is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions

N

; showing a bifurcation at

N=6

DOI: 10.26421/qic13.5-6-2

Related papers

Entanglement and output entropy of the diagonal map

Meik Hellmund

We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3 states $ω(z)$ and solve the case $z<0$ where $z$ is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions $N$ ; showing a bifurcation at N=6.We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3 states $ω(z)$ and solve the case $z<0$ where $z$ is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions $N$; showing a bifurcation at N=6.

arXiv:1206.6269Jun 27, 20120 citations

Concurrence and entanglement entropy of stochastic one-qubit maps

Meik Hellmund, Armin Uhlmann

Explicit expressions for the concurrence of all positive and trace-preserving (``stochastic'') one-qubit maps are presented. Our method allows to construct the relevant convex roof patterns. We conclude that two-component optimal decompositions always exist. Our results can be transferred to $2\ifmmode\times\else\texttimes\fi{}n$-quantum systems providing the concurrence for all rank-two density operators, as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of one-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.Explicit expressions for the concurrence of all positive and trace-preserving (``stochastic'') one-qubit maps are presented. Our method allows to construct the relevant convex roof patterns. We conclude that two-component optimal decompositions always exist. Our results can be transferred to $2\ifmmode\times\else\texttimes\fi{}n$-quantum systems providing the concurrence for all rank-two density operators, as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of one-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.

arXiv:0903.1340Physical Review AMay 15, 20097 citations

On properties of the space of quantum states and their application to the construction of entanglement monotones

Максим Евгеньевич Широков

We consider two properties of the set of quantum states as a convex topological space and some their implications concerning the notions of a convex hull and of a convex roof of a function defined on a subset of quantum states. By using these results we analyze two infinite-dimensional versions (discrete and continuous) of the convex roof construction of entanglement monotones, which is widely used in finite dimensions. It is shown that the discrete version may be 'false' in the sense that the resulting functions may not possess the main property of entanglement monotones while the continuous version can be considered as a 'true' generalized convex roof construction. We give several examples of entanglement monotones produced by this construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.We consider two properties of the set of quantum states as a convex topological space and some their implications concerning the notions of a convex hull and of a convex roof of a function defined on a subset of quantum states. By using these results we analyze two infinite-dimensional versions (discrete and continuous) of the convex roof construction of entanglement monotones, which is widely used in finite dimensions. It is shown that the discrete version may be 'false' in the sense that the resulting functions may not possess the main property of entanglement monotones while the continuous version can be considered as a 'true' generalized convex roof construction. We give several examples of entanglement monotones produced by this construction. In particular, we consider an infinite-dimensional generalization of the notion of Entanglement of Formation and study its properties.

arXiv:0804.1515Izvestiya MathematicsSep 7, 201032 citations

Entropy and optimal decompositions of states relative to a maximal commutative subalgebra

Armin Uhlmann

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix.To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix.

arXiv:quant-ph/9704017Apr 8, 199739 citations

Tangles of superpositions and the convex-roof extension

Andreas Osterloh, Jens Siewert, Armin Uhlmann

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $\mathrm{SL}(2,\mathbb{C})$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $\mathrm{SL}(2,\mathbb{C})$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.

arXiv:0710.5909Physical Review AMar 6, 200861 citations

Monogamy of the entanglement of formation

Yu Jing Guo, Gilad Gour

We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) is monogamous also on mixed tripartite states. To prove our results, we use the definition of monogamy without inequalities recently put forward [Gour and Guo, Quantum 2, 81 (2018)]. Our results promote the concept that monogamy of entanglement is a property of quantum entanglement and not an attribute of some particular measures of entanglement.We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) is monogamous also on mixed tripartite states. To prove our results, we use the definition of monogamy without inequalities recently put forward [Gour and Guo, Quantum 2, 81 (2018)]. Our results promote the concept that monogamy of entanglement is a property of quantum entanglement and not an attribute of some particular measures of entanglement.

arXiv:1809.08532Physical Review AApr 1, 201946 citations

N=3

states

\omega(z)

and solve the case

z<0

where

z

is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions

N

; showing a bifurcation at

N=6

ω(z)

and solve the case

z<0

where

z

is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions

N

; showing a bifurcation at N=6.

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is,

\mathrm{SL}(2,\mathbb{C})

invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.