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I've been reading some of the literature on "open quantum systems" and it looks like the following physical interpretations are made:

  • Reversible dynamics of a closed quantum system are represented by a group of automorphisms (of a $C^*$-algebra, von Neumann algebra, or $B(H)$, depending on whom you read)
  • Irreversible (but still deterministic) dynamics of a closed system are represented by a semigroup of endomorphisms
  • The dynamics of an open system (which are non-deterministic, hence also irreversible) are represented by a semigroup of unital completely positive maps.

I'm trying to motivate this interpretation for someone (in this case, me!) with just a basic familiarity with quantum theory. Using the formalism of QM where states are represented by unit vectors in a Hilbert space, the story goes something like this:

  • In the "Schrodinger picture," operators are fixed and states evolve in time. There are two sub-cases: First, a system that starts in a state $| \phi \rangle$ and isn't measured will, at time $t$ later, be in the state $e^{-itH/\hbar} | \phi \rangle$, where $H$ is a self-adjoint operator representing the Hamiltonian of the system. Second, when a system in state $| \phi \rangle$ undergoes a measurement (modeled as an instantaneous process) corresponding to a self-adjoint operator $X$ (pretend it's bounded with discrete spectrum for simplicity), the system will randomly end up in one of the eigenstates $\frac{P_\lambda | \phi \rangle}{\sqrt{\langle \phi | P_\lambda | \phi \rangle}}$, each with probability $\langle \phi | P_\lambda | \phi \rangle$, where $P_\lambda$ denotes projection onto the eigenspace of $X$ for the eigenvalue $\lambda$.
  • In the "Heisenberg picture," states are fixed and operators evolve in time. Corresponding to the first case above, one has the transformation $X \mapsto e^{itH/\hbar} X e^{-itH/\hbar}$, which is an automorphism of $B(H)$; corresponding to the second case above, one applies, with probabilities $\langle \phi | X | \phi \rangle$, one of the transformations $X \mapsto \frac{P_\lambda X P_\lambda}{\langle \phi | P_\lambda | \phi \rangle}$, each of which is a conditional expectation (in particular, a completely positive map) on $B(H)$.

A couple of things I'm stuck on. First, based on the above, it looks like the second case (measurement) corresponds to a random selection among several non-unital CP maps, rather than a single unital CP map. Second, I don't see the motivation for why an irreversible yet deterministic evolution of a closed system would correspond to a non-invertible endomorphism. What am I missing?

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Maybe this is more suitable for the theoretical physics site? –  Guess who it is. Mar 28 '12 at 21:26
Thanks, I hadn't seen that site! It looks like the density-operator formulation of QM takes care of my first question, but I'm still trying to figure out where endomorphisms come into the picture. Anyway, the question is now at… –  Dave Gaebler Mar 29 '12 at 16:37

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