I thought they were typically wavefunction to wavefunction maps, and they need some sort of sandwiching to apply to density matrices?
Yes, this is correct. My mistake, it does indeed need the sandwiching like this ρnew=aρolda†.
From your talk on tensors, I am sure it will not surprise you at all to know that the sandwhich thing itself (mapping from operators to operators) is often called a superoperator.
I think the reason it is as it is is their isn’t a clear line between operators that modify the state and those that represent measurements. For example, the Hamiltonian operator evolves the state with time. But, taking the trace of the Hamiltonian operator applied to the state gives the expectation value of the energy.
From your talk on tensors, I am sure it will not surprise you at all to know that the sandwhich thing itself (mapping from operators to operators) is often called a superoperator.
Oh it does surprise me, superoperators are a physics term but I just know linear algebra and dabble in physics, so I didn’t know that one. Like I’d think of it as the functor over vector spaces that maps V↦V⊗V.
I think the reason it is as it is is their isn’t a clear line between operators that modify the state and those that represent measurements. For example, the Hamiltonian operator evolves the state with time. But, taking the trace of the Hamiltonian operator applied to the state gives the expectation value of the energy.
Hm, I guess it’s true that we’d usually think of the matrix exponential as mapping V⊸V to V⊸V, rather than as mapping V⊗V⊸C to V⊸V. I guess it’s easy enough to set up a differential equation for the latter, but it’s much less elegant than the usual form.
In some papers people write density operators using an enhanced “double ket” Dirac notation, where eg. density operators are written to look like |x>>, with two “>”’s. They do this exactly because the differential equations look more elegant.
I think in this notation measurements look like <<m|, but am not sure about that. The QuTiP software (which is very common in quantum modelling) uses something like this under-the-hood, where operators (eg density operators) are stored internally using 1d vectors, and the super-operators (maps from operators to operators) are stored as matrices.
So structuring the notation in other ways does happen, in ways that look quite reminiscent of your tensors (maybe the same).
Yes, this is correct. My mistake, it does indeed need the sandwiching like this ρnew=aρolda†.
From your talk on tensors, I am sure it will not surprise you at all to know that the sandwhich thing itself (mapping from operators to operators) is often called a superoperator.
I think the reason it is as it is is their isn’t a clear line between operators that modify the state and those that represent measurements. For example, the Hamiltonian operator evolves the state with time. But, taking the trace of the Hamiltonian operator applied to the state gives the expectation value of the energy.
Oh it does surprise me, superoperators are a physics term but I just know linear algebra and dabble in physics, so I didn’t know that one. Like I’d think of it as the functor over vector spaces that maps V↦V⊗V.
Hm, I guess it’s true that we’d usually think of the matrix exponential as mapping V⊸V to V⊸V, rather than as mapping V⊗V⊸C to V⊸V. I guess it’s easy enough to set up a differential equation for the latter, but it’s much less elegant than the usual form.
In some papers people write density operators using an enhanced “double ket” Dirac notation, where eg. density operators are written to look like |x>>, with two “>”’s. They do this exactly because the differential equations look more elegant.
I think in this notation measurements look like <<m|, but am not sure about that. The QuTiP software (which is very common in quantum modelling) uses something like this under-the-hood, where operators (eg density operators) are stored internally using 1d vectors, and the super-operators (maps from operators to operators) are stored as matrices.
So structuring the notation in other ways does happen, in ways that look quite reminiscent of your tensors (maybe the same).