nLab Kadison-Singer problem

Idea

Given a separable Hilbert space $\mathcal{H}$, such as the sequence space $\mathcal{l}_2$, write $B(\mathcal{H})$ for its C*-algebra of bounded linear operators and $D(\mathcal{H})$ for the subalgebra of diagonal operators.

The Kadison-Singer problem is the question:

Does every pure state $\psi$ on $D(\mathcal{H})$ extend to a state $\rho$ on $B(\mathcal{H})$?

$$\array{ D(\mathcal{H}) &\overset{\psi}{\longrightarrow}& \mathbb{C} \\ \downarrow & \nearrow_{\mathrlap{\exists \rho}} \\ B(\mathcal{H}) }$$

This was proven to be the case in Marcus-Spielman-Srivastava 13.

Related entries

Other theorems about the foundations and interpretation of quantum mechanics include:

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Fell's theorem

• Gleason's theorem

• Wigner theorem

• Bell's theorem

• Bub-Clifton theorem

References

The problem was stated in

• Richard V. Kadison, Isadore Singer, Extensions of pure states, American Journal of Mathematics, 81(2):383–400, 1959 (jstor:2372748)

and its solution was proven in

• Adam Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem, 2013. (arXiv:1306.3969)

Review includes

• Nicholas Harvey, An introduction to the Kadison-Singer conjecture and the paving conjecture (pdf)