There are some non-obvious issues with saying “the wavefunction really exists, but the density matrix is only a representation of our own ignorance”. Its a perfectly defensible viewpoint, but I think it is interesting to look at some of its potential problems:
A process or machine prepares either |0> or |1> at random, each with 50% probability. Another machine prepares either |+> or |-> based on a coin flick, where |+> = (|0> + |1>)/root2, and |+> = (|0> - |1>)/root2. In your ontology these are actually different machines that produce different states. In contrast, in the density matrix formulation these are alternative descriptions of the same machine. In any possible experiment, the two machines are identical. Exactly how much of a problem this is for believing in wavefuntions but not density matrices is debatable—“two things can look the same, big deal” vs “but, experiments are the ultimate arbiters of truth, if experiemnt says they are the same thing then they must be and the theory needs fixing.”
There are many different mathematical representations of quantum theory. For example, instead of states in Hilbert space we can use quasi-probability distributions in phase space, or path integrals. The relevance to this discussion is that the quasi-probability distributions in phase space are equivalent to density matrices, not wavefunctions. To exaggerate the case, imagine that we have a large number of different ways of putting quantum physics into a mathematical language, [A, B, C, D....] and so on. All of them are physically the same theory, just couched in different mathematics language, a bit like say, [“Hello”, “Hola”, “Bonjour”, “Ciao”...] all mean the same thing in different languages. But, wavefunctions only exist as an entity separable from density matrices in some of those descriptions. If you had never seen another language maybe the fact that the word “Hello” contains the word “Hell” as a substring might seem to possibly correspond to something fundamental about what a greeting is (after all, “Hell is other people”). But its just a feature of English, and languages with an equal ability to greet don’t have it. Within the Hilbert space language it looks like wavefunctions might have a level of existence that is higher than that of density matrices, but why are you privileging that specific language over others?
In a wavefunction-only ontology we have two types of randomness, that is normal ignorance and the weird fundamental quantum uncertainty. In the density matrix ontology we have the total probability, plus some weird quantum thing called “coherence” that means some portion of that probability can cancel out when we might otherwise expect it to add together. Taking another analogy (I love those), the split you like is [100ml water + 100ml oil], (but water is just my ignorance and doesn’t really exist), and you don’t like the density matrix representation of [200ml fluid total, oil content 50%]. Their is no “problem” here per se but I think it helps underline how the two descriptions seem equally valid. When someone else measures your state they either kill its coherence (drop oil % to zero), or they transform its oil into water. Equivalent descriptions.
All of that said, your position is fully reasonable, I am just trying to point out that the way density matrices are usually introduced in teaching or textbooks does make the issue seem a lot more clear cut than I think it really is.
A process or machine prepares either |0> or |1> at random, each with 50% probability. Another machine prepares either |+> or |-> based on a coin flick, where |+> = (|0> + |1>)/root2, and |+> = (|0> - |1>)/root2. In your ontology these are actually different machines that produce different states.
I wonder if this can be resolved by treating the randomness of the machines quantum mechanically, rather than having this semi-classical picture where you start with some randomness handed down from God. Suppose these machines use quantum mechanics to do the randomization in the simplest possible way—they have a hidden particle in state |left>+|right> (pretend I normalize), they mechanically measure it (which from the outside will look like getting entangled with it) and if it’s on the left they emit their first option (|0> or |+> depending on the machine) and vice versa.
So one system, seen from the outside, goes into the state |L,0>+|R,1>, the other one into the state |L,0>+|R,0>+|L,1>-|R,1>. These have different density matrices. The way you get down to identical density matrices is to say you can’t get the hidden information (it’s been shot into outer space or something). And then when you assume that and trace out the hidden particle, you get the same representation no matter your philosophical opinion on whether to think of the un-traced state as a bare state or as a density matrix. If on the other hand you had some chance of eventually finding the hidden particle, you’d apply common sense and keep the states or density matrices different.
Anyhow, yeah, broadly agree. Like I said, there’s a practical use for saying what’s “real” when you want to predict future physics. But you don’t always have to be doing that.
You are completely correct in the “how does the machine work inside?” question. As you point out that density matrix has the exact form of something that is entangled with something else.
I think its very important to be discussing what is real, although as we always have a nonzero inferential distance between ourselves and the real the discussion has to be a little bit caveated and pragmatic.
A process or machine prepares either |0> or |1> at random, each with 50% probability. Another machine prepares either |+> or |-> based on a coin flick, where |+> = (|0> + |1>)/root2, and |+> = (|0> - |1>)/root2. In your ontology these are actually different machines that produce different states. In contrast, in the density matrix formulation these are alternative descriptions of the same machine. In any possible experiment, the two machines are identical. Exactly how much of a problem this is for believing in wavefuntions but not density matrices is debatable—“two things can look the same, big deal” vs “but, experiments are the ultimate arbiters of truth, if experiemnt says they are the same thing then they must be and the theory needs fixing.”
I like “different machines that produce different states”. I would bring up an example where we replace the coin by a pseudorandom number generator with seed 93762. If the recipient of the photons happens to know that the seed is 93762, then she can put every photon into state |0> with no losses. If the recipient of the photons does not know that the random seed is 93762, then she has to treat the photons as unpolarized light, which cannot be polarized without 50% loss.
So for this machine, there’s no getting away from saying things like: “There’s a fact of the matter about what the state of each output photon is. And for any particular experiment, that fact-of-the-matter might or might not be known and acted upon. And if it isn’t known and acted upon, then we should start talking about probabilistic ensembles, and we may well want to use density matrices to make those calculations easier.”
I think it’s weird and unhelpful to say that the nature of the machine itself is dependent on who is measuring its output photons much later on, and how, right?
Yes, in your example a recipient who doesn’t know the seed models the light as unpolarised, and one who does as say, H-polarised in a given run. But for everyone who doesn’t see the random seed its the same density matrix.
Lets replace that first machine with a similar one that produces a polarisation entangled photon pair, |HH> + |VV> (ignoring normalisation). If you have one of those photons it looks unpolarised (essentially your “ignorance of the random seed” can be thought of as your ignorance of the polarisation of the other photon).
If someone else (possibly outside your light cone) measures the other photon in the HV basis then half the time they will project your photon into |H> and half the time into |V>, each with 50% probability. This 50⁄50 appears in the density matrix, not the wavefunction, so is “ignorance probability”.
In this case, by what I understand to be your position, the fact of the matter is either (1) that the photon is still entangled with a distant photon, or (2) that it has been projected into a specific polarisation by a measurement on that distant photon. Its not clear when the transformation from (1) to (2) takes place (if its instant, then in which reference frame?).
So, in the bigger context of this conversation, OP: “You live in the density matrices (Neo)” Charlie :”No, a density matrix incorporates my own ignorance so is not a sensible picture of the fundamental reality. I can use them mathematically, but the underlying reality is built of quantum states, and that randomness when I subject them to measurements is fundamentally part of the territory, not the map. Lets not mix the two things up.” Me: “Whether a given unit of randomness is in the map (IE ignorance), or the territory is subtle. Things that randomly combine quantum states (my first machine) have a symmetry over which underlying quantum states are being mixed that looks meaningful. Plus (this post), the randomness can move abruptly from the territory to the map due to events outside your own light cone (although the amount of randomness is conserved), so maybe worrying too much about the distinction isn’t that helpful.
There are some non-obvious issues with saying “the wavefunction really exists, but the density matrix is only a representation of our own ignorance”. Its a perfectly defensible viewpoint, but I think it is interesting to look at some of its potential problems:
A process or machine prepares either |0> or |1> at random, each with 50% probability. Another machine prepares either |+> or |-> based on a coin flick, where |+> = (|0> + |1>)/root2, and |+> = (|0> - |1>)/root2. In your ontology these are actually different machines that produce different states. In contrast, in the density matrix formulation these are alternative descriptions of the same machine. In any possible experiment, the two machines are identical. Exactly how much of a problem this is for believing in wavefuntions but not density matrices is debatable—“two things can look the same, big deal” vs “but, experiments are the ultimate arbiters of truth, if experiemnt says they are the same thing then they must be and the theory needs fixing.”
There are many different mathematical representations of quantum theory. For example, instead of states in Hilbert space we can use quasi-probability distributions in phase space, or path integrals. The relevance to this discussion is that the quasi-probability distributions in phase space are equivalent to density matrices, not wavefunctions. To exaggerate the case, imagine that we have a large number of different ways of putting quantum physics into a mathematical language, [A, B, C, D....] and so on. All of them are physically the same theory, just couched in different mathematics language, a bit like say, [“Hello”, “Hola”, “Bonjour”, “Ciao”...] all mean the same thing in different languages. But, wavefunctions only exist as an entity separable from density matrices in some of those descriptions. If you had never seen another language maybe the fact that the word “Hello” contains the word “Hell” as a substring might seem to possibly correspond to something fundamental about what a greeting is (after all, “Hell is other people”). But its just a feature of English, and languages with an equal ability to greet don’t have it. Within the Hilbert space language it looks like wavefunctions might have a level of existence that is higher than that of density matrices, but why are you privileging that specific language over others?
In a wavefunction-only ontology we have two types of randomness, that is normal ignorance and the weird fundamental quantum uncertainty. In the density matrix ontology we have the total probability, plus some weird quantum thing called “coherence” that means some portion of that probability can cancel out when we might otherwise expect it to add together. Taking another analogy (I love those), the split you like is [100ml water + 100ml oil], (but water is just my ignorance and doesn’t really exist), and you don’t like the density matrix representation of [200ml fluid total, oil content 50%]. Their is no “problem” here per se but I think it helps underline how the two descriptions seem equally valid. When someone else measures your state they either kill its coherence (drop oil % to zero), or they transform its oil into water. Equivalent descriptions.
All of that said, your position is fully reasonable, I am just trying to point out that the way density matrices are usually introduced in teaching or textbooks does make the issue seem a lot more clear cut than I think it really is.
I wonder if this can be resolved by treating the randomness of the machines quantum mechanically, rather than having this semi-classical picture where you start with some randomness handed down from God. Suppose these machines use quantum mechanics to do the randomization in the simplest possible way—they have a hidden particle in state |left>+|right> (pretend I normalize), they mechanically measure it (which from the outside will look like getting entangled with it) and if it’s on the left they emit their first option (|0> or |+> depending on the machine) and vice versa.
So one system, seen from the outside, goes into the state |L,0>+|R,1>, the other one into the state |L,0>+|R,0>+|L,1>-|R,1>. These have different density matrices. The way you get down to identical density matrices is to say you can’t get the hidden information (it’s been shot into outer space or something). And then when you assume that and trace out the hidden particle, you get the same representation no matter your philosophical opinion on whether to think of the un-traced state as a bare state or as a density matrix. If on the other hand you had some chance of eventually finding the hidden particle, you’d apply common sense and keep the states or density matrices different.
Anyhow, yeah, broadly agree. Like I said, there’s a practical use for saying what’s “real” when you want to predict future physics. But you don’t always have to be doing that.
You are completely correct in the “how does the machine work inside?” question. As you point out that density matrix has the exact form of something that is entangled with something else.
I think its very important to be discussing what is real, although as we always have a nonzero inferential distance between ourselves and the real the discussion has to be a little bit caveated and pragmatic.
I like “different machines that produce different states”. I would bring up an example where we replace the coin by a pseudorandom number generator with seed 93762. If the recipient of the photons happens to know that the seed is 93762, then she can put every photon into state |0> with no losses. If the recipient of the photons does not know that the random seed is 93762, then she has to treat the photons as unpolarized light, which cannot be polarized without 50% loss.
So for this machine, there’s no getting away from saying things like: “There’s a fact of the matter about what the state of each output photon is. And for any particular experiment, that fact-of-the-matter might or might not be known and acted upon. And if it isn’t known and acted upon, then we should start talking about probabilistic ensembles, and we may well want to use density matrices to make those calculations easier.”
I think it’s weird and unhelpful to say that the nature of the machine itself is dependent on who is measuring its output photons much later on, and how, right?
Yes, in your example a recipient who doesn’t know the seed models the light as unpolarised, and one who does as say, H-polarised in a given run. But for everyone who doesn’t see the random seed its the same density matrix.
Lets replace that first machine with a similar one that produces a polarisation entangled photon pair, |HH> + |VV> (ignoring normalisation). If you have one of those photons it looks unpolarised (essentially your “ignorance of the random seed” can be thought of as your ignorance of the polarisation of the other photon).
If someone else (possibly outside your light cone) measures the other photon in the HV basis then half the time they will project your photon into |H> and half the time into |V>, each with 50% probability. This 50⁄50 appears in the density matrix, not the wavefunction, so is “ignorance probability”.
In this case, by what I understand to be your position, the fact of the matter is either (1) that the photon is still entangled with a distant photon, or (2) that it has been projected into a specific polarisation by a measurement on that distant photon. Its not clear when the transformation from (1) to (2) takes place (if its instant, then in which reference frame?).
So, in the bigger context of this conversation,
OP: “You live in the density matrices (Neo)”
Charlie :”No, a density matrix incorporates my own ignorance so is not a sensible picture of the fundamental reality. I can use them mathematically, but the underlying reality is built of quantum states, and that randomness when I subject them to measurements is fundamentally part of the territory, not the map. Lets not mix the two things up.”
Me: “Whether a given unit of randomness is in the map (IE ignorance), or the territory is subtle. Things that randomly combine quantum states (my first machine) have a symmetry over which underlying quantum states are being mixed that looks meaningful. Plus (this post), the randomness can move abruptly from the territory to the map due to events outside your own light cone (although the amount of randomness is conserved), so maybe worrying too much about the distinction isn’t that helpful.