IIUC, complex numbers are a number system that supports rotations -- one representation is as an angle and a magnitude. As such they work well at describing systems that have rotational components. This makes them useful for working with waves like in QM (light, etc.) and Fourier transformations/analysis (sine waves) which is why they are used in QM.
If you exclude non-real operations and states you are removing part of the system such that it becomes impossible to work with certain cases -- like handling non-real roots of ax^2 + bx + c polynomials.
It is possible to represent complex numbers as 2x2 matrices as those can encode 2D rotations. With the matrix formulation you are not dealing with imaginary numbers -- or you are, but they are not encoded with i = sqrt(-1) but as a 45deg rotation. IIRC, there is a formulation of Dirac's QED (Quantum ElectroDynamics) using matrices.
A function (which is an isomorphism) from complex numbers a+bi to matrices is a+bi |-> [[a,-b],[b,a]] where the matrix is listed by rows. So i is sent to the matrix R with a 0 in the top left, 1 in the bottom left, 0 in the bottom right and a -1 in the top right. R is a 90 degree rotation, you can check that it sends the unit vector [1,0] on the x-axis to [0,1], and the unit vector [0,1] on the y-axis to [-1,0].
Even simpler, complex numbers are really 2D vectors with addition and multiplication defined: a field. There's nothing "imaginary" about that second dimension, very frustrating to see them defined that way because it makes people think of it as an "escape hatch" out of real numbers. When you're working with complex numbers, you are working with a different system: `5 + 0i` is still a complex number because it's really `(5, 0)`.
My mental model is that complex numbers are the first of the basic number systems that no longer has a total ordering. That alone is super useful for it.
Quantum is an odd one, as the name indicates that it deals in quantums. Minimum values that can't be divided. The difficult parts seems more to be in systems that have a probability space more than an analytical model that describes them. Which, fair, it is not a number system.
So this is obviously an incredibly technical post. And I can't claim to understand half of it. But I do have one question that may or may not be intelligent. Given that preexisting entanglement is the issue, does that entanglement get "used up" or not? Will it be possible to drain it all by testing for long enough?
I find that cases like this represent one of the biggest problems in today’s research: once someone falsifies something, an entire branch of research gets cut off completely as nobody wants to pursue that path anymore, understandably. But if the “proof” is in fact wrong, then you actually just hid a big part of the research surface to everybody. And usually that’s also where progress is made: when, despite proof, research is pursued because of a gut feeling. Stay skeptic!
What was wrong with the proof in this case? The paper explicitly states and acknowledges the issue raised by this article before the author was aware of it. The author of the article just contends that it is an experimental issue to set up unentangled initial states which are required for the experiment, and indeed someone who was going to perform the experiment needs to convincing demonstrate the assumptions are met.
The author even admits this "is better than doing no test at all".
Nothing, except the perception of what was said and what was actually said. (The same happend to Bells inequality actually)
“ (…) you can just mimic the behavior of complex numbers using pairs of real numbers (and appropriately tweaked definitions of operations).
(…) What Renou et al are actually claiming is that if you start with quantum mechanics, and then remove all operations and states involving non-real numbers, and then try to emulate what was lost using what remains, you will fail in an experimentally detectable way”
Meaning it’s actually totally possible to only use reals to encode the complex Numbers, but not to also remove all operators which do the same things as the complex numbers would.
Quantum computing research feels like one of those things whose greatest effort would likely be classified research. In fact, you could argue the article in the OP looks like well-poisoning based on the author's conclusions.
Frankly I am so tired of this whole branch of research where people try to be foundational about "quantum theory" but at the same time boil it down to qubits, gates, bell tests and, well, two-by-two matrices.
Here is my viewpoint, which somehow some people find controversial: quantum theory is first and foremost a description of individual particles. To describe their time evolution, we use the Schrodinger equation:
i d_t Psi = H Psi
What is that "i" there? Oh right, the imaginary unit. So... quantum theory uses complex numbers.
Now you are free to search for another theory without the "i", and perhaps even find something that is somehow mathematically consistent. But that theory either describes experiments just as well as ordinary quantum theory, in which case it is physically equivalent and of no advantage (except to those with strong allergies to complex numbers), or it does not, and then it is wrong.
Of course the last logical possibility is that your theory might do better than quantum theory... but that is the dream only of those who do not known quantum field theory.
There is really nothing to the appearance of complex numbers in QM. In QM we must design wave functions which do the double duty of representing the probability of measurement outcomes AND capture the symmetries implicit in the system related to the fact that there are degrees of freedom between preparation of a state and measurement (for example, we may rotate our detector any way we wish before we make a measurement of a particle in a given prepared spin state). To accomplish this we need some number-like objects to denote our wave function in that square to real numbers but have enough structure to represent (in this case) the rotations.
As you venture further into the universe of QFT you find that you need even more exotic number like objects like spinors with their own peculiar structures, but the essence is the same: they must serve the purpose of representing probabilities and symmetries. The complex numbers in QM mean nothing at all except in that they serve these purposes.
If we wish to speak informally and wave our hands a bit we can say that it isn't so surprising that we find the complex numbers and related number like objects because the complex numbers are a promise to square something at a later date and recover a real number, which is what we need to satisfy the requirement to represent probabilities.
In fact, we can formulate classical probabilistic mechanics with complex numbers (the Koopman von Neuman operator theory) and again, they appear because we want to operate on objects living in a nice Hilbert space which also square to probabilities. In only took me 20 years to understand this, so I can sympathize with confusion.
It's a long time since I read it, but there's a book called "The Structure and Interpretation of Quantum Mechanics" [1] by R. I. G. Hughes. The "Structure" part of it begins by building up most of the mathematical framework (including use of complex numbers, Hilbert spaces, operators, etc), motivated only by the desire to build a physical theory that is probabilistic in nature. It then shows how you can add one extra ingredient that turns the framework into that used for quantum mechanics [2]. I assume that everything discussed up to that point applies equally to Koopman-von Neumann.
It's a really nice book, very self-contained. I think anyone with a basic mathematical education (A-Level or equivalent) could get through it without having to read other things to acquire prerequisites, though they should be prepared to think quite hard.
1. The resemblance to the titles of Gerald Jay Sussman's "Structure and Interpretation" books appears to be coincidental. The title is meant literally: the book is split into two sections, one on the (mathematical) structure of QM and one on its (philosophical) interpretation. There are no similarities in style, pedagogy or subject matter to Sussmann's books and no use of, or reference to, programming. The author was a professor of philosophy at the University of South Carolina.
2. He actually lists a collection of alternatives for that extra ingredient, any one of which has the same effect when added.
Yes, the post is focusing on the overall effect of operations (unitaries) rather than their continuous trajectories (hamiltonians acting on system via Schrodinger equation) (analogous to working with impulses rather than forces).
To make the continuous case interesting as a compilation problem, you'd need some alternate formulation of the Schrodinger equation, e.g. based on the limit of small powers of unitaries rather than on the matrix exponential, so that deleting i didn't delete literally all processes. Or you could arbitrarily declare real-only hamiltonians are permitted, despite the Schrodinger equation saying "i". But that'd be kinda lame, imo.
The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness.
Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way.
I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.
The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data.
Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.
And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity.
Quantum mechanics likewise is just an approximation of quantum field theories.
It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.
More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were.
If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.
The phase space formulation of QM uses less complex numbers than the Schrodinger one: it models states using quasi-probability distributions, where the "probabilities" behave in all the usual ways except they can go negative. Interestingly, the classical limit of this (that is, when h goes to zero) still has negative probabilities in it.
IIUC, complex numbers are a number system that supports rotations -- one representation is as an angle and a magnitude. As such they work well at describing systems that have rotational components. This makes them useful for working with waves like in QM (light, etc.) and Fourier transformations/analysis (sine waves) which is why they are used in QM.
If you exclude non-real operations and states you are removing part of the system such that it becomes impossible to work with certain cases -- like handling non-real roots of ax^2 + bx + c polynomials.
It is possible to represent complex numbers as 2x2 matrices as those can encode 2D rotations. With the matrix formulation you are not dealing with imaginary numbers -- or you are, but they are not encoded with i = sqrt(-1) but as a 45deg rotation. IIRC, there is a formulation of Dirac's QED (Quantum ElectroDynamics) using matrices.
A function (which is an isomorphism) from complex numbers a+bi to matrices is a+bi |-> [[a,-b],[b,a]] where the matrix is listed by rows. So i is sent to the matrix R with a 0 in the top left, 1 in the bottom left, 0 in the bottom right and a -1 in the top right. R is a 90 degree rotation, you can check that it sends the unit vector [1,0] on the x-axis to [0,1], and the unit vector [0,1] on the y-axis to [-1,0].
Even simpler, complex numbers are really 2D vectors with addition and multiplication defined: a field. There's nothing "imaginary" about that second dimension, very frustrating to see them defined that way because it makes people think of it as an "escape hatch" out of real numbers. When you're working with complex numbers, you are working with a different system: `5 + 0i` is still a complex number because it's really `(5, 0)`.
My mental model is that complex numbers are the first of the basic number systems that no longer has a total ordering. That alone is super useful for it.
Quantum is an odd one, as the name indicates that it deals in quantums. Minimum values that can't be divided. The difficult parts seems more to be in systems that have a probability space more than an analytical model that describes them. Which, fair, it is not a number system.
So this is obviously an incredibly technical post. And I can't claim to understand half of it. But I do have one question that may or may not be intelligent. Given that preexisting entanglement is the issue, does that entanglement get "used up" or not? Will it be possible to drain it all by testing for long enough?
No, the pre-shared states are never consumed. They are catalysts, not fuel.
I find that cases like this represent one of the biggest problems in today’s research: once someone falsifies something, an entire branch of research gets cut off completely as nobody wants to pursue that path anymore, understandably. But if the “proof” is in fact wrong, then you actually just hid a big part of the research surface to everybody. And usually that’s also where progress is made: when, despite proof, research is pursued because of a gut feeling. Stay skeptic!
What was wrong with the proof in this case? The paper explicitly states and acknowledges the issue raised by this article before the author was aware of it. The author of the article just contends that it is an experimental issue to set up unentangled initial states which are required for the experiment, and indeed someone who was going to perform the experiment needs to convincing demonstrate the assumptions are met.
The author even admits this "is better than doing no test at all".
Nothing, except the perception of what was said and what was actually said. (The same happend to Bells inequality actually)
“ (…) you can just mimic the behavior of complex numbers using pairs of real numbers (and appropriately tweaked definitions of operations). (…) What Renou et al are actually claiming is that if you start with quantum mechanics, and then remove all operations and states involving non-real numbers, and then try to emulate what was lost using what remains, you will fail in an experimentally detectable way”
Meaning it’s actually totally possible to only use reals to encode the complex Numbers, but not to also remove all operators which do the same things as the complex numbers would.
Quantum computing research feels like one of those things whose greatest effort would likely be classified research. In fact, you could argue the article in the OP looks like well-poisoning based on the author's conclusions.
Frankly I am so tired of this whole branch of research where people try to be foundational about "quantum theory" but at the same time boil it down to qubits, gates, bell tests and, well, two-by-two matrices.
Here is my viewpoint, which somehow some people find controversial: quantum theory is first and foremost a description of individual particles. To describe their time evolution, we use the Schrodinger equation:
i d_t Psi = H Psi
What is that "i" there? Oh right, the imaginary unit. So... quantum theory uses complex numbers.
Now you are free to search for another theory without the "i", and perhaps even find something that is somehow mathematically consistent. But that theory either describes experiments just as well as ordinary quantum theory, in which case it is physically equivalent and of no advantage (except to those with strong allergies to complex numbers), or it does not, and then it is wrong.
Of course the last logical possibility is that your theory might do better than quantum theory... but that is the dream only of those who do not known quantum field theory.
/rant, with apologies
There is really nothing to the appearance of complex numbers in QM. In QM we must design wave functions which do the double duty of representing the probability of measurement outcomes AND capture the symmetries implicit in the system related to the fact that there are degrees of freedom between preparation of a state and measurement (for example, we may rotate our detector any way we wish before we make a measurement of a particle in a given prepared spin state). To accomplish this we need some number-like objects to denote our wave function in that square to real numbers but have enough structure to represent (in this case) the rotations.
As you venture further into the universe of QFT you find that you need even more exotic number like objects like spinors with their own peculiar structures, but the essence is the same: they must serve the purpose of representing probabilities and symmetries. The complex numbers in QM mean nothing at all except in that they serve these purposes.
If we wish to speak informally and wave our hands a bit we can say that it isn't so surprising that we find the complex numbers and related number like objects because the complex numbers are a promise to square something at a later date and recover a real number, which is what we need to satisfy the requirement to represent probabilities.
In fact, we can formulate classical probabilistic mechanics with complex numbers (the Koopman von Neuman operator theory) and again, they appear because we want to operate on objects living in a nice Hilbert space which also square to probabilities. In only took me 20 years to understand this, so I can sympathize with confusion.
It's a long time since I read it, but there's a book called "The Structure and Interpretation of Quantum Mechanics" [1] by R. I. G. Hughes. The "Structure" part of it begins by building up most of the mathematical framework (including use of complex numbers, Hilbert spaces, operators, etc), motivated only by the desire to build a physical theory that is probabilistic in nature. It then shows how you can add one extra ingredient that turns the framework into that used for quantum mechanics [2]. I assume that everything discussed up to that point applies equally to Koopman-von Neumann.
It's a really nice book, very self-contained. I think anyone with a basic mathematical education (A-Level or equivalent) could get through it without having to read other things to acquire prerequisites, though they should be prepared to think quite hard.
1. The resemblance to the titles of Gerald Jay Sussman's "Structure and Interpretation" books appears to be coincidental. The title is meant literally: the book is split into two sections, one on the (mathematical) structure of QM and one on its (philosophical) interpretation. There are no similarities in style, pedagogy or subject matter to Sussmann's books and no use of, or reference to, programming. The author was a professor of philosophy at the University of South Carolina.
2. He actually lists a collection of alternatives for that extra ingredient, any one of which has the same effect when added.
It is one of my favorites.
Yes, the post is focusing on the overall effect of operations (unitaries) rather than their continuous trajectories (hamiltonians acting on system via Schrodinger equation) (analogous to working with impulses rather than forces).
To make the continuous case interesting as a compilation problem, you'd need some alternate formulation of the Schrodinger equation, e.g. based on the limit of small powers of unitaries rather than on the matrix exponential, so that deleting i didn't delete literally all processes. Or you could arbitrarily declare real-only hamiltonians are permitted, despite the Schrodinger equation saying "i". But that'd be kinda lame, imo.
(Note: am author of post)
The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness.
Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way.
I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.
The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data.
Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.
And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity.
Quantum mechanics likewise is just an approximation of quantum field theories.
It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.
More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were.
If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.
The phase space formulation of QM uses less complex numbers than the Schrodinger one: it models states using quasi-probability distributions, where the "probabilities" behave in all the usual ways except they can go negative. Interestingly, the classical limit of this (that is, when h goes to zero) still has negative probabilities in it.
>Not allowing the players to come into the game with entangled states is really, really strange.
I think i saw such a warning on a casino door in LV.
How did they check?
They just observe you, then you're good to come in.