On the shape of correlation matrices for unitaries
DOI:
https://doi.org/10.7146/math.scand.a-142800Abstract
For a positive integer $n$, we study the collection $\mathcal {F}_{\mathrm {fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\leq i,j\leq n$, can be written as $a_{ij}=\tau (U_j^*U_i)$ for some $n$-tuple $U_1, U_2, …, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal {M}$ with tracial state τ. We show that $\mathcal {F}_{\mathrm {fin}}(n)$ is not closed for every $n\geq 8$. This improves a result by Musat and R{ø}rdam which states the same for $n\geq 11$.
References
Bhat, B. V. S., Nayak, S., and Shankar, P., On products of symmetries in von Neumann algebras, J. Operator Theory, to appear. https://doi.org/10.48550/arXiv.2204.00009
Dykema, K., and Juschenko, K., Matrices of unitary moments, Math. Scand. 109 (2011), no. 2, 225–239. https://doi.org/10.7146/math.scand.a-15186
Dykema, K., Paulsen, V., and Prakash, J., Non-closure of the set of quantum correlations via graphs, Comm. Math. Phys. 365 (2019), no. 3, 1125–1142. https://doi.org/10.1007/s00220-019-03301-1
Ji, Z., Natarajan, A., Vidick, T., Wright, J., and Yuen, H., MIP^*=RE, Preprint, arXiv:2001.04383. https://doi.org/10.48550/arXiv.2001.04383
Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. https://doi.org/10.1007/BF01232444
Kruglyak, S. A., Rabanovich, V. I., and Samou ılenko, Y. S., On sums of projections, Funktsional. Anal. i Prilozhen. 36 (2002), no. 3, 20–35, 96; translation in Funct. Anal. Appl. 36 (2002), no. 3, 182–195. https://doi.org/10.1023/A:1020193804109
Musat, M., and Rørdam, M., Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla, with an appendix by Narutaka Ozawa, Comm. Math. Phys. 375 (2020), no. 3, 1761–1776. https://doi.org/10.1007/s00220-019-03449-w
Russell, T. B. Two-outcome synchronous correlation sets and Connes' embedding problem, Quantum Inf. Comput. 20 (2020), no. 5–6, 361–374.
