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I'm studying how to build quantum circuits and understand qubit entanglement. The following are my two confusing questions. I would greatly appreciate any assistance or insights regarding these questions. Thank you.

  1. Building Quantum Circuits from Matrices vs. Vectors:

    Can someone explain the main differences between making quantum circuits from unitary matrices and from quantum state vectors? I think they're pretty similar, but building circuits from the state vector is a special case where all inputs are zero. However, I'm not entirely certain, and I feel like I might be overlooking something.

  2. Detecting Qubit Entanglement from Given Data:

    When given a quantum state vector or a unitary matrix representing a quantum operation, is there a reliable method or algorithm to determine which qubits are entangled and to what degree? I'm particularly interested in understanding if it's possible to extract information about qubit entanglement directly from the provided state vector or unitary matrix. Are there established techniques or emerging methodologies that address this challenge? Furthermore, how effective are these methods in practice, especially when dealing with mixed states or containing numerous qubits?

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  1. If you build a quantum circuit you are always building a matrix, which can transform an arbitrary input state to an output state. If you analyze a quantum circuit you can do it for arbitrary input vectors, or only for a limited set of input vectors, or even just for one particular input vector. But usually that input vector will not contain only zeroes (it can be, but wouldn't often be a case useful in practice).
  2. To see which qubits are entangled there are actually quite a number of criteria and measures of entanglement. You will have to study the subject further: Multipartite entanglement. See also question 17580.
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