Shtetl-Optimized

Last night, two papers appeared on the quantum physics arXiv that my coauthors and I have been working on for more than a year, and that I’m pretty happy about.

The first paper, with Guy Rothblum, is Gentle Measurement of Quantum States and Differential Privacy (85 pages, to appear in STOC’2019). This is Guy’s first paper that has anything to do with quantum, and also my first paper that has anything to do with privacy. (What do I care about privacy? I just share everything on this blog…) The paper has its origin when I gave a talk at the Weizmann Institute about “shadow tomography” (a task where you have to measure quantum states very carefully to avoid destroying them), and Guy was in the audience, and he got all excited that the techniques sounded just like what they use to ensure privacy in data-mining, and I figured it was just some wacky coincidence and brushed him off, but he persisted, and it turned out that he was 100% right, and our two fields were often studying the same problems from different angles and we could prove it. Anyway, here’s the abstract:

In differential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O(α)-DP, and any product measurement that is ε-DP is also O(ε√n)-gentle.

Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown d-dimensional quantum state ρ, as well as known two-outcome measurements E₁,…,E_m, shadow tomography asks us to estimate Pr[E_i accepts ρ], for every i∈[m], by measuring few copies of ρ. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)²(log d)²) copies of ρ, compared to Aaronson’s previous bound of ~O((log m)⁴(log d)). Our protocol has the advantages of being online (that is, the E_i‘s are processed one at a time), gentle, and conceptually simple.

Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.

The second paper, with Robin Kothari, UT Austin PhD student William Kretschmer, and Justin Thaler, is Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. Here’s the abstract:

Given only a membership oracle for S, it is well-known that approximate counting takes Θ(√(N/|S|)) quantum queries. But what if a quantum algorithm is also given “QSamples”—i.e., copies of the state |S⟩=Σ_i∈S|i⟩—or even the ability to apply reflections about |S⟩? Our first main result is that, even then, the algorithm needs either Θ(√(N/|S|)) queries or else Θ(min{|S|^1/3,√(N/|S|)}) reflections or samples. We also give matching upper bounds.

We prove the lower bound using a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents. We lower-bound Laurent polynomial degree using two methods: a new “explosion argument” that pits the positive- and negative-degree parts of the polynomial against each other, and a new formulation of the dual polynomials method.

Our second main result rules out the possibility of a black-box Quantum Merlin-Arthur (or QMA) protocol for proving that a set is large. More precisely, we show that, even if Arthur can make T quantum queries to the set S⊆[N], and also receives an m-qubit quantum witness from Merlin in support of S being large, we have Tm=Ω(min{|S|,√(N/|S|)}). This resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.

Note that QMA is “stronger” than the queries+QSamples model in that Merlin’s witness can be anything, rather than just the specific state |S⟩, but also “weaker” in that Merlin’s witness cannot be trusted. Intriguingly, Laurent polynomials also play a crucial role in our QMA lower bound, but in a completely different manner than in the queries+QSamples lower bound. This suggests that the “Laurent polynomial method” might be broadly useful in complexity theory.

I need to get ready for our family’s Seder now, but after that, I’m happy to answer any questions about either of these papers in the comments.

Meantime, the biggest breakthrough in quantum complexity theory of the past month isn’t either of the above: it’s the paper by Anand Natarajan and John Wright showing that MIP*, or multi-prover interactive proof systems with entangled provers, contains NEEXP, or nondeterministic doubly-exponential time (!!). I’ll try to blog about this later, but if you can’t wait, check out this excellent post by Thomas Vidick.

Recently a Twitter account started called justsaysinmice. The only thing this account does, is to repost breathless news articles about medical research breakthroughs that fail to mention that the effect in question was only observed in mice, and then add the words “IN MICE” to them. Simple concept, but it already seems to be changing the conversation about science reporting.

It occurred to me that we could do something analogous for quantum computing. While my own deep-seated aversion to Twitter prevents me from doing it myself, which of my readers is up for starting an account that just reposts one overhyped QC article after another, while appending the words “A CLASSICAL COMPUTER COULD ALSO DO THIS” to each one?

The purpose of this post is mostly just to signal-boost Konstantin Kakaes’s article in MIT Technology Review, entitled “No, scientists didn’t just ‘reverse time’ with a quantum computer.” The title pretty much says it all—but if you want more, you should read the piece, which includes the following droll quote from some guy calling himself “Director of the Quantum Information Center at the University of Texas at Austin”:

If you’re simulating a time-reversible process on your computer, then you can ‘reverse the direction of time’ by simply reversing the direction of your simulation. From a quick look at the paper, I confess that I didn’t understand how this becomes more profound if the simulation is being done on IBM’s quantum computer.

Incredibly, the time-reversal claim has now gotten uncritical attention in Newsweek, Discover, Cosmopolitan, my Facebook feed, and elsewhere—hence this blog post, which has basically no content except “the claim to have ‘reversed time,’ by running a simulation backwards, is exactly as true and as earth-shattering as a layperson might think it is.”

If there’s anything interesting here, I suppose it’s just that “scientists use a quantum computer to reverse time” is one of the purest examples I’ve ever seen of a scientific claim that basically amounts to a mind-virus or meme optimized for sharing on social media—discarding all nontrivial “science payload” as irrelevant to its propagation.

Manolis Kellis is a computational biologist at MIT, known as one of the leaders in applying big data to genomics and gene regulatory networks. Throughout my 9 years at MIT, Manolis was one of my best friends there, even though our research styles and interests might seem distant. He and I were in the same PECASE class; see if you can spot us both in this photo (in the rows behind America’s last sentient president). My and Manolis’s families also became close after we both got married and had kids. We still keep in touch.

Today Manolis will be celebrating his 42nd birthday, with a symposium on the meaning of life (!). He asked his friends and colleagues to contribute talks and videos reflecting on that weighty topic.

Here’s a 15-minute video interview that Manolis and I recorded last night, where he asks me to pontificate about the implications of quantum mechanics for consciousness and free will and whether the universe is a computer simulation—and also about, uh, how to balance blogging with work and family.

Also, here’s a 2-minute birthday video that I made for Manolis before I really understood what he wanted. Unlike the first video, this one has no academic content, but it does involve me wearing a cowboy hat and swinging a makeshift “lasso.”

Happy birthday Manolis!

A few weeks ago, I was at QIP’2019 in Boulder, CO. This week I was at SQuInT’2019 in Albuquerque, NM. There were lots of amazing talks—feel free to ask in the comments section.

There’s an interview with me at the website “GigaOm,” conducted by Byron Reese and entitled Quantum Computing: Capabilities and Limits. I didn’t proofread the transcript and it has some errors in it, but hopefully the meaning comes through. In other interview news, if you were interested in my podcast with Adam Ford in Melbourne but don’t like YouTube, Adam has helpfully prepared transcripts of the two longest segments: The Ghost in the Quantum Turing Machine and The Winding Road to Quantum Supremacy.

The New York Times ran an article entitled The Hard Part of Computer Science? Getting Into Class, about the surge in computer science majors all over the US, and the shortage of professors to teach them. The article’s go-to example of a university where this is happening is UT Austin, and there’s extensive commentary from my department chair, Don Fussell.

The STOC’2019 accepted papers list is finally out. Lots of cool stuff!

Greetings from QIP’2019 in Boulder, Colorado! Obvious highlights of the conference include Urmila Mahadev’s opening plenary talk on her verification protocol for quantum computation (which I blogged about here), and Avishay Tal’s upcoming plenary on his and Ran Raz’s oracle separation between BQP and PH (which I blogged about here). If you care, here are the slides for the talk I just gave, on the paper “Online Learning of Quantum States” by me, Xinyi Chen, Elad Hazan, Satyen Kale, and Ashwin Nayak. Feel free to ask in the comments about what else is going on.

I returned a few days ago from my whirlwind Australia tour, which included Melbourne and Sydney; a Persian wedding that happened to be held next to a pirate ship (the Steve Irwin, used to harass whalers and adorned with a huge Jolly Roger); meetings and lectures graciously arranged by friends at UTS; a quantum computing lab tour personally conducted by 2018 “Australian of the Year” Michelle Simmons; three meetups with readers of this blog (or more often, readers of the other Scott A’s blog who graciously settled for the discount Scott A); and an excursion to Grampians National Park to see wild kangaroos, wallabies, koalas, and emus.

But the thing that happened in Australia that provided the actual occassion for this post is this: I was interviewed by Adam Ford in Carlton Gardens in Melbourne, about quantum supremacy, AI risk, Integrated Information Theory, whether the universe is discrete or continuous, and to be honest I don’t remember what else. You can watch the first segment, the one about the prospects for quantum supremacy, here on YouTube. My only complaint is that Adam’s video camera somehow made me look like an out-of-shape slob who needs to hit the gym or something.

Update (Jan. 16): Adam has now posted a second video on YouTube, wherein I talk about my “Ghost in the Quantum Turing Machine” paper, my critique of Integrated Information Theory, and more.

And now Adam has posted yet a third segment, in which I talk about small, lighthearted things like existential threats to civilization and the prospects for superintelligent AI.

And a fourth, in which I talk about whether reality is discrete or continuous.

Related to the “free will / consciousness” segment of the interview: the biologist Jerry Coyne, whose blog “Why Evolution Is True” I’ve intermittently enjoyed over the years, yesterday announced my existence to his readers, with a post that mostly criticizes my views about free will and predictability, as I expressed them years ago in a clip that’s on YouTube (at the time, Coyne hadn’t seen GIQTM or my other writings on the subject). Coyne also took the opportunity to poke fun at this weird character he just came across whose “life is devoted to computing” and who even mistakes tips for change at airport smoothie stands. Some friends here at QIP had a good laugh over the fact that, for the world beyond theoretical computer science and quantum information, this is what 23 years of research, teaching, and writing apparently boil down to: an 8.5-minute video clip where I spouted about free will, and also my having been arrested once in a comic mix-up at Philadelphia airport. Anyway, since then I had a very pleasant email exchange with Coyne—someone with whom I find myself in agreement much more often than not, and who I’d love to have an extended conversation with sometime despite the odd way our interaction started.

[By prior agreement, this post will be cross-posted on Microsoft’s Q# blog, even though it has nothing to do with the Q# programming language.  It does, however, contain many examples that might be fun to implement in Q#!]

Why should Nature have been quantum-mechanical?  It’s totally unclear what would count as an answer to such a question, and also totally clear that people will never stop asking it.

Short of an ultimate answer, we can at least try to explain why, if you want this or that piece of quantum mechanics, then the rest of the structure is inevitable: why quantum mechanics is an “island in theoryspace,” as I put it in 2003.

In this post, I’d like to focus on a question that any “explanation” for QM at some point needs to address, in a non-question-begging way: why should amplitudes have been complex numbers?  When I was a grad student, it was his relentless focus on that question, and on others in its vicinity, that made me a lifelong fan of Chris Fuchs (see for example his samizdat), despite my philosophical differences with him.

It’s not that complex numbers are a bad choice for the foundation of the deepest known description of the physical universe—far from it!  (They’re a field, they’re algebraically closed, they’ve got a norm, how much more could you want?)  It’s just that they seem like a specific choice, and not the only possible one.  There are also the real numbers, for starters, and in the other direction, the quaternions.

Quantum mechanics over the reals or the quaternions still has constructive and destructive interference among amplitudes, and unitary transformations, and probabilities that are absolute squares of amplitudes.  Moreover, these variants turn out to lead to precisely the same power for quantum computers—namely, the class BQP—as “standard” quantum mechanics, the one over the complex numbers.  So none of those are relevant differences.

Indeed, having just finished teaching an undergrad Intro to Quantum Information course, I can attest that the complex nature of amplitudes is needed only rarely—shockingly rarely, one might say—in quantum computing and information.  Real amplitudes typically suffice.  Teleportation, superdense coding, the Bell inequality, quantum money, quantum key distribution, the Deutsch-Jozsa and Bernstein-Vazirani and Simon and Grover algorithms, quantum error-correction: all of those and more can be fully explained without using a single i that’s not a summation index.  (Shor’s factoring algorithm is an exception; it’s much more natural with complex amplitudes.  But as the previous paragraph implied, their use is removable even there.)

It’s true that, if you look at even the simplest “real” examples of quantum systems—or as a software engineer might put it, at the application layers built on top of the quantum OS—then complex numbers are everywhere, in a way that seems impossible to remove.  The Schrödinger equation, energy eigenstates, the position/momentum commutation relation, the state space of a spin-1/2 particle in 3-dimensional space: none of these make much sense without complex numbers (though it can be fun to try).

But from a sufficiently Olympian remove, it feels circular to use any of this as a “reason” for why quantum mechanics should’ve involved complex amplitudes in the first place.  It’s like, once your OS provides a certain core functionality (in this case, complex numbers), it’d be surprising if the application layer didn’t exploit that functionality to the hilt—especially if we’re talking about fundamental physics, where we’d like to imagine that nothing is wasted or superfluous (hence Rabi’s famous question about the muon: “who ordered that?”).

But why should the quantum OS have provided complex-number functionality at all?  Is it possible to answer that question purely in terms of the OS’s internal logic (i.e., abstract quantum information), making minimal reference to how the OS will eventually get used?  Maybe not—but if so, then that itself would seem worthwhile to know.

If we stick to abstract quantum information language, then the most “obvious, elementary” argument for why amplitudes should be complex numbers is one that I spelled out in Quantum Computing Since Democritus, as well as my Is quantum mechanics an island in theoryspace? paper.  Namely, it seems desirable to be able to implement a “fraction” of any unitary operation U: for example, some V such that V²=U, or V³=U.  With complex numbers, this is trivial: we can simply diagonalize U, or use the Hamiltonian picture (i.e., take e^-iH/2 where U=e^-iH), both of which ultimately depend on the complex numbers being algebraically closed.  Over the reals, by contrast, a 2×2 orthogonal matrix like $$ U = \left(\begin{array}[c]{cc}1 & 0\\0 & -1\end{array}\right)$$

has no 2×2 orthogonal square root, as follows immediately from its determinant being -1.  If we want a square root of U (or rather, of something that acts like U on a subspace) while sticking to real numbers only, then we need to add another dimension, like so: $$ \left(\begin{array}[c]{ccc}1 & 0 & 0\\0 & -1 & 0\\0 & 0&-1\end{array}\right)=\left(\begin{array}[c]{ccc}1 & 0 & 0\\0 & 0 & 1\\0 & -1 & 0\end{array}\right) ^{2} $$

This is directly related to the fact that there’s no way for a Flatlander to “reflect herself” (i.e., switch her left and right sides while leaving everything else unchanged) by any continuous motion, unless she can lift off the plane and rotate herself through the third dimension.  Similarly, for us to reflect ourselves would require rotating through a fourth dimension.

One could reasonably ask: is that it?  Aren’t there any “deeper” reasons in quantum information for why amplitudes should be complex numbers?

Indeed, there are certain phenomena in quantum information that, slightly mysteriously, work out more elegantly if amplitudes are complex than if they’re real.  (By “mysteriously,” I mean not that these phenomena can’t be 100% verified by explicit calculations, but simply that I don’t know of any deep principle by which the results of those calculations could’ve been predicted in advance.)

One famous example of such a phenomenon is due to Bill Wootters: if you take a uniformly random pure state in d dimensions, and then you measure it in an orthonormal basis, what will the probability distribution (p₁,…,p_d) over the d possible measurement outcomes look like?  The answer, amazingly, is that you’ll get a uniformly random probability distribution: that is, a uniformly random point on the simplex defined by p_i≥0 and p₁+…+p_d=1.  This fact, which I’ve used in several papers, is closely related to Archimedes’ Hat-Box Theorem, beloved by friend-of-the-blog Greg Kuperberg.  But here’s the kicker: it only works if amplitudes are complex numbers.  If amplitudes are real, then the resulting distribution over distributions will be too bunched up near the corners of the probability simplex; if they’re quaternions, it will be too bunched up near the middle.

There’s an even more famous example of such a Goldilocks coincidence—one that’s been elevated, over the past two decades, to exalted titles like “the Axiom of Local Tomography.”  Namely: suppose we have an unknown finite-dimensional mixed state ρ, shared by two players Alice and Bob.  For example, ρ might be an EPR pair, or a correlated classical bit, or simply two qubits both in the state |0⟩.  We imagine that Alice and Bob share many identical copies of ρ, so that they can learn more and more about it by measuring this copy in this basis, that copy in that basis, and so on.

We then ask: can ρ be fully determined from the joint statistics of product measurements—that is, measurements that Alice and Bob can apply separately and locally to their respective subsystems, with no communication between them needed?  A good example here would be the set of measurements that arise in a Bell experiment—measurements that, despite being local, certify that Alice and Bob must share an entangled state.

If we asked the analogous question for classical probability distributions, the answer is clearly “yes.”  That is, once you’ve specified the individual marginals, and you’ve also specified all the possible correlations among the players, you’ve fixed your distribution; there’s nothing further to specify.

For quantum mixed states, the answer again turns out to be yes, but only because amplitudes are complex numbers!  In quantum mechanics over the reals, you could have a 2-qubit state like $$ \rho=\frac{1}{4}\left(\begin{array}[c]{cccc}1 & 0 & 0 & -1\\0 & 1 & 1 & 0\\0 & 1 & 1 & 0\\-1& 0 & 0 & 1\end{array}\right) ,$$

which clearly isn’t the maximally mixed state, yet which is indistinguishable from the maximally mixed state by any local measurement that can be specified using real numbers only.  (Proof: exercise!)

In quantum mechanics over the quaternions, something even “worse” happens: namely, the tensor product of two Hermitian matrices need not be Hermitian.  Alice’s measurement results might be described by the 2×2 quaternionic density matrix $$ \rho_{A}=\frac{1}{2}\left(\begin{array}[c]{cc}1 & -i\\i & 1\end{array}\right), $$

and Bob’s results might be described by the 2×2 quaternionic density matrix $$ \rho_{B}=\frac{1}{2}\left(\begin{array}[c]{cc}1 & -j\\j & 1\end{array}\right), $$

and yet there might not be (and in this case, isn’t) any 4×4 quaternionic density matrix corresponding to ρ_A⊗ρ_B, which would explain both results separately.

What’s going on here?  Why do the local measurement statistics underdetermine the global quantum state with real amplitudes, and overdetermine it with quaternionic amplitudes, being in one-to-one correspondence with it only when amplitudes are complex?

We can get some insight by looking at the number of independent real parameters needed to specify a d-dimensional Hermitian matrix.  Over the complex numbers, the number is exactly d²: we need 1 parameter for each of the d diagonal entries, and 2 (a real part and an imaginary part) for each of the d(d-1)/2 upper off-diagonal entries (the lower off-diagonal entries being determined by the upper ones).  Over the real numbers, by contrast, “Hermitian matrices” are just real symmetric matrices, so the number of independent real parameters is only d(d+1)/2.  And over the quaternions, the number is d+4[d(d-1)/2] = 2d(d-1).

Now, it turns out that the Goldilocks phenomenon that we saw above—with local measurement statistics determining a unique global quantum state when and only when amplitudes are complex numbers—ultimately boils down to the simple fact that $$ (d_A d_B)^2 = d_A^2 d_B^2, $$

but $$\frac{d_A d_B (d_A d_B + 1)}{2} > \frac{d_A (d_A + 1)}{2} \cdot \frac{d_B (d_B + 1)}{2},$$

and conversely $$ 2 d_A d_B (d_A d_B – 1) < 2 d_A (d_A – 1) \cdot 2 d_B (d_B – 1).$$

In other words, only with complex numbers does the number of real parameters needed to specify a “global” Hermitian operator, exactly match the product of the number of parameters needed to specify an operator on Alice’s subsystem, and the number of parameters needed to specify an operator on Bob’s.  With real numbers it overcounts, and with quaternions it undercounts.

A major research goal in quantum foundations, since at least the early 2000s, has been to “derive” the formalism of QM purely from “intuitive-sounding, information-theoretic” postulates—analogous to how, in 1905, some guy whose name I forget derived the otherwise strange-looking Lorentz transformations purely from the assumption that the laws of physics (including a fixed, finite value for the speed of light) take the same form in every inertial frame.  There have been some nontrivial successes of this program: most notably, the “axiomatic derivations” of QM due to Lucien Hardy and (more recently) Chiribella et al.  Starting from axioms that sound suitably general and nontechnical (if sometimes unmotivated and weird), these derivations perform the impressive magic trick of deriving the full mathematical structure of QM: complex amplitudes, unitary transformations, tensor products, the Born rule, everything.

However, in every such derivation that I know of, some axiom needs to get introduced to capture “local tomography”: i.e., the “principle” that composite systems must be uniquely determined by the statistics of local measurements.  And while this principle might sound vague and unobjectionable, to those in the business, it’s obvious what it’s going to be used for the second it’s introduced.  Namely, it’s going to be used to rule out quantum mechanics over the real numbers, which would otherwise be a model for the axioms, and thus to “explain” why amplitudes have to be complex.

I confess that I was always dissatisfied with this.  For I kept asking myself: would I have ever formulated the “Principle of Local Tomography” in the first place—or if someone else had proposed it, would I have ever accepted it as intuitive or natural—if I didn’t already know that QM over the complex numbers just happens to satisfy it?  And I could never honestly answer “yes.”  It always felt to me like a textbook example of drawing the target around where the arrow landed—i.e., of handpicking your axioms so that they yield a predetermined conclusion, which is then no more “explained” than it was at the beginning.

Two months ago, something changed for me: namely, I smacked into the “Principle of Local Tomography,” and its reliance on complex numbers, in my own research, when I hadn’t in any sense set out to look for it.  This still doesn’t convince me that the principle is any sort of a-priori necessity.  But it at least convinces me that it’s, you know, the sort of thing you can smack into when you’re not looking for it.

The aforementioned smacking occurred while I was writing up a small part of a huge paper with Guy Rothblum, about a new connection between so-called “gentle measurements” of quantum states (that is, measurements that don’t damage the states much), and the subfield of classical CS called differential privacy.  That connection is a story in itself; let me know if you’d like me to blog about it separately.  Our paper should be on the arXiv any day now; in the meantime, here are some PowerPoint slides.

Anyway, for the paper with Guy, it was of interest to know the following: suppose we have a two-outcome measurement E (let’s say, on n qubits), and suppose it accepts every product state with the same probability p.  Must E then accept every entangled state with probability p as well?  Or, a closely-related question: suppose we know E’s acceptance probabilities on every product state.  Is that enough to determine its acceptance probabilities on all n-qubit states?

I’m embarrassed to admit that I dithered around with these questions, finding complicated proofs for special cases, before I finally stumbled on the one-paragraph, obvious-in-retrospect “Proof from the Book” that slays them in complete generality.

Here it is: if E accepts every product state with probability p, then clearly it accepts every separable mixed state (i.e., every convex combination of product states) with the same probability p.  Now, a well-known result of Braunstein et al., from 1998, states that (surprisingly enough) the separable mixed states have nonzero density within the set of all mixed states, in any given finite dimension.  Also, the probability that E accepts ρ can be written as f(ρ)=Tr(Eρ), which is linear in the entries of ρ.  OK, but a linear function that’s determined on a subset of nonzero density is determined everywhere.  And in particular, if f is constant on that subset then it’s constant everywhere, QED.

But what does any of this have to do with why amplitudes are complex numbers?  Well, it turns out that the 1998 Braunstein et al. result, which was the linchpin of the above argument, only works in complex QM, not in real QM.  We can see its failure in real QM by simply counting parameters, similarly to what we did before.  An n-qubit density matrix requires 4ⁿ real parameters to specify (OK, 4ⁿ-1, if we demand that the trace is 1).  Even if we restrict to n-qubit density matrices with real entries only, we still need 2ⁿ(2ⁿ+1)/2 parameters.  By contrast, it’s not hard to show that an n-qubit real separable density matrix can be specified using only 3ⁿ real parameters—and indeed, that any such density matrix lies in a 3ⁿ-dimensional subspace of the full 2ⁿ(2ⁿ+1)/2-dimensional space of 2ⁿ×2ⁿ symmetric matrices.  (This is simply the subspace spanned by all possible tensor products of n Pauli I, X, and Z matrices—excluding the Y matrix, which is the one that involves imaginary numbers.)

But it’s not only the Braunstein et al. result that fails in real QM: the fact that I wanted for my paper with Guy fails as well.  As a counterexample, consider the 2-qubit measurement that accepts the state ρ with probability Tr(Eρ), where $$ E=\frac{1}{2}\left(\begin{array}[c]{cccc}1 & 0 & 0 & -1\\0 & 1 & 1 & 0\\0 & 1 & 1 & 0\\-1 & 0 & 0 & 1\end{array}\right).$$

I invite you to check that this measurement, which we specified using a real matrix, accepts every product state (a|0⟩+b|1⟩)(c|0⟩+d|1⟩), where a,b,c,d are real, with the same probability, namely 1/2—just like the “measurement” that simply returns a coin flip without even looking at the state at all.  And yet the measurement can clearly be nontrivial on entangled states: for example, it always rejects $$\frac{\left|00\right\rangle+\left|11\right\rangle}{\sqrt{2}},$$ and it always accepts $$ \frac{\left|00\right\rangle-\left|11\right\rangle}{\sqrt{2}}.$$

Is it a coincidence that we used exactly the same 4×4 matrix (up to scaling) to produce a counterexample to the real-QM version of Local Tomography, and also to the real-QM version of the property I wanted for the paper with Guy?  Is anything ever a coincidence in this sort of discussion?

I claim that, looked at the right way, Local Tomography and the property I wanted are the same property, their truth in complex QM is the same truth, and their falsehood in real QM is the same falsehood.  Why?  Simply because Tr(Eρ), the probability that the measurement E accepts the mixed state ρ, is a function of two Hermitian matrices E and ρ (both of which can be either “product” or “entangled”), and—crucially—is symmetric under the interchange of E and ρ.

Now it’s time for another confession.  We’ve identified an elegant property of quantum mechanics that’s true but only because amplitudes are complex numbers: namely, if you know the probability that your quantum circuit accepts every product state, then you also know the probability that it accepts an arbitrary state.  Yet, despite its elegance, this property turns out to be nearly useless for “real-world applications” in quantum information and computing.  The reason for the uselessness is that, for the property to kick in, you really do need to know the probabilities on product states almost exactly—meaning (say) to 1/exp(n) accuracy for an n-qubit state.

Once again a simple example illustrates the point.  Suppose n is even, and suppose our measurement simply projects the n-qubit state onto a tensor product of n/2 Bell pairs.  Clearly, this measurement accepts every n-qubit product state with exponentially small probability, even as it accepts the entangled state 
$$\left(\frac{\left|00\right\rangle+\left|11\right\rangle}{\sqrt{2}}\right)^{\otimes n/2}$$

with probability 1.  But this implies that noticing the nontriviality on entangled states, would require knowing the acceptance probabilities on product states to exponential accuracy.

In a sense, then, I come back full circle to my original puzzlement: why should Local Tomography, or (alternatively) the-determination-of-a-circuit’s-behavior-on-arbitrary-states-from-its-behavior-on-product-states, have been important principles for Nature’s laws to satisfy?  Especially given that, in practice, the exponential accuracy required makes it difficult or impossible to exploit these principles anyway?  How could we have known a-priori that these principles would be important—if indeed they are important, and are not just mathematical spandrels?

But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about.  Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.

In the literature, when the question of “why not quaternionic amplitudes?” is discussed at all, you’ll typically read things about how the parameter-counting doesn’t quite work out (just like it doesn’t for real QM), or how the tensor product of quaternionic Hermitian matrices need not be Hermitian.  In this paper by McKague, you’ll read that the CHSH game is winnable with probability 1 in quaternionic QM, while in this paper by Fernandez and Schneeberger, you’ll read that the non-commutativity of the quaternions introduces an order-dependence even for spacelike-separated operations.

But none of that does justice to the enormity of the problem.  To put it bluntly: unless something clever is done to fix it, quaternionic QM allows superluminal signaling.  This is easy to demonstrate: suppose Alice holds a qubit in the state |1⟩, while Bob holds a qubit in the state |+⟩ (yes, this will work even for unentangled states!)  Also, let $$U=\left(\begin{array}[c]{cc}1 & 0\\0 & j\end{array}\right) ,~~~V=\left(\begin{array}[c]{cc}1 & 0\\0& i\end{array}\right).$$

We can calculate that, if Alice applies U to her qubit and then Bob applies V to his qubit, Bob will be left with the state $$ \frac{j \left|0\right\rangle +
k \left|1\right\rangle}{\sqrt{2}}.$$

By contrast, if Alice decided to apply U only after Bob applied V, Bob would be left with the state 
$$ \frac{j \left|0\right\rangle – k \left|1\right\rangle}{\sqrt{2}}.$$

But Bob can distinguish these two states with certainty, for example by applying the unitary $$ \frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}j & k\\k & j\end{array}\right). $$

Therefore Alice communicated a bit to Bob.

I’m aware that there’s a whole literature on quaternionic QM, including for example a book by Adler.  Would anyone who knows that literature be kind enough to enlighten us on how it proposes to escape the signaling problem?  Regardless of the answer, though, it seems worth knowing that the “naïve” version of quaternionic QM—i.e., the version that gets invoked in quantum information discussions like the ones I mentioned above—is just immediately blasted to smithereens by the signaling problem, without the need for any subtle considerations like the ones that differentiate real from complex QM.

Update (Dec. 20): In response to this post, Stephen Adler was kind enough to email me with further details about his quaternionic QM proposal, and to allow me to share them here. Briefly, Adler completely agrees that quaternionic QM inevitably leads to superluminal signaling—but in his proposal, the surprising and nontrivial part is that quaternionic QM would reduce to standard, complex QM at large distances. In particular, the strength of a superluminal signal would fall off exponentially with distance, quickly becoming negligible beyond the Planck or grand unification scales. Despite this, Adler says that he eventually abandoned his proposal for quaternionic QM, since he was unable to make specific particle physics ideas work out (but the quaternionic QM proposal then influenced his later work).

Unrelated Update (Dec. 18): Probably many of you have already seen it, and/or already know what it covers, but the NYT profile of Donald Knuth (entitled “The Yoda of Silicon Valley”) is enjoyable and nicely written.

Happy Thanksgiving!

People have sometimes asked me: “how do you do it?  how do you do your research, write papers, teach classes, mentor grad students, build up the quantum center at UT, travel and give talks every week or two, serve on program committees, raise two rambunctious young kids, and also blog and also participate in the comments and also get depressed about people saying mean things on social media?”  The answer is that increasingly I don’t.  Something has to give, and this semester, alas, that something has often been blogging.

And that’s why, today, I’m delighted to have a special guest post by my good friend Terry Rudolph.  Terry, who happens to be Erwin Schrödinger’s grandson, has done lots of fascinating work over the years in quantum computing and the foundations of quantum mechanics, and previously came up on this blog in the context of the PBR (Pusey-Barrett-Rudolph) Theorem.  Today, he’s a cofounder and chief architect at PsiQuantum, a startup in Palo Alto that’s trying to build silicon-photonic quantum computers.

Terry’s guest post is about the prospects for teaching quantum theory at the junior high school level—something he thought about a lot in the context of writing his interesting recent book Q is for Quantum.  I should stress that the opinions in this post are Terry’s, and don’t necessarily reflect the official editorial positions of Shtetl-Optimized.  Personally, I have taught the basics of quantum information to sharp junior high and high school students, so I certainly know that it’s possible.  (By undergrad, it’s not only possible, but maybe should become standard for both physics and CS majors.)  But I would also say that, given the current state of junior high and high school education in the US, it would be a huge step up if most students graduated fully understanding what’s a probability, what’s a classical bit, what’s a complex number, and any of dozens of other topics that feed into quantum information—so why not start by teaching the simpler stuff well?  And also, if students don’t learn the rules of classical probability first, then how will they be properly shocked when they come to quantum? 🙂

But without further ado, here’s Terry—who’s also graciously agreed to stick around and answer some comments.

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Can we/should we teach Quantum Theory in Junior High?

by Terry Rudolph

Should we?

Reasons which suggest the answer is “yes” include:

Economic: We are apparently into a labor market shortage in quantum engineers.  We should not, however, need the recent hype around quantum computing to make the economic case – the frontier of many disparate regions of the modern science and technology landscape is quantum.  Surely if students do decide to drop out of school at 16 they should at least be equipped to get an entry-level job as a quantum physicist?

Educational: If young peoples’ first exposures to science are counterintuitive and “cutting edge,” it could help excite them into STEM.  The strong modern quantum information theoretic connections between quantum physics, computer science and math can help all three subjects constructively generate common interest.

Pseudo-Philosophical: Perhaps our issues with understanding/accepting quantum theory are because we come to it late and have lost the mental plasticity for a “quantum reset” of our brain when we eventually require it late in an undergraduate degree.  It may be easier to achieve fluency in the “language of quantum” with early exposure.

Can we?

There are two distinct aspects to this question: Firstly, is it possible at the level of “fitting it in” – training teachers, adjusting curricula and so on?  Secondly, can a nontrivial, worthwhile fraction of quantum theory even be taught at all to pre-calculus students?

With regards to the first question, as the child of two schoolteachers I am very aware that an academic advocating for such disruption will not be viewed kindly by all.  As I don’t have relevant experience to say anything useful about this aspect, I have to leave it for others to consider.

Let me focus for the remainder of this post on the second aspect, namely whether it is even possible to appropriately simplify the content of the theory.  This month it is exactly 20 years since I lectured the first of many varied quantum courses I have taught at multiple universities. For most of that period I would have said it simply wasn’t possible to teach any but the most precocious of high school students nontrivial technical content of quantum theory – despite some brave attempts like Feynman’s use of arrows in QED: The Strange Theory of Light and Matter (a technique that cannot easily get at the mysteries of two-particle quantum theory, which is where the fun really starts).  I now believe, however, that it is actually possible.

A pedagogical method covering nontrivial quantum theory using only basic arithmetic

My experience talking about quantum theory to 12-15 year olds has only been in the idealized setting of spending a few hours with them at science fairs, camps and similar.  In fact it was on the way to a math camp for very young students, desperately trying to plan something non-trivial to engage them with, that I came up with a pedagogical method which I (and a few colleagues) have found does work.

I eventually wrote the method into a short book Q is for Quantum, but if you don’t want to purchase the book then here is a pdf of Part I,, which takes a student knowing only the rules of basic arithmetic through to learning enough quantum computing they can understand the Deutsch–Jozsa algorithm.  In fact not only can they do a calculation to see how it works in detail, they can appreciate conceptual nuances often under-appreciated in popular expositions, such as why gate speed doesn’t matter – it’s all about the number of steps, why classical computing also can have exponential growth in “possible states” so interference is critical, why quantum computers do not compute the uncomputable and so on.

Before pointing out a few features of the approach, here are some rules I set myself while writing the book:

• No analogies, no jargon – if it can’t be explained quantitatively then leave it out.
• No math more than basic arithmetic and distribution across brackets.
• Keep clear the distinction between mathematical objects and the observed physical events they are describing.
• Be interpretationally neutral.
• No soap opera: Motivate by intriguing with science, not by regurgitating quasi-mythological stories about the founders of the theory.
• No using the word “quantum” in the main text! This was partly to amuse myself, but I also thought if I was succeeding in the other points then I should be able to avoid a word almost synonymous with “hard and mysterious.”

One of the main issues to confront is how to represent and explain superposition.  It is typical in popular expositions to draw analogies between a superposition of, say, a cat which is dead and a cat which is alive by saying it is dead “and” alive.  But if superposition was equivalent to logical “and”, or, for that matter, logical “or”, then quantum computing wouldn’t be interesting, and in this and other ways the analogy is ultimately misleading.  The approach I use is closer to the latter – an unordered list of possible states for a system (which is most like an “or”) can be used to represent a superposition. Using a list has some advantages – it is natural to apply a transformation to all elements of a list, for instance doubling the list of ingredients in a recipe.  More critically, given two independent lists of possibilities the new joint list of combined possibilities is a natural concept.  This makes teaching the equivalent of the Kronecker (tensor) product for multiple systems easy, something often a bit tricky even for undergrads to become comfortable with.

Conceptually the weirdest part of the whole construction, particularly for someone biased by the standard formalism, is that I use a standard mathematical object (a negative or minus sign) applied to a diagram of a physical object (a black or white ball).  Moreover, positive and negative balls in a diagram can cancel out (interfere).  This greatly simplifies the exposition, by removing a whole level of abstraction in the standard theory (we do not need to use a vector containing entries whose specific ordering must be remembered in order to equate them to the physical objects).  While it initially seemed odd to me personally to do this, I have yet to have any young person think of it as any more weird than using the negative sign on a number.  And if it is always kept clear that drawing and manipulating the whole diagram is an abstract thing we do, which may or may not have any correspondence to what is “really going on” in the physical setups we are describing, then there really is no difference.

There are some subtleties about the whole approach – while the formalism is universal for quantum computing, it can only make use of unitary evolution which is proportional to a matrix with integer entries.  Thus the Hadamard gate (PETE box) is ok, the Controlled-NOT and Toffoli likewise, but a seemingly innocuous gate like the controlled-Hadamard is not capable of being incorporated (without adding a whole bunch of unintuitive and unjustified rules).  The fact the approach covers a universal gate set means some amazing things can be explained in this simple diagrammatic language.  For example, the recent paper Quantum theory cannot consistently describe the use of itself, which led to considerable discussion on this blog, can be fully reproduced.  That is, a high school student can in principle understand the technical details of a contemporary argument between professional physicists.  I find this amazing.

Based on communication with readers I have come to realize the people at most risk of being confused by the book are actually those already with a little knowledge – someone who has done a year or two’s worth of undergraduate quantum courses, or someone who has taken things they read in pop-sci books too literally.  Initially, as I was developing the method, I thought it would be easy to keep “touching base” with the standard vector space formalism.  But in fact it becomes very messy to do so (and irrelevant for someone learning quantum theory for the first time).  In the end I dropped that goal, but now realize I need to develop some supplementary notes to help someone in that situation.

Q is for Quantum is certainly not designed to be used as a classroom text – if nothing else my particular style and choice of topics will not be to others’ tastes, and I haven’t included all the many, many simple examples and exercises I have students doing along with me in class when I actually teach this stuff.  It should be thought of as more a “proof of principle,” that the expository challenge can be met.  Several colleagues have used parts of these ideas already for teaching, and they have given me some great feedback.  As such I am planning on doing a revised and slightly expanded version at some point, so if you read it and have thoughts for improvement please send me them.

If you like quantum, complexity, etc., then please read to the end! I’ve gotten a bunch of emails lately of the form “why haven’t you ever blogged about such-and-such?,” when it turned out that I damn well did blog about it; it was just somewhere down in a multi-item post.

1. Like many of you, I watched the US midterm election results with (mostly…) disappointment and dismay.  I think that history will judge us harshly for not totally and unequivocally rebuking everything Trump stands for and every politician associated with him.  But that’s not what I wanted to blog about today.

2. There was a breakthrough in communication complexity by Arkadev Chattopadhyay, Nikhil Mande, and Suhail Sherif: the first exponential separation between randomized communication complexity and log approximate rank for a total Boolean function f.  This falsifies the longstanding conjecture that these measures are polynomially related (though it doesn’t resolve the original log rank conjecture).  For those of you keeping score at home, the quantum communication complexity of f is sandwiched in between randomized CC and log approximate rank.  So, at least one of the following must now be true: either randomized CC is exponentially separated from quantum CC, or else quantum CC is exponentially separated from log approximate rank.  My money’s on the latter.

3. Ewin Tang, who achieved fame with a quantum-inspired classical algorithm for recommendation systems (which I blogged about in July), is now back with quantum-inspired classical algorithms for principal component analysis and supervised clustering.  Well, with the announcements of such algorithms; details of the analysis are to come later.

4. A bunch of people asked me about the paper by Sergey Bravyi, David Gosset, and Robert Koenig, Quantum advantage with shallow circuits.  tl;dr: it’s great!  And it was deservedly a highlight of the QIP conference back in January!  That’s why it confused me when everyone started asking about it a couple weeks ago.  The resolution is that the paper was just recently published in Science magazine, which led to popular coverage like this, which in turn led to people asking me whether this result unconditionally proves P≠BQP (that is, quantum computers can solve more problems in polynomial time than classical computers), and if not why not.  The answer is no: the paper proves an unconditional separation, but one that’s a long long way from P≠BQP, or anything else that would entail solving the central open problems of complexity theory like P vs. PSPACE.  Basically, it shows there are problems solvable in constant time with a quantum computer that aren’t solvable in constant time classically, for suitable meanings of “problem” (namely, a relation problem) and “in constant time” (namely, NC⁰ circuits, in which each output bit depends on only a constant number of input bits).  I understand that a stronger separation has since been achieved, between quantum NC⁰ and classical AC⁰, in work that’s not yet on the arXiv.  The problems in question, however, are all easy to solve in P, or even in classical logarithmic time, given a polynomial number of parallel processors.

5. A bunch of people also asked me about the paper by Xun Gao and Luming Duan, Efficient classical simulation of noisy quantum computation.  This paper tries to formalize something that many of us have suspected/feared for years, namely that random quantum circuits (the whole thing is specific to random circuits) can tolerate only a tiny amount of noise and decoherence before they become efficiently simulable classically.  If true, this has obvious implications for the sampling-based quantum supremacy experiments that Google and others are planning for the next few years: namely, that all the engineering effort they’ve already been investing anyway to push down the noise rate, will actually be necessary!  However, correspondence with the authors revealed that there’s a significant gap in the analysis as it currently stands: namely, the current proof applies only to closed quantum systems, which would (for example) rule out all the techniques that people eventually hope to use to achieve quantum fault-tolerance—all of which are based on constantly measuring subsets of the qubits, doing essentially error-free classical computation on the measurement outcomes, throwing away noisy qubits, and pumping in fresh qubits.  Xun and Duan say that they’re currently working on an extension to open systems; in my personal view, such an extension seems essential for this interesting result to have the informal interpretation that the authors want.

6. My friend Bram Cohen asked me to announce that his company, Chia, has launched a competition for best implementation of its Verifiable Delay Functions (VDFs), with real money rewards.  You can find the details at this Github page.

7. The second Q2B (“Quantum 2 Business”) conference, organized by QC Ware Corp., will be held this coming December 10-12, at the Computer History Museum in Mountain View.  There will be two keynote addresses, one by John Preskill and the other by me.  I hope I’ll get a chance to meet some of you there!

8. Longtime colleague and friend-of-the-blog Ashwin Nayak asked me to announce that the 2019 Conference on Computational Complexity, to be held July 18-20 in exciting New Brunswick, NJ, is now accepting submissions.  I hope to be there!

9. OK, what the hell: the 21st annual, and nearly impossible to capitalize correctly, SQuInT (Southwest Quantum Information and Technology) workshop will be held February 2019 in Albuquerque, NM.  UT Austin is now a node of the SQuInT network, and I’ll hopefully be attending along with a posse of students and postdocs.  The deadline for abstract submission is coming up soon: Monday November 12!

10. I went to morning Shabbat services in Austin this past weekend, exactly one week after the tragedy in Pittsburgh.  There was massively increased security, with armed guards interrogating us, Israeli-style, about why we had no membership sticker on our car, whether we knew the name of the rabbi, etc.  Attendance was maybe a factor of three higher than usual.  About thirty priests, ministers, and Christian seminary students, and one Muslim, came up to the bima to say a prayer of solidarity with Jews.  The mayor of Austin, Steve Adler, was also on hand to give a speech.  Then the rabbi read a letter to the synagogue by Sen. Ted Cruz denouncing antisemitism (well, parts of it; he said the letter was really long).  There were murmurs of disapproval from the crowd when Cruz’s name was mentioned, but then everyone quieted down and listened.  The thing is, the US and large parts of the world are now so far outside the norms of liberal democracy, in territory so terrifyingly uncharted since the end of World War II, that shooting up synagogues is bad is actually something that it’s better than not for powerful people to affirm explicitly.  Anyway, while I’m neither a believer nor much of a synagogue-goer, I found the show of unity moving.

“Unperformed measurements have no results.” —Asher Peres

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With two looming paper deadlines, two rambunctious kids, an undergrad class, program committee work, faculty recruiting, and an imminent trip to Capitol Hill to answer congressional staffers’ questions about quantum computing (and for good measure, to give talks at UMD and Johns Hopkins), the only sensible thing to do is to spend my time writing a blog post.

So: a bunch of people asked for my reaction to the new Nature Communications paper by Daniela Frauchiger and Renato Renner, provocatively titled “Quantum theory cannot consistently describe the use of itself.”  Here’s the abstract:

Quantum theory provides an extremely accurate description of fundamental processes in physics.  It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which it has been tested experimentally.  Here, we propose a Gedankenexperiment to investigate the question whether quantum theory can, in principle, have universal validity.  The idea is that, if the answer was yes, it must be possible to employ quantum theory to model complex systems that include agents who are themselves using quantum theory.  Analysing the experiment under this presumption, we find that one agent, upon observing a particular measurement outcome, must conclude that another agent has predicted the opposite outcome with certainty.  The agents’ conclusions, although all derived within quantum theory, are thus inconsistent.  This indicates that quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner.

I first encountered Frauchiger and Renner’s argument back in July, when Renner (who I’ve known for years, and who has many beautiful results in quantum information) presented it at a summer school in Boulder, CO where I was also lecturing.  I was sufficiently interested (or annoyed?) that I pulled an all-nighter working through the argument, then discussed it at lunch with Renner as well as John Preskill.  I enjoyed figuring out exactly where I get off Frauchiger and Renner’s train—since I do get off their train.  While I found their paper thought-provoking, I reject the contention that there’s any new problem with QM’s logical consistency: for reasons I’ll explain, I think there’s only the same quantum weirdness that (to put it mildly) we’ve known about for quite some time.

In more detail, the paper makes a big deal about how the new argument rests on just three assumptions (briefly, QM works, measurements have definite outcomes, and the “transitivity of knowledge”); and how if you reject the argument, then you must reject at least one of the three assumptions; and how different interpretations (Copenhagen, Many-Worlds, Bohmian mechanics, etc.) make different choices about what to reject.

But I reject an assumption that Frauchiger and Renner never formalize.  That assumption is, basically: “it makes sense to chain together statements that involve superposed agents measuring each other’s brains in different incompatible bases, as if the statements still referred to a world where these measurements weren’t being done.”  I say: in QM, even statements that look “certain” in isolation might really mean something like “if measurement X is performed, then Y will certainly be a property of the outcome.”  The trouble arises when we have multiple such statements, involving different measurements X₁, X₂, …, and (let’s say) performing X₁ destroys the original situation in which we were talking about performing X₂.

But I’m getting ahead of myself.  The first thing to understand about Frauchiger and Renner’s argument is that, as they acknowledge, it’s not entirely new.  As Preskill helped me realize, the argument can be understood as simply the “Wigner’s-friendification” of Hardy’s Paradox.  In other words, the new paradox is exactly what you get if you take Hardy’s paradox from 1992, and promote its entangled qubits to the status of conscious observers who are in superpositions over thinking different thoughts.  Having talked to Renner about it, I don’t think he fully endorses the preceding statement.  But since I fully endorse it, let me explain the two ingredients that I think are getting combined here—starting with Hardy’s paradox, which I confess I didn’t know (despite knowing Lucien Hardy himself!) before the Frauchiger-Renner paper forced me to learn it.

Hardy’s paradox involves the two-qubit entangled state

$$\left|\psi\right\rangle = \frac{\left|00\right\rangle + \left|01\right\rangle + \left|10\right\rangle}{\sqrt{3}}.$$

And it involves two agents, Alice and Bob, who measure the left and right qubits respectively, both in the {|+〉,|-〉} basis.  Using the Born rule, we can straightforwardly calculate the probability that Alice and Bob both see the outcome |-〉 as 1/12.

So what’s the paradox?  Well, let me now “prove” to you that Alice and Bob can never both get |-〉.  Looking at |ψ〉, we see that conditioned on Alice’s qubit being in the state |0〉, Bob’s qubit is in the state |+〉, so Bob can never see |-〉.  And conversely, conditioned on Bob’s qubit being in the state |0〉, Alice’s qubit is in the state |+〉, so Alice can never see |-〉.  OK, but since |ψ〉 has no |11〉 component, at least one of the two qubits must be in the state |0〉, so therefore at least one of Alice and Bob must see |+〉!

When it’s spelled out so plainly, the error is apparent.  Namely, what do we even mean by a phrase like “conditioned on Bob’s qubit being in the state |0〉,” unless Bob actually measured his qubit in the {|0〉,|1〉} basis?  But if Bob measured his qubit in the {|0〉,|1〉} basis, then we’d be talking about a different, counterfactual experiment.  In the actual experiment, Bob measures his qubit only in the {|+〉,|-〉} basis, and Alice does likewise.  As Asher Peres put it, “unperformed measurements have no results.”

Anyway, as I said, if you strip away the words and look only at the actual setup, it seems to me that Frauchiger and Renner’s contribution is basically to combine Hardy’s paradox with the earlier Wigner’s friend paradox.  They thereby create something that doesn’t involve counterfactuals quite as obviously as Hardy’s paradox does, and so requires a new discussion.

But to back up: what is Wigner’s friend?  Well, it’s basically just Schrödinger’s cat, except that now it’s no longer a cat being maintained in coherent superposition but a person, and we’re emphatic in demanding that this person be treated as a quantum-mechanical observer.  Thus, suppose Wigner entangles his friend with a qubit, like so:

$$ \left|\psi\right\rangle = \frac{\left|0\right\rangle \left|FriendSeeing0\right\rangle + \left|1\right\rangle \left|FriendSeeing1\right\rangle}{\sqrt{2}}. $$

From the friend’s perspective, the qubit has been measured and has collapsed to either |0〉 or |1〉.  From Wigner’s perspective, no such thing has happened—there’s only been unitary evolution—and in principle, Wigner could even confirm that by measuring |ψ〉 in a basis that included |ψ〉 as one of the basis vectors.  But how can they both be right?

Many-Worlders will yawn at this question, since for them, of course “the collapse of the wavefunction” is just an illusion created by the branching worlds, and with sufficiently advanced technology, one observer might experience the illusion even while a nearby observer doesn’t.  Ironically, the neo-Copenhagenists / Quantum Bayesians / whatever they now call themselves, though they consider themselves diametrically opposed to the Many-Worlders (and vice versa), will also yawn at the question, since their whole philosophy is about how physics is observer-relative and it’s sinful even to think about an objective, God-given “quantum state of the universe.”  If, on the other hand, you believed both that

1. collapse is an objective physical event, and
2. human mental states can be superposed just like anything else in the physical universe,

then Wigner’s thought experiment probably should rock your world.

OK, but how do we Wigner’s-friendify Hardy’s paradox?  Simple: in the state

$$\left|\psi\right\rangle = \frac{\left|00\right\rangle + \left|01\right\rangle + \left|10\right\rangle}{\sqrt{3}},$$

we “promote” Alice’s and Bob’s entangled qubits to two conscious observers, call them Charlie and Diane respectively, who can think two different thoughts that we represent by the states |0〉 and |1〉.  Using far-future technology, Charlie and Diane have been not merely placed into coherent superpositions over mental states but also entangled with each other.

Then, as before, Alice will measure Charlie’s brain in the {|+〉,|-〉} basis, and Bob will measure Diane’s brain in the {|+〉,|-〉} basis.  Since the whole setup is mathematically identical to that of Hardy’s paradox, the probability that Alice and Bob both get the outcome |-〉 is again 1/12.

Ah, but now we can reason as follows:

1. Whenever Alice gets the outcome |-〉, she knows that Diane must be in the |1〉 state (since, if Diane were in the |0〉 state, then Alice would’ve certainly seen |+〉).
2. Whenever Diane is in the |1〉 state, she knows that Charlie must be in the |0〉 state (since there’s no |11〉 component).
3. Whenever Charlie is in the |0〉 state, she knows that Diane is in the |+〉 state, and hence Bob can’t possibly see the outcome |-〉 when he measures Diane’s brain in the {|+〉,|-〉} basis.

So to summarize, Alice knows that Diane knows that Charlie knows that Bob can’t possibly see the outcome |-〉.  By the “transitivity of knowledge,” this implies that Alice herself knows that Bob can’t possibly see |-〉.  And yet, as we pointed out before, quantum mechanics predicts that Bob can see |-〉, even when Alice has also seen |-〉.  And Alice and Bob could even do the experiment, and compare notes, and see that their “certain knowledge” was false.  Ergo, “quantum theory can’t consistently describe its own use”!

You might wonder: compared to Hardy’s original paradox, what have we gained by waving a magic wand over our two entangled qubits, and calling them “conscious observers”?  Frauchiger and Renner’s central claim is that, by this gambit, they’ve gotten rid of the illegal counterfactual reasoning that we needed to reach a contradiction in our analysis of Hardy’s paradox.  After all, they say, none of the steps in their argument involve any measurements that aren’t actually performed!  But clearly, even if no one literally measures Charlie in the {|0〉,|1〉} basis, he’s still there, thinking either the thought corresponding to |0〉 or the thought corresponding to |1〉.  And likewise Diane.  Just as much as Alice and Bob, Charlie and Diane both exist even if no one measures them, and they can reason about what they know and what they know that others know.  So then we’re free to chain together the “certainties” of Alice, Bob, Charlie, and Diane in order to produce our contradiction.

As I already indicated, I reject this line of reasoning.  Specifically, I get off the train at what I called step 3 above.  Why?  Because the inference from Charlie being in the |0〉 state to Bob seeing the outcome |+〉 holds for the original state |ψ〉, but in my view it ceases to hold once we know that Alice is going to measure Charlie in the {|+〉,|-〉} basis, which would involve a drastic unitary transformation (specifically, a “Hadamard”) on the quantum state of Charlie’s brain.  I.e., I don’t accept that we can take knowledge inferences that would hold in a hypothetical world where |ψ〉 remained unmeasured, with a particular “branching structure” (as a Many-Worlder might put it), and extend them to the situation where Alice performs a rather violent measurement on |ψ〉 that changes the branching structure by scrambling Charlie’s brain.

In quantum mechanics, measure or measure not: there is no if you hadn’t measured.

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Unrelated Announcement: My awesome former PhD student Michael Forbes, who’s now on the faculty at the University of Illinois Urbana-Champaign, asked me to advertise that the UIUC CS department is hiring this year in all areas, emphatically including quantum computing. And, well, I guess my desire to do Michael a solid outweighed my fear of being tried for treason by my own department’s recruiting committee…

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Another Unrelated Announcement: As of Sept. 25, 2018, it is the official editorial stance of Shtetl-Optimized that the Riemann Hypothesis and the abc conjecture both remain open problems.