# Schur transform and the outcome probabilities for a particular type of state

I was reading about the Schur transform and its applications in knowing about an unknown quantum state.

Consider $$\rho^{\otimes k}$$, which means $$k$$ copies of an unknown $$n$$ qubit quantum density matrix $$\rho$$.

Let's say I measure $$\rho^{\otimes k}$$ in the Schur basis --- that is, I apply the Schur transform to this state and measure in the standard basis. An example of this can be found in equations $$13$$, $$14$$, and $$15$$ of this paper.

Can we (non trivially and as tightly as possible) upper bound the probability of getting each of the $$2^{nk}$$ outcomes after our measurement is done? I am looking for an upper bound that depends on the specific properties of the Schur transform (and not just the fact that it is a unitary transform.)

I have seen it remarked in different papers that knowing the spectrum of $$\rho$$ is sufficient to put bounds on the probability of getting each outcome after measuring $$\rho^{\otimes k}$$ in the Schur basis --- but it wasn't too clear how.

For simplicity, let us say we are given a guarantee on the spectrum of $$\rho$$ --- each eigenvalue $$\lambda_i$$ of $$\rho$$ satisfies

$$$$\lambda_i \leq \frac{1}{2^{n}} + \frac{1}{2^{\text{poly}(n)}}.$$$$

Note that if $$\rho$$ is maximally mixed, then by a trivial argument, each outcome is equally likely.

• I am not quite sure I understand what outcomes you are asking for? Are you looking for the probability of getting a certain partition $\lambda$? Certainly, unitaries don't change the probabilities of states... Oct 22 at 21:40
• Yes, let's say I am looking for the probability of getting a certain partition $\lambda$. Oct 22 at 23:54
• Yeah ok cool. I think the state of the art on this might be found in arxiv.org/abs/1612.00034 Oct 23 at 16:22