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How much classical information is carried by a quantum state? An approach inspired by Kolmogorov complexity

In quantum mechanics, a state is an element of a Hilbert space whose dimension exponentially grows with the increase of the number of particles (or qubits, in quantum computing). The vague question "is this huge Hilbert space really there?" has been rigorously formalized inside the computational complexity theory; the research suggests a positive answer to the question. Along this line, I give a definition of the (classical) information content of a quantum state, taking inspiration from the Kolmogorov complexity. I show that, for some well-known quantum circuits (having a number of gates polynomial in the number of qubits), the information content of the output state, evaluated according to my definition, is polynomial in the number of qubits. On the other hand, applying known results, it is possible to devise quantum circuits that generate much more complex states, having an exponentially-growing information content. A huge amount of classical information can be really present inside a quantum state, however, I show that this property is not necessarily exploited by quantum computers, not even by quantum algorithms showing an exponential speed-up with respect to classical computation.