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### Why does it matter that Schmidt number is invariant under unitary transformations?

Crucially, "the Schmidt number is preserved under unitary transformations on system A or system B alone". So Schmidt numbers are invariant under local operations $U_A\otimes U_B$, but not ...
• 1,991
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### How to calculate the Schmidt decomposition of a state without SVD

When a question says "without using the SVD" it may be implying that there are easier ways, at least in this specific case. It's always worth just having a stare at the state for a minute (...
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### How to write Schmidt decomposition for pure tripartite state?

Assuming the Schmidt decomposition exists, it would be written as: $$\sum_{i=1}^rp_i\left|i_A\right\rangle\left|i_C\right\rangle\left|i_B\right\rangle$$ With $\left\{\left|i_A\right\rangle\right\}_i$, ...
• 6,692
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### Why does Schmidt decomposition (2 qubits) requires density matrix of each system?

The reason why they go from the pure state to the individual density matrices of the subsystems is that this is going to help you find the Schmidt coefficients and the orthonormal bases for both sites....
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