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Discrimination of dead-alive superpositions allows resurrection

...but the confusion about these simple insights shows some people's trouble with the state-dependence and indeed, with the universal rules of quantum mechanics, too...

In a new, 12-page-long quant-ph preprint,

On the Hardness of Detecting Macroscopic Superpositions,
Aaronson and Atia (Austin) and Susskind (Stanford & the Google Evil Corporation) make an interesting observation, and I think that it is basically a correct one:
If you had the measurement of Schrödinger-cat-like superpositions \(\ket{{\rm dead}} + \ket{{\rm alive}}\) under full control, you could slightly extend your gadget and the extended gadget would also allow you to resurrect the cat.
In this form, I think that the statement is correct. The detection of complicated superpositions is mostly equivalent to the reversal of decoherence; and it is also, perhaps less trivially, equivalent to the ability to transform the distinct states onto each other (which means to turning dead cats into alive ones).

These dudes dedicate 12 rather complicated pages to the demonstration of this statement – and its generalization which says that if one of the things may be done partially, the others may also be done partially. But I am just annoyed by the degree of obfuscation and the rather striking sloppiness in their language.

So how does it work? Imagine that you have two very specific microstates of a cat, \(\ket{{\rm dead}}\) and \(\ket{{\rm alive}}\), and you play with the 2D Hilbert space of the superpositions. This is already an utterly unrealistic assumption because a cat carries a huge entropy, whether it is alive or dead (so there are exponentially many microstates), and even if you managed to pick a unique microstate for an alive cat, a cat may be killed in exponentially many ways, so there would be lots of the dead microstates. But let's assume that only two states are relevant. The real point is that the superpositions are \[ \ket \psi = a\ket{{\rm alive}}+b \ket{{\rm dead}} \] where \(a,b\in\CC\) are complex numbers. It seems that they want to write their preprint as a partially popular one and the language of the abstract is just terribly anti-quantum-mechanical. They wrote:
if one had a quantum circuit to determine if a system was in an equal superposition of two orthogonal states...
Sorry but if the coefficients above obey \(|a|^2 = |b|^2\), then it is right to call the state \(\ket{\psi}\) "an equal superposition" of the (orthogonal) dead and alive basis vectors. There is absolutely nothing special about the situation when the "relative phase is positive real". In fact, the phase of all these vectors is a pure convention and there doesn't exist any preferred choice of this convention. Not only that, all the phases brutally evolve with time (typically at different rates for "dead" and for "alive"). And because the "dead" and "alive" states are almost certainly not energy eigenstates, they evolve much more than by changing their prefactor (the phase) which is why any "fixed" two-dimensional Hilbert space will be insufficient for the full monitoring of (and manipulation with) the cat.

So their suggestion that the sum "dead plus alive" is an equal superposition while "dead minus alive" is not an equal superposition is a brutal misinterpretation of the role of the relative phases in quantum mechanics (the phases, including signs, are "unphysical" which is how they differ from the probabilities themselves whose sign change or phase change would be rather dramatic, and impossible). Whether two states' superposition is an "equal superposition" surely mustn't depend on the relative phase. But more generally, they talk about the measurement apparatus ("quantum circuit") that does a certain measurement. And they imagine that the measurement decides
...if a system was in an equal superposition of two...
This is a very weird way of organizing quantum mechanics because it implicitly suggests that we may "measure the state" directly. In quantum mechanics, we don't measure states (wave functions aren't observable in the colloquial sense which is almost the same statement as "wave functions aren't observables" in the technical mathematical sense), we measure observables (Hermitian linear operators). Great, in this case, we may construct the operator that their "quantum circuit" is supposed to measure by the assumption:\[ L = \frac{ (\ket{{\rm dead}} + \ket{{\rm alive}}) (\bra{{\rm dead}} + \bra{{\rm alive}}) }{2} \] which is a Hermitian linear operator, a projection operator on the state that is the "sum" superposition. But indeed, as soon as you write the operator \(L\) in this way, you see that your "quantum circuit" has some amazing abilities. It not only measures the relative phase between the "dead" and "alive" states because it acts as \(1\) on the "sum superposition" but as \(0\) on the "difference superposition". This operator \(L\) also kills all the other microstates of the cat, whether they behave as dead or alive cats (or dogs).

Much more natural measurements obviously never look like \(L\) above. This \(L\) is only "supported" by the highly cherry-picked two-dimensional subspace of the Hilbert space and the cherry-picking of this two-dimensional space is at least as "impossible in practice" as all the other operations that they and we discuss; well, in the "sum-difference" basis, the operator is only supported by a one-dimensional Hilbert space because it is a projection operator on a single particular state (or the one-dimensional space spanned by it). But regular gadgets also "do measure" something about many other microstates you may enter and they will generically produce nonzero results for most of them.

My point is that the operator \(L\) doesn't really "measure if the object is in a particular superposition". Instead, it is an operator that measures the relative phase between "dead" and "alive" at a given moment. And indeed, it is totally trivial to see that once you have a gadget that can measure the "relative phase", basically the same gadget may also switch one state to the other – it may transform "dead" into "alive" and vice versa.

Why is it so? This is just a damn simple two-dimensional Hilbert space. Every undergrad should have known everything about 2-dimensional Hilbert spaces e.g. from the Feynman Lectures on Physics. So such a space is equivalent e.g. to the spin-1/2 particle's spin. And all the observables are real combinations of \(1,\sigma_x,\sigma_y,\sigma_z\), the (identity and) Pauli matrices! If you have a gadget that can measure the relative phase, it is a gadget that measures \(\sigma_x\) or \(\sigma_y\) or their combinations, the "totally off-diagonal" \(2\times 2 \) matrices.

Without a loss of generality, you may say that such a gadget or circuit measures \(\sigma_x\), or it transforms the system so that the measurement of \(\sigma_x\) is transformed to the much easier measurement of \(\sigma_z\). But if you can manipulate the 2-component state in this fine way, you may obviously turn it into a gadget that rotates the state by 90 degrees in the dead-alive plan, i.e. that maps \(a\to b \) and \(b\to a\), among other things. Why? Because the matrix \(\sigma_x\) is simply exchanging the two amplitudes! So yes, this gadget turns alive cats into dead cats and, more impressively, vice versa.

Well, we may adjust the argument in the previous argument to their "quantum circuit" as they imagined it. They actually imagine that they have a quantum circuit that may turn the "difficult" measurement of \(\sigma_x\) to the "easy" measurement of the diagonal \(\sigma_z\). What does this circuit do in the 2-dimensional space? Well, it must rotate the coordinates \(a,b\) into \((a+b)/\sqrt 2, (a-b)/\sqrt 2\). But that's nothing else than some form of a rotation of the 2 coordinates by 45 degrees. That doesn't cripple your ability to do the step from the previous paragraph. If you repeat a rotation by 45 degrees twice, you get a rotation by 90 degrees, e.g. \(a\to b\), \(b\to -a\).

Mathematically, all these would-be impressive operations are exactly analogous to the mundane measurement of the electron's spin with respect to another axis such as \(x\). If your gadget can only measure the \(z\)-component of the spin, you may measure the \(x\)-component by simply rotating the electron by 90 degrees at the beginning (around the \(y\)-axis, for example). You can achieve it by adding the magnetic field \(\vec B\) in the direction of the \(y\)-axis. Needless to say, this magnetic field is pretty much sufficient for measuring \(\sigma_y\) itself: it sends the electron along one of the two trajectories depending on the \(y\)-component of the spin. I suppose that you will learn or recall how the components of the electron's spin are measured and manipulated. They don't do anything else; they just give fancy names to these trivial problems – while they omit all the extra difficulties that would arise if you dealt with a cat and not an electron.

So yes, indeed, the "circuits" that can rotate the Hilbert space (e.g. by 45 degrees) along some difficult axis "for the purpose of measurements" may also rotate the Hilbert space "for the purpose of resurrection". This is a totally trivial consequence of the basic quantum mechanical axioms or thinking. And we don't really need the 45-degree rotations to make the general point. The exchange of the two coordinates is simpler and has the same consequence. If you can measure \(\sigma_x\), your gadget to measure \(\sigma_x\) also has the effect of swapping "up" and "down" or, as they are called here, "alive" and "dead". In particular, it resurrects a dead cat.

I just find it silly to spend many pages with this trivial point. They spend so much time with this trivial point exactly because they add lots of the obfuscation that makes their text "always a little bit wrong", as Feynman would say about the "energy makes it go" physics textbooks. They want to present the quantum mechanical measurements as if they were classical measurements. And when it is done so, the demonstration of their trivial statement becomes both more convoluted; and the very trivial statement looks more mysterious, too.

They claim that their insights have consequences for the state dependence in quantum gravity. I am sorry but everyone who discusses difficult things such as state dependence in quantum gravity should first understand the basic mathematics and physics of two-dimensional Hilbert spaces in quantum mechanics. Aaronson, Atia, and Susskind are clearly still confused about that basic undergraduate material of quantum mechanics. Their paper doesn't talk about any field-like operators including the metric tensor which is why it has nothing whatsoever to say about "quantum gravity", at least not about the parts of "quantum gravity" that go beyond the "elementary quantum mechanics".

It is surely true that the construction of gadgets to measure or transform states just like \(\sigma_x\) in large Hilbert spaces describing complex objects becomes hard. But I don't think that there is any totally universal definition of the "hardness" (like some physics-universal "complexity" quantity). A way to refine the notion of "hardness" is to count "the number of components that a gadget may have". But the resulting number of components obviously depends on the "list of component types that you allow" and the rules how they can be added together. In the context of a realistic theory of quantum gravity – which contains both gravity and other forces, as the swampland reasoning basically implies – this is a totally "vacuum-dependent" or, more generally, "environment-dependent" question. There can't be any definition of the hardness that produces the right result for "all superselection sectors of quantum gravity" simultaneously.

That is a general reasoning why I find the bulk of Susskind's and similar papers dealing with "complexity as a measure of something in quantum gravity" incorrect. He just assumes that the complexity-like quantities are completely universal across the quantum gravity Hilbert spaces. But they are actually totally dependent on the vacuum. This dependence could actually be said to be at least "morally the same thing" as the state dependence of quantum gravity. But to show that Susskind's reasoning is misguided, it is enough to realize much more innocent facts such as the fact that (see The Elegant Universe LOL which is totally enough here): Dualities typically transform very difficult problems to very easy ones.

The funny ability of the string dualities is that some objects that looks like insanely complex bound states or composites of the simplest objects become "elementary and simple" in a dual description but they may be used just like the original "elementary" objects. So you may construct your "intermediate" complex states using totally different "elementary" objects and which composites are "simple" and which are "complex" gets totally rearranged, and in many situations, the complexity is largely rotated upside down.

One must obviously be careful about "which list of elementary particles or fields or observables" is allowed when we want to discuss a proposed quantity such as "the complexity of a state". Everyone who understands the very meaning of dualities must agree. Because the basic allowed observables are normally understood as some excitations of the empty space (the vacuum), the complexity function defined for whole Hilbert spaces is always state-dependent. A black hole microstate does look like the empty space "in most of the volume" and in combination with the previous statement, it means that any notion of complexity must depend on the chosen pure state, too. At some level, this is really another complete proof of the state dependence.

I found it increasingly clear that some people's opposition to the "state dependence" was nothing else than another manifestation of their highly imperfect (to put it mildly) understanding of the universal rules of quantum mechanics. Some people want the field operators etc. to be "state-independent" for the very same reason why they want observations to be "observer-independent". But all observations in quantum mechanics are unavoidably "observer-dependent": they simply want to keep on thinking about the Universe classically. A priori, all observables (Hemitian linear operators on a Hilbert space) are "equally allowed" as all others. Whether one operator looks simpler because it is "more diagonal" than others totally depends on the chosen basis (or Fock-like construction) of the Hilbert space and none of the bases may be said to be "universally preferred" across the landscape of string theory, or similarly across the Hilbert space of some black hole microstates.

So the possible quantitatively well-defined schemes to measure the "complexity of states" are as numerous as the possible environments that allow observers to do things and mentally divide them to elementary particles or elementary processes. Those are equivalent to "systems of field operators within quantum gravity" and there are many of them, basically as many of them as the number of "string vacua". The general qualitative philosophical insights, e.g. that some measurement apparatuses look simpler than others, are valid. But no quantitative definitions of the complexity and similar things may ever be state-independent or observer-independent. The entropy (the log of the dimension of a relevant Hilbert space) is the only characteristic of a "physical system" that doesn't depend on the choice of the observer and his interpretation "what the empty space is". But even the entropy becomes debatable once you realize that every observer may always imagine that there's a "totally disconnected component of the space" which carries some extra entropy. The normal desire to "subtract this entropy and the inaccessible degrees of freedom" depends on the statement that they're inaccessible. But the inaccessibility still depends on the Hamiltonian which is (at least when it comes to details) dependent on the chosen vacuum and gauges, too.

Susskind wants to turn the "most important ideas in physics" to something that is independent of the algebra of the field-like operators (which includes the information about the mutual relationships of these operators and their evolution – which is given by their relationship with some Hamiltonian operator, anyway). But everything in quantum physics depends on the field-like operators, otherwise it is not quantum physics although it may be the basis of some computer science papers or at least mathematical masturbation papers inspired by physics and by computer science.

And that is the memo.

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