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I've been using the phrase "quantum information" for some time, but am beginning to wonder if the phrase means the same thing to me as to the QM community. Specifically, my understanding is that 1) the amount of quantum information in a system is not measurable, because it would require measuring the wavefunction of the system (the relative probability for all the possible states of the system); 2) IF the wavefunction could be measured, the amount of quantum information it contains would relate to the unlikelihood of that particular wavefunction (which might be equated to the negentropy of the wavefunction).

In a past question I asked about "conservation of uncertainty" and posed the question poorly. By "uncertainty", I meant an integral of the probability density of the wavefunction over phase space - which, I assume, is always equal to 1. Per my understanding (which I'm questioning), the information in a wavefunction described in phase space would relate to distortion of the phase space probability distribution away from an "equilibrium" distribution. I realize I am probably using some of these words in nonstandard ways and will appreciate thoughtful corrections.

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    $\begingroup$ What is precisely your question? $\endgroup$ Jan 25, 2021 at 10:45
  • $\begingroup$ Looking for a useful definition of "quantum information" as applied to, e.g., a small number of interacting particles. $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 16:50
  • $\begingroup$ "quantum information" as a research field? As a notion of information? As a notion of ... well, what? It is used in very ambiguous ways - for a field, for a qualitative notion of some kind of information, for a quantitative notion, ... $\endgroup$ Jan 25, 2021 at 19:53

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According to wikipedia:

Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

For me personally, I tend to think "quantum information" as a non-precise term referring to a field of study. If, in context, it was being used in a technical way, I would interpret "information" as "entropy", meaning $-Tr[\rho \log \rho]$ where $\rho$ is the density matrix.

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  • $\begingroup$ Isn't "negentropy" usually associated with information? $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 16:52
  • $\begingroup$ I've never heard of "negentropy" (I guess it is $+Tr \rho \log \rho$ instead of $-Tr \rho \log \rho$?), but systems with lower entropy are more "informative" in the sense that in a system with minimum entropy, you need more bits of information to specify which state the system is in, while you don't need many bits to describe a system in a maximum entropy state. $\endgroup$
    – Andrew
    Jan 25, 2021 at 18:14
  • $\begingroup$ Negentropy is defined in Wikipedia and other places online. $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 18:50
  • $\begingroup$ Well, if you find it useful, go for it! $\endgroup$
    – Andrew
    Jan 25, 2021 at 19:55
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In terms of classical probability, information refers to the number of binary questions that need to be answered before you have complete certainty of the result, that's the reason why it's usually defined as $$-\log_2(p_i).$$ In particular, some of the possibilities require more answers than others, yet you can compute the mean value of the information and you would get one of the definitions of entropy or a multiple of it.

In quantum systems you get pure states after a measurement of a complete set of observables, and any pure state can be associated to the measurement of a complete set of observables. Thus, there is no information in pure states, because all questions that could be answered simultaneously were already answered. Yet, this is not the end of the story because we could still consider mixed states represented with a density matrix such as $$\rho = \frac{1}{2}|\psi\rangle\langle\psi |+\frac{1}{2}|\phi\rangle\langle\phi |.$$ Información could then be extended to density matrices, such that the diagonal elements coincide with the usual definition: $I= -\log_2(\rho)$... yet, this definition has problems: it is undefined for the null eigenvalues of the density matrix. But this can be solved with some smart regularization such as $$I= -\lim_{a\to 0^+}\rho^a\log_2(\rho).$$

This regularization becomes unnecessary when you refer to its mean value we get a multiple of Von Neumann's entropy $$S= -tr\{\rho\log_2(\rho)\}.$$

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The term quantum information is often used in the expression "conservation of quantum information". So what does it mean in that context? Conservation of information simply means that if you know the final state you can in principle reconstruct the initial state.

So in this case quantum information is not some quantitative property of a specific quantum state. If we want a quantitative measure of quantum information, it would have to be a property of an ensemble. Here is a simple scenario to illustrate this:

Suppose the initial state is an ensemble of 256 copies of our system, each in a different state. Conservation of information implies that the final state of the ensemble will also consist of 256 different states (no two states can start out different but end up the same). In terms of information, we can say that we need exactly 8 bits to describe a specific member of the initial ensemble. And because of conservation of information we still need exactly 8 bits to describe a specific member of the final ensemble.

So we can restate the principle of conservation of information like this: the amount of information necessary to specify a specific member of the ensemble doesn't change with time. More succinctly, the ensemble entropy does not change with time.

So, at least in this context, quantum information, i.e. the thing that is conserved, is synonymous with the amount of information "stored" in an ensemble, i.e. its entropy. It's a property of an ensemble, not of a specific quantum state.

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  • $\begingroup$ That makes sense. Suppose, though, that we have two particles, one which (A) is known to be spin "up" or 1:0 likelihood of spin up vs down, and the other of which (B) is known to have an indeterminate spin orientation, or 50:50 likelihood of spin up vs down. Let A and B interact. Out of the interaction come A' and B', with new probability distributions of up and down. I "think" that there are constraints on what those probability distributions can be, in the form of correlations between the wavefunctions of A' and B'; and that the constraints amount to conservation of information. Right? $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 20:32
  • $\begingroup$ I will assume that you mean the second particle is in a 50-50 superposition. Then quantum mechanics does not impose any constraints on what the final state can be, it can be any element in the Hilbert space without violating the unitary nature of time evolution.. $\endgroup$ Jan 25, 2021 at 21:55
  • $\begingroup$ But presumably not A' being 1:0 and B' being 1,0, or A' being 50:50 and B' being 50:50. $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 23:57
  • $\begingroup$ Those possibilities are not excluded by principles of quantum mechanics either, assuming, again, that when you say 50-50 you mean a 50-50 superposition. $\endgroup$ Jan 26, 2021 at 3:26
  • $\begingroup$ I hope @Qmechanic will chime in. $\endgroup$
    – S. McGrew
    Jan 27, 2021 at 18:13
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Here's my own view: roughly speaking, if we have a state expressed in a basis of "classical" states

$$|\Psi\rangle = \sum a_j |x_j\rangle,$$

then the system will eventually decohere into one of the states $|x_j\rangle$ with probability $|a_j|^2$. This probability distribution is not very special, it might just be due to classical entropy, ie. there is a mixed density matrix with the same probability for each outcome upon a measurement in the $x$ basis. The only truly quantum information in the wavefunction is the relative phases of the $a_j$, which is where special effects like the diffraction pattern in the double-slit experiment come from.

What you're talking about in phase space reminds me a bit of squeezed states. The uncertainty principle is definitely a part of quantum mechanics (especially since h-bar appears explicitly there), but it doesn't really capture quantum information, in my view. There are some mundane uncertainty principles, such as trying to measure the time-dependent Fourier transform of a signal. You can't have perfect resolution simultaneously in both the time and frequency domain.

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Quantum mechanics does not describe reality itself, but describes what we know of reality. States in quantum mechanics describe the information we have from measurement. They are states of knowledge not physical states.

We start with an initial, known, state, $|f\rangle$ which describes the information we have from measurement (and preparation) at time $t=0$ without loss of generality. The state evolves to $U(t)|f\rangle$ at time $t$, during which time there is no measurement of the state and consequently there is no change in the information we have about the state. This is a vital constraint because it can be shown that it means that $U$ must be a unitary operator obeying the conditions for Stone's theorem (1932) and from this we can show that the general form of Schrodinger's equation necessarily follows.

When we do a measurement at time $t$ there is a change in information as a result of the new measurement. This change is often known as wave function collapse.

Confusion arises when people conflate the wave function with something physical, leading to well known an paradoxical behaviours. These paradoxes are resolved when it is seen that quantum mechanics describes knowledge, or information. Quantum information simply means that information is encoded into the wave function, rather than being presented as definite measurement results.

This definition is in line with modern forms of the Copenhagen interpretation, which follow Dirac and von Neumann in discarding wave-particle duality and is made formal in their axiomatic treatments. It extends to multiparticle states through the construction of Fock space, which enables us to treat multiparticle space within essentially the same mathematical structure, in which interactions are likewise constrained by the requirement of unitarity so that a Schrodinger equation is obeyed. One can develop all of qft and the standard model of particle physics from this point of view, though physicists usually shortcut the maths and develop qft in a heuristic manner, justifying it through correct experimental results rather than mathematical rigour.

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  • $\begingroup$ Your last sentence is a quasi-definition, but "encoded" suggests that the information was deliberately put there in the wave function. I'm hoping for a definition that would apply to, e.g., a small number of interacting particles. $\endgroup$
    – S. McGrew
    Jan 25, 2021 at 16:55
  • $\begingroup$ @S.McGrew, information is only "deliberately" put into the wave function in the sense that the wave function is mathematically constrained to conserve the information which defines the initial state. The definition extends to multiparticle states through the construction of Fock space, and interactions are likewise constrained by the requirement of unitarity. One can develop all of qft and the standard model of particle physics from this point of view, though physicists usually shortcut the maths and develop qft phenomenologically. $\endgroup$ Jan 25, 2021 at 19:31

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