## What additional conditions are there to make POVM measurement same as projective measurement?

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We know that POVMs are applied in the more general cases where the system is not necessarily closed. So mathematically, how does going from open to closed system change the scenario in case of POVM so that it becomes a projective measurement?

I don't understand the question. What do you mean exactly with "how does going from open to closed change the scenario"? – glS – 2021-02-16T09:25:40.463

@glS.: We can't apply projective measurement when the system is not closed. In an open system the evolution of a state can't be described by unitary operation. We often hear that a general measurement will be projective measurement if it follows the additional condition that $$M_{m}^\daggerM_{m'}= \delta_{m,m'}M_{m}$$. I want to know if there is some relation holds between these two. – user27286 – 2021-02-16T11:51:20.557

Are you aware of the concept of a Naimark dilation? – Rammus – 2021-02-16T11:59:19.747

@Rammus.: Heard first time in my life a min ago from you. – user27286 – 2021-02-16T12:01:43.387

sure you can do projective measurements when the system is not closed. The system being "open" or "closed" refers to its dynamics, i.e. how it evolves. That doesn't affect the types of measurements you can perform on the state itself. About the relation between POVMs and projective measurements, have you seen https://quantumcomputing.stackexchange.com/q/7004/55? See also https://quantumcomputing.stackexchange.com/q/12275/55

– glS – 2021-02-16T12:26:36.163