We give a general solution to the question of when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density operators. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k -entangled states are also considered as examples of our techniques.We give a general solution to the question of when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density operators. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by the properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of the largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k -entangled states are also considered as examples of our techniques.

arXiv:1306.3187Journal of Physics A Mathematical and TheoreticalOct 7, 20133 citations

Cones with a mapping cone symmetry in the finite-dimensional case

Łukasz Skowronek

Linear Algebra and its ApplicationsFeb 16, 201115 citations

From Finite Distinguishability to Canonical Geometry: The John–Löwner Ellipsoid of a Comparison Body

Gustav Stieger

A finite public comparison doctrine supplies finite distinctions between states, not a metric. This paper shows how such finite comparison data nevertheless determine a canonical positive quadratic comparison form. Given a finite-dimensional real comparison-germ quotient $E_{\mathrm{pub}}$ with its chosen public comparison-difference normalization, form the closed centrally symmetric convex hull $K_{\mathrm{pub}}$ of those difference vectors. When $K_{\mathrm{pub}}$ is compact and spans $E_{\mathrm{pub}}$ , the John--L\"owner extremal ellipsoid theorem supplies a unique minimal-volume ellipsoid enclosing $K_{\mathrm{pub}}$ . Its unit balldefines a positive definite quadratic form $g_{\mathrm{pub}}$ on $E_{\mathrm{pub}}$ . The construction is functorial with respect to public symmetries: every linear automorphism preserving the comparison body $K_{\mathrm{pub}}$ preserves the quadratic form $g_{\mathrm{pub}}$ . Thus the metric is not postulated. It is the canonical quadratic envelope of the finite public distinguishability body. For the maximal comparison doctrine on a finite set, the construction yields the standard Euclidean form on the zero-sum comparison space, with the scale fixed by the public difference radius. This is the affine comparison geometry underlying the classical simplex before probabilities are added. A final outlook explains how the same convex-body mechanism interfaces with quantum comparison bodies and information geometry.A finite public comparison doctrine supplies finite distinctions between states, not a metric. This paper shows how such finite comparison data nevertheless determine a canonical positive quadratic comparison form. Given a finite-dimensional real comparison-germ quotient $E_{\mathrm{pub}}$ with its chosen public comparison-difference normalization, form the closed centrally symmetric convex hull $K_{\mathrm{pub}}$ of those difference vectors. When $K_{\mathrm{pub}}$ is compact and spans $E_{\mathrm{pub}}$, the John--L\"owner extremal ellipsoid theorem supplies a unique minimal-volume ellipsoid enclosing $K_{\mathrm{pub}}$. Its unit balldefines a positive definite quadratic form $g_{\mathrm{pub}}$ on $E_{\mathrm{pub}}$. The construction is functorial with respect to public symmetries: every linear automorphism preserving the comparison body $K_{\mathrm{pub}}$ preserves the quadratic form $g_{\mathrm{pub}}$. Thus the metric is not postulated. It is the canonical quadratic envelope of the finite public distinguishability body. For the maximal comparison doctrine on a finite set, the construction yields the standard Euclidean form on the zero-sum comparison space, with the scale fixed by the public difference radius. This is the affine comparison geometry underlying the classical simplex before probabilities are added. A final outlook explains how the same convex-body mechanism interfaces with quantum comparison bodies and information geometry.

May 9, 20260 citations

Generalized channels: Channels for convex subsets of the state space

Anna Jenčová

Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).

arXiv:1105.1899Journal of Mathematical PhysicsJan 1, 201235 citations

Quadratic convex reformulations for a class of complex quadratic programming problems

Cheng Zhu Lu, Gaojian Kang, Guangtai Qu +1 more

Computational Optimization and ApplicationsMar 12, 20250 citations

E_{\mathrm{pub}}

with its chosen public comparison-difference normalization, form the closed centrally symmetric convex hull

K_{\mathrm{pub}}

of those difference vectors. When

K_{\mathrm{pub}}

is compact and spans

E_{\mathrm{pub}}

, the John--L\"owner extremal ellipsoid theorem supplies a unique minimal-volume ellipsoid enclosing

K_{\mathrm{pub}}

. Its unit balldefines a positive definite quadratic form

g_{\mathrm{pub}}

E_{\mathrm{pub}}

. The construction is functorial with respect to public symmetries: every linear automorphism preserving the comparison body

K_{\mathrm{pub}}

preserves the quadratic form

g_{\mathrm{pub}}

. Thus the metric is not postulated. It is the canonical quadratic envelope of the finite public distinguishability body. For the maximal comparison doctrine on a finite set, the construction yields the standard Euclidean form on the zero-sum comparison space, with the scale fixed by the public difference radius. This is the affine comparison geometry underlying the classical simplex before probabilities are added. A final outlook explains how the same convex-body mechanism interfaces with quantum comparison bodies and information geometry.