[Physics] a local operator in quantum mechanics

localitynon-localityoperatorsquantum mechanicssymmetry

In quantum mechanics, what exactly is meant by "local" operator?

What about a "global" or a "non-local" operator? Are these the same?

Can you also also help me understand what exactly is a local perturbation and a local symmetry?

Best Answer

If you are dealing with a multipartite state $\lvert\Psi\rangle$, then the distinction between local and non-local operations is an important one for example for the study of entanglement.

Consider for simplicity a bipartite state $\lvert\Psi\rangle$, that is, a state that can be written as $\lvert\Psi\rangle=\sum_{ij}c_{ij}\lvert i\rangle\otimes\lvert j\rangle$. A local operation $A\equiv (A'\otimes I)$ is one that only acts on one part of the system. For example, an operation of the form $$A\lvert\Psi\rangle=\sum_{ij} c_{ij}(A'\lvert i\rangle)\otimes\lvert j\rangle$$ is local in that it only affects the first part of the system. As another example, an operation of the form $A'\otimes B'$ is also local, and its action on a product state $\lvert\psi\rangle\otimes\lvert\phi\rangle$ would read $$\lvert\psi\rangle\otimes\lvert\phi\rangle\to (A'\lvert\psi\rangle)\otimes(B'\lvert\phi\rangle).$$ One notable property of local (unitary) operations is that they do not affect the entanglement (in a way that can be made precise).

Global operations are operations that are not local.

For a two-qubit system, an example of a global operation can be a CNOT, while a local operation is any single-qubit gate.

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