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Conical Designs

Matthew A. Graydon, D. M. Appleby

Published Jul 19, 2015• 0 citations
OpenAlex

Abstract

Complex projective t-designs, particularly SICs and full sets of MUBs, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They include arbitrary rank symmetric informationally complete measurements (SIMs) and full sets of arbitrary rank mutually unbiased measurements (MUMs). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical 2-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture SIMs and full sets of MUMs form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical 2-design. This suggests a way to search for new kinds of projective 2-design.

Related papers

Quantum conical designs

Matthew A. Graydon, D. M. Appleby

Abstract Complex projective t -designs, particularly sic s and full sets of mutually unbiased bases, play an important role in quantum information. We introduce a generalization which we call conical t -designs. They include arbitrary rank symmetric informationally complete measurements ( sim s) and full sets of arbitrary rank mutually unbiased measurements ( mum s). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical two-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture sim s and full sets of mum s form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical two-design. This suggests a way to search for new kinds of projective two-design.

arXiv:1507.05323•Journal of Physics A Mathematical and Theoretical•Jan 18, 2016•39 citations

Entanglement and designs

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Abstract We describe a connection between entanglement and designs. It involves the conical two-designs introduced in a previous paper. These are a generalization of projective two-designs which includes full sets of arbitrary rank mutually unbiased measurements ( mum s) and arbitrary rank symmetric informationally complete measurements ( sim s), as well as the more familiar mub s and sic s. We show that a povm is a conical two-design if and only if there exists what we call a regular entanglement monotone whose restriction to the pure states is a function of the norm of the probability vector. In that case the concurrence is such a monotone. We also generalize and develop previous work on designs and entanglement detection.

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Equivalence Relations Between Conical 2-Designs and Mutually Unbiased Generalized Equiangular Tight Frames

Katarzyna Siudzińska

Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually unbiased bases (MUBs). In this paper, we establish a one-to-one correspondence between conical 2-designs and mutually unbiased generalized equiangular tight frames, both of which are common generalizations of SIC POVMs and MUBs to operators of arbitrary rank. It turns out that there exist rich families of operators that belong to only one of those two classes. This raises important questions about which symmetries have to be preserved for applicational prominence.

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Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs

Jakub Czartowski, Dardo Goyeneche, Markus Grassl +1 more

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.

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QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY

Gerhard Zauner

This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).

International Journal of Quantum Information•Feb 1, 2011•289 citations

Mutually unbiased bases, spherical designs, and frames

Andreas Klappenecker, Martin Roetteler

The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 18, 2005•24 citations
Complex projective t-designs, particularly SICs and full sets of MUBs, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They include arbitrary rank symmetric informationally complete measurements (SIMs) and full sets of arbitrary rank mutually unbiased measurements (MUMs). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical 2-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture SIMs and full sets of MUMs form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical 2-design. This suggests a way to search for new kinds of projective 2-design.
Abstract Complex projective t -designs, particularly sic s and full sets of mutually unbiased bases, play an important role in quantum information. We introduce a generalization which we call conical t -designs. They include arbitrary rank symmetric informationally complete measurements ( sim s) and full sets of arbitrary rank mutually unbiased measurements ( mum s). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical two-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture sim s and full sets of mum s form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical two-design. This suggests a way to search for new kinds of projective two-design.
Abstract We describe a connection between entanglement and designs. It involves the conical two-designs introduced in a previous paper. These are a generalization of projective two-designs which includes full sets of arbitrary rank mutually unbiased measurements ( mum s) and arbitrary rank symmetric informationally complete measurements ( sim s), as well as the more familiar mub s and sic s. We show that a povm is a conical two-design if and only if there exists what we call a regular entanglement monotone whose restriction to the pure states is a function of the norm of the probability vector. In that case the concurrence is such a monotone. We also generalize and develop previous work on designs and entanglement detection.
Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually unbiased bases (MUBs). In this paper, we establish a one-to-one correspondence between conical 2-designs and mutually unbiased generalized equiangular tight frames, both of which are common generalizations of SIC POVMs and MUBs to operators of arbitrary rank. It turns out that there exist rich families of operators that belong to only one of those two classes. This raises important questions about which symmetries have to be preserved for applicational prominence.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.
This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.