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book reviews
doorstopper scientific booksreviewed by T. Nelson |
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book reviews
doorstopper scientific booksreviewed by T. Nelson |
Reviewed by T. Nelson
The old quantum mechanics (QM) that we learned years ago is now a mere footnote to the exciting, new and improved quantum mechanics, which is all about entanglement, qubits, quantum teleportation, and the EPR paradox.
As physicist Alain Aspect pointed out, we owe that to the questions Einstein raised about QM in the famous 1935 Einstein, Podolsky, and Rosen paper, now known simply as EPR. If it had been written by anyone but Einstein, it would have been ignored and no one would have ever heard of qubits.
Part I is a course on traditional quantum mechanics. Part II covers entanglement and qubits. Part III covers entropy. The three sections are essentially different books that can be read independently.
There’s not much to say about Part I other than that it’s clearly written. Understanding the basics in these 317 pages is essential for moving on to Part II.
Part II discusses the authors' primary interests: superposition, entanglement, and qubits, which are hot topics thanks to all those scare articles about AI and the coming cyberhacking armageddon.
There are several other good books on qubits, but the discussion here is more complete and more physics-oriented. Drawbacks to the BB84 protocol and the Ekert-91 protocol for quantum key distribution, which others gloss over, are readily apparent. As the present authors say, in BB84 an eavesdropper could easily siphon off enough photons to reconstruct part of the key. It’s true that the proverbial Bob and Alice would detect it but it would be too late: their love note has already been intercepted and maybe even tampered with.
This book also explains entanglement entropy more clearly than other books. The only drawback is that casual readers may feel like they’re drowning in theories by the time they get to the Choi-Jamiołkowski isomorphism (which is important in quantum teleportation). Even so, they’ll learn useful math tricks, like a shortcut for calculating eigenvalues from a 4×4 matrix. They’ll also learn the weaknesses of the theories. Admission that theories are imperfect is part of what makes science credible and it’s something other books ignore.
In the section on the famous ‘quantum socks’ caper the authors discuss what if anything all of this might mean. According to Bertlmann and Friis, the biggest challenge of the Copenhagen interpretation was the so-called ‘measurement problem.’ Most physicists now say that including the measuring apparatus in the calculations means the question of how a wave function could collapse when observed (as in the famous Schrödinger cat) is no longer a mystery. Others, including Časlav Brukner and Anton Zeilinger, say the wave function is merely information:
For Brukner and Zeilinger the wave function that describes the state of the system is just the mathematical representation of our knowledge of the system. In a measurement the abrupt collapse of the wave function corresponds merely to our sudden change of knowledge and does not correspond to a real physical process.
On the other hand there’s the Pusey-Barrett-Rudolf theorem, which says:
Any model in which a quantum state represents mere information . . . must make predictions that contradict those of quantum theory. [p.402]
A few physicists even say the wavefunction isn’t real but merely a tool for understanding, so the expression “wave function collapse” has no physical meaning. Bertlmann says it is either not ‘mere’ information or it is not realism.
The debate goes on. Whatever they come up with, it's clear that multiverses and quantum consciousness are no longer needed.
I spent a fair amount of time trying to understand the authors’ proof of the Kochen-Specker theorem, which is a diagrammatic scheme for disproving the hidden variables theory. I finally realized that what they wrote was not a proof at all but a contrived example used to help understand the real proof, which is somewhere in the literature and very complicated.
A graphical proof would be a hard sell. It’s probably wise not to believe any theories unless they’ve been empirically verified. That goes double for a mere diagram. What was needed was an experiment.
And so a major strength of Part II is its descriptions of those experiments. For instance, there is a good description of Aspect’s Nobel-prize-winning refinements to the famous two-slit experiment. It closed the door on the hidden-variables theory, which hypothesized that particles had some as-yet undiscovered property that accounted for their ‘spooky’ behavior. These experiments make the topic more concrete and more convincing. They also help the reader understand where the theory is still weak and might even inspire new experiments.
Indeed, the discussion (p. 412) of polarizing beam splitters to generate superposed states makes it sound like doing quantum optics in your basement might be a fun thing to do; but whether you could afford the components (electro-optic modulator $3,094; single photon detector $5,260, compensator $3,705; laboratory-grade laser, don’t ask) is another question.
For every entangled pair there are many uncorrelated photons that get in the way. Coincidence detectors are essential to cut the background. For your basement lab you might have to build one yourself.
One important question is how to tell whether two qubits are entangled or ‘separable.’ The authors use a lot of space describing different schemes, none of which is entirely satisfactory, for classifying them. There is agreement that a negative eigenvalue or “negativity” always means entanglement. It’s calculated by a criterion called PPT, or positive partial transposition, where you tweak the matrices to see if you get a negative number. If so, the matrix becomes indefinite and you have an entangled state. But doing PPT is a lot of trouble, so a better way is needed.
The authors finally settle on entanglement witnesses, which means a state is entangled if you can find a Hermitian operator (i.e., a matrix), called a witness, for which the trace is minus. This comes from the Hahn-Banach theorem in math, which says you can always find a hyperplane to separate two convex sets. Witnesses are great because they’re Hermitian operators, which means you could measure them. You no longer need to know the exact quantum state.
The authors praise Walter Thirring, who found that whether a state is entangled or not can depend on how you calculate it. You can get opposite answers for any quantum state. An infallible measure of entanglement is still needed.
Bell inequalities are propositions that must be true if quanta are like objects in everyday life. All separable states satisfy these inequalities; violation of them means entanglement. However, Reinhold Werner found that certain mixed entangled states can also satisfy a Bell inequality. The authors call these Werner states and conclude we still don’t know whether entanglement and nonlocality are synonymous or distinct concepts.
To complicate matters, some quantum states can change from satisfying to violating Bell after local filtering. This happens with states made from Gisin states and the phenomenon is called hidden non-locality.
When you get to higher-dimensional systems, say qutrits for 3 qubits and qudits for ≥ 3, the authors say it's ‘notoriously difficult’ to tell whether you have entanglement. This is because at higher dimensions not every point on a Bloch sphere (a graphical way of visualizing qubits) corresponds to a physical state. Instead of a sphere you get a ‘hyperball with holes’ and the math gets very complicated.
There are different kinds of qubits and another important question is how to interconvert them. Some can be converted and some can’t, in which case you need ‘entanglement catalysis’, a sort of chemical reaction like |ψ⟩ + |χ⟩ → |𝜙⟩ + |χ⟩, to make it happen. Another chemistry-inspired task is distillation, where many weakly-entangled particles are reacted together to form a handful of highly entangled ones, such as the singlet ψ− which is ‘pure’ and useful in quantum communication.
Quantum teleportation, the authors say, is not really matter transfer as the popular media portray it; it’s just a transfer of information from one place to another. According to the no-cloning theorem it is not and can never be a perfect copy of the original state. So converting yourself to energy and transmitting yourself to Mars would be theoretically impossible. The authors make teleportation sound easy, but if you want to teleport your particle it would probably be easier just to put a stamp on it and mail it to Bob c/o the publisher.
Every scientific book has a Part III, where authors put all their leftover wacky ideas.
Part III, “Advanced Topics,” discusses familiar things like physical entropy, Shannon entropy, mutual information, and quantum entropy, then goes back to atomic physics and the relationship between qubits and photon states and even subatomic particles.
In the movie Oppenheimer the most interesting scenes were the ones showing the formulas on the blackboard. (Most people probably didn’t notice, but some of them were for the wrong thing.) Bertlmann would be a blockbuster.
jul 31 2026