# A tight Cram\'er-Rao bound for joint parameter estimation with a pure two-mode squeezed probe

## TL;DR

This paper derives the exact quantum Cramér-Rao bound for estimating displacement in a two-mode squeezed vacuum state, showing that a dual homodyne measurement achieves this bound, confirming its optimality.

## Contribution

It provides a tight, explicit Cramér-Rao bound for joint displacement estimation using two-mode squeezed states, demonstrating the optimality of dual homodyne measurement.

## Key findings

- The Holevo Cramér-Rao bound equals 4exp(-2r).
- Dual homodyne measurement achieves the bound.
- The bound is tight and optimal.

## Abstract

We calculate the Holevo Cram\'er-Rao bound for estimation of the displacement experienced by one mode of an two-mode squeezed vacuum state with squeezing r and find that it is equal to 4exp(-2r). This equals the sum of the mean squared error obtained from a dual homodyne measurement, indicating that the bound is tight and that the dual homodyne measurement is optimal.