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

arXiv:1606.02390 · Physical Review A

Volume monogamy of quantum steering ellipsoids for multiqubit systems

Shuming Cheng, Antony Milne, Michael J. W. Hall, Howard M. Wiseman

Published Oct 6, 2016• 51 citations
arXivHTMLPDF

Abstract

The quantum steering ellipsoid can be used to visualize 2-qubit states, and thus provides a generalization of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication. We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states (further conjectured to hold for the mixed case), and generalize our 3-qubit inequality to $n$ qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any 2-qubit state is nonincreasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multiqubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.

DOI
10.1103/physreva.94.042105

Related papers

Quantum steering ellipsoids, extremal physical states and monogamy

Antony Milne, Sania Jevtic, David H. Jennings +2 more

Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman–Kundu–Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.

arXiv:1403.0418•New Journal of Physics•Aug 7, 2014•92 citations

Experimental Validation of Quantum Steering Ellipsoids and Tests of Volume Monogamy Relations

Chao Zhang, Shuming Cheng, Li Li +8 more

The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid-called the quantum steering ellipsoid-in the Bloch ball. This ellipsoid provides a simple visual characterization of the initial two-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarization-entangled photonic qubit states. Moreover, for pure three-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarization and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.

arXiv:1809.08011•Physical Review Letters•Feb 22, 2019•51 citations

Quantum steering ellipsoids

Antony Milne

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.

•Aug 1, 2016•0 citations

Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering

B. G. Divyamani, I. Reena, Prasanta Kumar Panigrahi +2 more

Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and is useful for obtaining their correlation properties. The steering ellipsoids of two-qubit states that have undergone local operations on both the qubits, in order to bring the state to its canonical form, are the so-called canonical steering ellipsoids. The steering ellipsoids corresponding to the two-qubit subsystems of permutation-symmetric $N$-qubit states are considered here. We construct and analyze the geometric features of the canonical steering ellipsoids corresponding to pure permutation-symmetric $N$-qubit states with two distinct spinors. Depending on the degeneracy of the two spinors in the pure symmetric $N$-qubit state, several families arise which cannot be converted into one another through stochastic local operations and classical communication (SLOCC). The canonical steering ellipsoids of the two-qubit states drawn from the pure symmetric $N$-qubit states with two distinct spinors allow for a geometric visualization of the SLOCC-equivalent class of states. We show that the states belonging to the $W$ class correspond to oblate spheroids centered at $(0,0,1/(N\ensuremath{-}1))$ with fixed semiaxis lengths $1/\sqrt{N\ensuremath{-}1}$ and $1/(N\ensuremath{-}1)$. The states belonging to all other SLOCC-inequivalent families correspond to ellipsoids centered at the origin of the Bloch sphere. We also explore volume monogamy relations of states belonging to these families, mainly the $W$ class of states.

arXiv:2301.01714•Physical Review A•Apr 11, 2023•6 citations

Monogamy relations for quantum steering of multi-qubit states

Shuming Cheng, Xiaozhen Ge, Zairong Xi

Quantum steering plays a fundamental role in quantum information processing regime. In particular, there are two steering-related concepts: one is the quantum steering ellipsoid, which has been used to visualize the two-qubit state, and the other is the Einstein-Podolsky-Rosen (EPR) steering that is a nonlocal correlation between entanglement and Bell nonlocality. In this work, we study the monogamy phenomenon for these two steering properties of the general n-qubit states with n > 4. First, we conjecture that there is a universal tradeoff relation in terms of the spin correlation matrices of reduced bipartite states. From this relation, we can derive the similar monogamy relation for the volume of quantum steering ellipsoids and the linear EPR-steering witness.

Chinese Control Conference (CCC)•Jul 1, 2017•0 citations

Multipartite Gaussian steering: Monogamy constraints and quantum cryptography applications

Yu Xiang, Ioannis Kogias, Gerardo Adesso +1 more

We derive laws for the distribution of quantum steering among different parties in multipartite Gaussian states under Gaussian measurements. We prove that a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality holds quantitatively for a recently introduced measure of Gaussian steering. We then define the residual Gaussian steering, stemming from the monogamy inequality, as an indicator of collective steering-type correlations. For pure three-mode Gaussian states, the residual acts as a quantifier of genuine multipartite steering, and is interpreted operationally in terms of the guaranteed key rate in the task of secure quantum secret sharing. Optimal resource states for the latter protocol are identified, and their possible experimental implementation discussed. Our results pin down the role of multipartite steering for quantum communication.

arXiv:1603.08173•Physical Review A•Jan 12, 2017•181 citations
The quantum steering ellipsoid can be used to visualize 2-qubit states, and thus provides a generalization of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication. We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states (further conjectured to hold for the mixed case), and generalize our 3-qubit inequality to nnn qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any 2-qubit state is nonincreasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multiqubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.
Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman–Kundu–Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.
The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid-called the quantum steering ellipsoid-in the Bloch ball. This ellipsoid provides a simple visual characterization of the initial two-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarization-entangled photonic qubit states. Moreover, for pure three-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarization and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.
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.
Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and is useful for obtaining their correlation properties. The steering ellipsoids of two-qubit states that have undergone local operations on both the qubits, in order to bring the state to its canonical form, are the so-called canonical steering ellipsoids. The steering ellipsoids corresponding to the two-qubit subsystems of permutation-symmetric NNN-qubit states are considered here. We construct and analyze the geometric features of the canonical steering ellipsoids corresponding to pure permutation-symmetric NNN-qubit states with two distinct spinors. Depending on the degeneracy of the two spinors in the pure symmetric NNN-qubit state, several families arise which cannot be converted into one another through stochastic local operations and classical communication (SLOCC). The canonical steering ellipsoids of the two-qubit states drawn from the pure symmetric NNN-qubit states with two distinct spinors allow for a geometric visualization of the SLOCC-equivalent class of states. We show that the states belonging to the WWW class correspond to oblate spheroids centered at $(0,0,1/(N\ensuremath{-}1))$ with fixed semiaxis lengths $1/\sqrt{N\ensuremath{-}1}$ and $1/(N\ensuremath{-}1)$. The states belonging to all other SLOCC-inequivalent families correspond to ellipsoids centered at the origin of the Bloch sphere. We also explore volume monogamy relations of states belonging to these families, mainly the WWW class of states.
Quantum steering plays a fundamental role in quantum information processing regime. In particular, there are two steering-related concepts: one is the quantum steering ellipsoid, which has been used to visualize the two-qubit state, and the other is the Einstein-Podolsky-Rosen (EPR) steering that is a nonlocal correlation between entanglement and Bell nonlocality. In this work, we study the monogamy phenomenon for these two steering properties of the general n-qubit states with n > 4. First, we conjecture that there is a universal tradeoff relation in terms of the spin correlation matrices of reduced bipartite states. From this relation, we can derive the similar monogamy relation for the volume of quantum steering ellipsoids and the linear EPR-steering witness.
We derive laws for the distribution of quantum steering among different parties in multipartite Gaussian states under Gaussian measurements. We prove that a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality holds quantitatively for a recently introduced measure of Gaussian steering. We then define the residual Gaussian steering, stemming from the monogamy inequality, as an indicator of collective steering-type correlations. For pure three-mode Gaussian states, the residual acts as a quantifier of genuine multipartite steering, and is interpreted operationally in terms of the guaranteed key rate in the task of secure quantum secret sharing. Optimal resource states for the latter protocol are identified, and their possible experimental implementation discussed. Our results pin down the role of multipartite steering for quantum communication.