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

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

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

  • $\begingroup$ 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... $\endgroup$
    – Condo
    Oct 22 at 21:40
  • $\begingroup$ Yes, let's say I am looking for the probability of getting a certain partition $\lambda$. $\endgroup$
    – BlackHat18
    Oct 22 at 23:54
  • $\begingroup$ Yeah ok cool. I think the state of the art on this might be found in arxiv.org/abs/1612.00034 $\endgroup$
    – Condo
    Oct 23 at 16:22

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