I am a beginner at QC, currently going through a paper on Quantum Anonymous Voting. Please clarify the orthogonal property described in the following scenario.
Consider $n$ voters $V_{0}, V_{1}, V_{2}, ..., V_{n-1}$. $V_{0}$ is preparing the following state and distributes $n-1$ particles to other $n-1$ players keeping 1st particle with itself.
$$\boxed{|S_{n}\rangle = \frac{1} {\sqrt{n!}} \sum_{S\in P_n^{n}} ( \,-1) \,^{\Gamma(S)}|s_{0}\rangle |s_{1}\rangle ....|s_{n-1}\rangle}$$
Here $P_n^{n}$ is the set of all permutations of $Z_n := \{0,1,··· ,n−1\}$, $S$ is a permutation (or sequence) in the form $S = s_0 s_1 ···s_{n−1}$. $\Gamma(S)$, named inverse number, is defined as the number of transpositions of pairs of elements of $S$ that must be composed to place the elements in canonical order, $012 · · · n−1$.
Consider the following security attack
Assume there are $l$ dishonest voters, $V_{i_{0}}, V_{i_{1}, V_{i_{2}}},...,V_{i_{l-1}}$. They first intercept some transmitted particles, entangle them with an auxiliary system prepared in advance and then return the operated particles to honest voters. The state of the whole composite system is denoted by $|\psi\rangle$. To elude the detection, it is required that all measurements outcomes should be distinct when measuring each particle held by honest voter in the Fourier basis, and thus $|\psi\rangle$ should be in the form
$$\boxed{|\psi\rangle = \sum_{S\in P_n^{n-l}} \frac{( \,-1) \,^{\Gamma(S)} F^{\otimes(n-l)} |S\rangle}{\sqrt{|P_n^{n-l}|}}\otimes |u_{S}\rangle}$$
where $S=s_{0}s_{j_{0}}...s_{j_{n-l-2}}$, $|u_{S}\rangle$ are the states of composite system of $l$ particles and auxiliary system, $P_{n}^{n-l}=\{x_{0}x_{1}...x_{n-l-1}| x_{0},x_{1},...,x_{n-l-1} \in Z_{n} \}$ and $|P_n^{n-l}| = \frac{n!}{l!}$ is its size. $P_n^{n-l}$ can be divided into $\frac{n!}{(n-1)!l!}$ subsets, each one corresponding to the set of all the (n-1)! permutations of a $n-l$ combination of $Z_{n}$.
Any two states $|u_{S_{0}}\rangle$ and $|u_{S_{1}}\rangle$ such that $S_{0} \in P_{n}^{n-l,w_{0}}$, $S_{1} \in P_{n}^{n-l,w_{1}}$, and $w_{0} \neq w_{1}$ should be orthogonal to each other, ie., $\langle$ $u_{S_{0}} | u_{S_{1}}$$\rangle$ = 0. if not the honest voters cannot deterministically know subset $P_{n}^{n-l,w}$ in which the honest voters announce the correct measurement outcomes to avoid being detected.
I want clarification on above highlighted sentence about orthogonality. If possible please help me with an example.
