# How can one estimate the von Neumann entropy of an unknown quantum state?

Given many copies of some unknown quantum state $$\rho$$, I would like to compute its von Neumann entropy $$S(\rho)$$. What algorithm could be used for this that minimizes the number of copies required? We require that the estimate of the entropy has to be $$\varepsilon-$$close and one will need more copies as $$\varepsilon\rightarrow 0$$.

The naive solution is to do tomography and obtain a classical description of the state. This would require exponentially many copies as we increase the dimension of $$\rho$$. But the classical description of the state has a lot more information so perhaps there is a smarter way?

• My guess is to look at whether there are methods to estimate the spectrum of the quantum state that are more efficient than full tomography. Actually, after a quick search I found this work Measuring Quantum Entropy which deals with the exact problem you state. – Rammus Sep 1 at 16:34
• That's a tricky question, and it depends also whether you want epsilon to scale with N etc.. You could think e.g. about a series expansion of the entropy function in rho. On the other hand, measuring tr(rho^k) is not very efficient since the relative error is typically large. Generally, the entropy is not so easy to estimate well, even given many copies ... – Norbert Schuch Sep 1 at 19:36

Similar ideas with Quantum PCA may be useful. Meaning, apply Quantum Phase Estimation on unitary $$e^{-i\rho t}$$ to obtain estimates of the eigenvalues of $$\rho$$ and finally estimate von Neumann entropy as $$S(\rho) = -\sum \tilde{\lambda_i} \text{log}\tilde{\lambda_i}$$. In the original paper there is the claim that you obtain estimates of the eigenvalues with accuracy $$O(\epsilon)$$ from $$O(\frac{1}{\epsilon^3})$$ copies of $$\rho$$.