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The monotonicity conjecture for N-volume quantum Fisher information

Fix N∈NN\in{\mathbb N}, ρ∈Dn1\rho\in{\cal D}_n^1, and A1,…,AN∈Mn,saA_1,\ldots,A_N\in M_{n,sa}. For f∈Fop rf\in{\cal F}_{op}^{\,r} define

V(f):=(f(0)2)N2Volρf(i[ρ,A1],…,i[ρ,AN]).V(f):=\left(\frac{f(0)}{2}\right)^{\frac{N}{2}}{\rm Vol}_{\rho}^f(i[\rho,A_1],\ldots,i[\rho,A_N]).

Here f~\widetilde f denotes the auxiliary transform used to order regular normalized symmetric operator monotone functions.

Monotonicity conjecture. For any f,g∈Fop rf,g\in{\cal F}_{op}^{\,r},

f~≤g~⟹V(f)≥V(g).\widetilde f\leq\widetilde g\quad\Longrightarrow\quad V(f)\geq V(g).

This predicts that the scaled quantum Fisher-information volume is order-reversing with respect to the transformed operator monotone functions. The source does not specify a resolution status.

Sources & referencesView supporting material

Primary source

P. Gibilisco, D. Imparato and T. Isola, “A volume inequality for quantum Fisher information and the uncertainty principle”, arXiv:0706.0791 (2007).

operator-monotone-functionsquantum-information

Progress summary

Refreshed
Partially solved

The conjecture remains unproved in general, although it is established for volumes involving at most three observables.

A 2007 preprint formulates the conjecture for arbitrary NN and self-adjoint matrices, asserting that the scaled quantum Fisher-information volume decreases under the stated order of the transformed functions.

Known results

  • The corresponding inequality was proved for N=1N=1 and N=2N=2 through joint work by several authors.
  • The 2007 preprint proves the real case N=3N=3.
  • A 2008 paper records a determinant inequality whose validity would imply the general conjecture, but does not establish it.
  • A related determinant-of-correlation-matrices proposition is reported, without being identified as a proof of the conjecture.

Current status (as of August 2026): The cases N=1,2,3N=1,2,3 are settled, while the conjecture for arbitrary NN remains open; no public proof, counterexample, or later resolution was found.

Sources

Solutions 0

No solutions have been posted yet.