehensive overview of entanglement, highlighting its crucial role in quantum mechanics. Our focus centers on the connection between entanglement, von Neumann entropy, a measure of the information content within quantum systems and the geometry of composite Hilbert spaces. We discuss various methods for quantifying and characterizing entanglement through a geometric perspective and elucidate how this connection unveils the nature of quantum entanglement, offering valuable insights into the underlying structure of quantum systems. This study underscores the significance of geometry as a key tool for understanding the rich landscape of quantum correlations and their implications across various domains of physics and information theory. An example of entanglement as an indispensable resource for the task of state teleportation is presented at the end.
Quantum entanglement, a fundamental phenomenon in quantum mechanics, presents significant challenges in information processing and secure communication. Current modeling approaches often struggle to adequately capture the intricate dynamics of entangled states, particularly when dealing with complex correlations. This paper proposes a novel geometric model for quantum entanglement, aiming to provide a more intuitive and mathematically rigorous framework for analyzing entanglement's evolution and measurement. We develop a model based on the concept of entangled states as geometric objects, utilizing a complex structure to represent and track the correlations between entangled particles. This model offers a potentially more powerful and flexible approach to understanding and manipulating quantum entanglement, with implications for quantum information processing.
Geometry is often a valuable guide to complex problems in physics. In this paper, we introduce a novel geometric quantity called quantum reactivity (QR) to probe quantum correlations in higher-dimensional quantum systems. Much like quantum discord, QR is not a measure of quantum entanglement but can be useful in quantum information processes where a notion of quantum correlation in higher dimensions is needed. Both quantum discord and QR are extendable to an arbitrarily large number of qubits; however, unlike discord, QR satisfies the invariance under unitary operations. Our approach parallels Schumacher's singlet state triangle inequality, which used an information geometry-based entropic distance. We use a generalization of information distance to area, volume, and higher-dimensional volumes and then use these to define a quantity that we call QR, which is the familiar ratio of surface area to volume. We examine a spectrum of multipartite states (Werner, W, GHZ, randomly generated density matrices, etc.) and demonstrate that QR can provide an ordering of these quantum states as to their degree of quantum correlation.
Quantum mechanics can produce correlations that are stronger than classically allowed. This stronger–than–classical correlation is the “fuel” for quantum computing. In 1991 Schumacher forwarded a beautiful geometric approach, analogous to the well-known result of Bell, to capture non-classicality of this correlation for a singlet state. He used a well-established information distance defined on an ensemble of identically–prepared states. He calculated that for certain detector settings used to measure the entangled state, the resulting geometry violated a triangle inequality — a violation that is not possible classically. This provided a novel information–based geometric Bell inequality in terms of a “covariance distance.” Here we experimentally reproduce his construction and demonstrate a definitive violation for a Bell state of two photons based on the usual spontaneous parametric down-conversion in a paired BBO crystal. The state we produced had a visibility of Vad=0.970. We discuss generalizations to higher dimensional multipartite quantum states.