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I am trying to prove that the $|S_{n}\rangle$ is $n$-lateral rotationally invariant, where $|S_{n}\rangle$ is defined as

$$|S_{n}\rangle=\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}|S\rangle\equiv\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}|s_{0}s_{1}.....s_{n-1}\rangle.$$

A $n$ dimensional qudit state on Hilbert space $H_{n}$ is the superposition of its computational basis $\{|i\rangle\}, i\in\{0,1,2,...,n-1\}$. Here, $|S_n\rangle$ is a state of $n$ such particles on $H_{n}^{\otimes n}$.

Consider now another basis $|i'\rangle$ connected with the computational basis by a unitary transformation $U$, where $$|i\rangle=\sum_{j} U_{ji}|i'\rangle.$$

Then, in this new basis, the state $|S_{n}\rangle$ takes the same form up to a global phase factor $\phi$. That is,

$$|S_{n}\rangle=e^{i\phi}\sum_{M \in P_{n}^{n}} (-)^{\tau(M)}|M'\rangle \equiv e^{i\phi}\sum_{M \in P_{n}^{n}} (-)^{\tau(M)}|m_{0}'m_{1}'...m_{n-1}'\rangle.$$

Here $P_{n}^{n} = \{x_{0}x_{1}x_{2}...x_{n-1}|x_{0},x_{1},x_{2},...,x_{n-1 }\in Z_{n}, \forall j \neq k, x_{j} \neq x_{k}\}$ and the phase factor is given by $$e^{i\phi} = \det(U).$$

**Proof**

$$|S_{n}\rangle=\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}\sum_{m_{0}=0}^{n-1}U_{m_{0},s_{0}}|m_{0}'\rangle\otimes...\otimes\sum_{m_{n-1}=0}^{n-1}U_{m_{n-1},s_{n-1}}|m_{n-1}'\rangle$$

$$=\left[\sum_{M \in P_{n}^{n}} + \sum_{M \not\in P_{n}^{n}}\right]\left[\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}U_{m_{0},s_{0}}U_{m_{1},s_{1}}...U_{m_{n-1},s_{n-1}}\right]|M\rangle$$

$$=\left[\sum_{M \in P_{n}^{n}} + \sum_{M \not\in P_{n}^{n}}\right]det(U_{m_{j}, s_{i}})|M\rangle$$

.... continued

Here how $\left[\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}U_{m_{0},s_{0}}U_{m_{1},s_{1}}...U_{m_{n-1},s_{n-1}}\right]= det(U_{m_{j}, s_{i}})$?

Please provide a reference if you have any. I have a basic idea about unitary transformations and their matrix representations.

$\tau(S)$, 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, $012 · · · n − 1$.

1Please provide a reference yourself! What is the map $\tau$ doing? – Marsl – 2019-11-28T08:12:10.573

@Marsl I provided the required reference – Adam Levine – 2019-11-28T11:21:34.267

1please try to use sensible titles for your questions. I tried to edit in a title actually related to the question being asked. Please check that I interpreted your question correctly. – glS – 2019-11-28T12:01:46.623

@AdamLevine $\tau(S)$ here is the parity of the permutation $S$, right?

– glS – 2019-11-28T12:11:34.870