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Equivalence relations between conical 2-designs and mutually unbiased generalized equiangular tight frames

Katarzyna Siudzińska

TL;DR

This work establishes a close formal link between conical 2-designs and generalized equiangular tight frames (GETFs), with a main focus on when these structures align with mutually unbiased GETFs (MU GETFs). It proves that any GETF with M = d^2 is a homogeneous conical 2-design and, conversely, that a conical 2-design of linearly independent operators yields a GETF, thereby giving a one-to-one correspondence in the homogeneous case. Extending to collections of GETFs (MU GETFs) and to informationally overcomplete constructions, the authors derive conditions under which MU GETFs are equidistant and form conical 2-designs, and they express κ_± in terms of conventional GETF parameters and the index of coincidence. They also show that equidistant MU GETFs can be used to realize conical 2-designs beyond the homogeneous setting, with exact relations between parameters κ_+, κ_− and μ, S that govern the overlaps and state-purity relations. The work reveals rich families of conical 2-designs that are not MU GETFs and lists open questions on broader constructions and applications to quantum information tasks such as entropic uncertainty and state tomography.

Abstract

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.

Equivalence relations between conical 2-designs and mutually unbiased generalized equiangular tight frames

TL;DR

This work establishes a close formal link between conical 2-designs and generalized equiangular tight frames (GETFs), with a main focus on when these structures align with mutually unbiased GETFs (MU GETFs). It proves that any GETF with M = d^2 is a homogeneous conical 2-design and, conversely, that a conical 2-design of linearly independent operators yields a GETF, thereby giving a one-to-one correspondence in the homogeneous case. Extending to collections of GETFs (MU GETFs) and to informationally overcomplete constructions, the authors derive conditions under which MU GETFs are equidistant and form conical 2-designs, and they express κ_± in terms of conventional GETF parameters and the index of coincidence. They also show that equidistant MU GETFs can be used to realize conical 2-designs beyond the homogeneous setting, with exact relations between parameters κ_+, κ_− and μ, S that govern the overlaps and state-purity relations. The work reveals rich families of conical 2-designs that are not MU GETFs and lists open questions on broader constructions and applications to quantum information tasks such as entropic uncertainty and state tomography.

Abstract

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.

Paper Structure

This paper contains 13 sections, 12 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Conical 2-designs
  3. Generalized equiangular tight frames
  4. Conical 2-designs vs. generalized equiangular tight frames
  5. Mutually unbiased generalized equiangular tight frames
  6. Conical 2-designs vs. mutually unbiased GETFs
  7. Conclusions
  8. Acknowledgements
  9. Properties of generalized equiangular tight frames
  10. Linearly independent subset of mutually unbiased GETFs
  11. Admissible range of $S$
  12. Trace parameters for conical 2-designs in Proposition 6
  13. Consequences of semi-positivity of $R_{\alpha,k}$

Key Result

Proposition 1

If $\mathcal{P}=\{P_k;\,k=1,\ldots,d^2\}$ is a generalized equiangular tight frame, then it is a conical 2-design.

Theorems & Definitions (28)

  • Definition 1
  • Example 1
  • Example 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 2
  • Proposition 3
  • ...and 18 more