Question

My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with k and l "bits of information", respectively, requires kl bits to fully describe it. A critical consequence of this is that the k+l bits in each of the individual subsystems are not sufficient to "span" their joint space, so that there must be states that are "entangled" (formally, cannot be expressed by the tensor product of the states of the subsystems). That much I (think) I understand.

But the text then seems to argue that this is a property that is exclusive to quantum systems. Is that true? Certainly there are classical systems where the state of one subsystem depends on the state of another. Doesn't any joint system that requires conditional probabilities to describe need "extra bits" beyond those necessary to describe the individual subsystems?

Perhaps I'm missing some subtlety about what constitutes a "sub system" or some unstated assumption about how systems are broken into parts. Perhaps a classical system that is "entangled" by conditional probabilities isn't thought of as consisting of valid "subsystems" in the same way that quantum system is.

Is "entanglement" unique to quantum systems? Are conditional probabilities just a kind of "classical entanglement"?

Please be pedagogical. This is not meant to be a deep or advanced mathematical question, just a simple conceptual and definitional one. Just imagine I'm having basic probability explained to me (without reference to quantum vs. classical, or even physics). If one struck out all occurrences of the word "quantum" from the linked text (sections 0.2. and 0.3), wouldn't that be a perfectly valid part of such an explanation?

Answer 1

You are correct that the description of a classical probability distribution of a joint system requires kl parameters. There indeed is a difference between classical and quantum systems in this sense, but it is more subtle.

Every classical probability distribution can be described as a probabilistic mixture of deterministic states. For these deterministic states (extreme points of the space of probability densities), the description complexity of a joint system can be described by k+l bits of information.

Every quantum density matrix can be described as a probabilistic mixture of pure states. For these pure states (extreme points of the of density matrices) the description requires kl parameters.

Thus, classical probabilistic systems can be described in terms of probability distributions over more fundamental objects: deterministic states. These deterministic states only require k+l parameters.

Quantum mixed states (the quantum analogue of probability distributions) can also be described in terms of probability distributions over more fundamental objects: pure states. However, these pure states now require kl parameters.