The Information Paradox and the Copenhagen Interpretation

Question

I'm curious why the so-called 'information paradox' is a paradox. I know that it's regarded as a paradox due to the fact that Hawking radiation seemingly violates the unitarity of the time-evolution of quantum states. But isn't it the case that QM already contradicts unitarity if the Copenhagen interpretation is accepted? If some linear combination of two states A and B collapses into, say, B with the probability given by the Born rule, you cannot recover the original state without already knowing it. So it seems that unitarity is already violated in QM given the Copenhagen interpretation. Then why is it a big deal if Hawking radiation violates unitarity?

Answer 1

Collapse of the wave-function, as required by the Copenhagen interpretation indeed requires a non-unitary evolution of the system. This, however, is also widely viewed as problem. It is one of the elements that makes the measurement problem into a problem.

The modern take on this problem involves quantum decoherence. In a nut-shell, quantum decoherence is a phenomenon in quantum mechanics where if a quantum system in an entangled state interacts with suitably large quantum environment, then if one is ignorant of the actual quantum state of the environment, then the effective state of the quantum system (obtained by "tracing out" the Hilbert space of the environment) will collapse to a classical mixed state. On its own such evolution would be non-unitary, but unitarity maintained if one considers the full system+environment as one quantum system.

So quantum decoherence allows us to understand one aspect of the apparent collapse of the wave function in a measurement of quantum system (which necessarily involves interacting with a classic system through the measurement). However, it does not fully solve the measurement problem, because we are left with a classical ensemble of possible outcomes, rather than the single outcome we experience in a typical measurement.

This is were the consistent histories approach to interpreting quantum mechanics typically comes into the modern discussion. I'll leave it to the Wikipedia article to explain what this entails exactly, and just suffice with noting that when combined with quantum decoherence, consistent histories become an elaborated more detailed version of the Copenhagen interpretation.

Interestingly, the origin of the black hole information paradox is in some way related to this discussion: Hawking radiation can be viewed as the result of the decoherence of the quantum vacuum after interacting with large number of quantum degrees of freedom of a black hole. It is therefore not surprising that quantum information theory plays an important role in contemporary attempts at solving the black hole information paradox.

Answer 2

It is a difference between open and closed systems.

A quantum system S undergoing measurement is an open system. The full system SAE (System+Apparatus+Environment) is a closed system that undergoes unitary evolution. But the SAE Hamiltonian couples the subsystems, so S alone evolves non-unitarily. The measurement process will never be described by a Hamiltonian for S alone.

On the other hand, a closed system is expected to evolve unitarily. By a closed system I mean one that is fully self-contained with no external interactions, i.e. a system with a hermitian Hamiltonian. In the information paradox the system is in some sense supposed to be "the whole universe", which one expects to be closed and therefore evolve unitarily. That being said, defining the correct "closed system" in the black hole case can be a little bit tricky, and you have to do it carefully. But that's a different story.