On quantum measurement (Part 3: No cloning allowed)

In the previous two parts, I told you how I became interested in the quantum measurement problem (Part I), and provided a bit of historical background (Part 2). Now we'll get to the heart of the matter. 

Note that I'm using MathJax to display equations in this blog. If your browser shows a bunch of dollar signs and gibberish where equations should appear, you probably have to figure out how to install MathJax on your browser. Don't email me: I know nothing about such intricacies.

Let me remind you that our hero John von Neumann described quantum measurement as a two-stage process. (No, I'm not showing his likeness.) The first stage is now commonly described as entanglement. This is what we'll discuss here. I'll get to the second process (the one where the wavefunction ostensibly collapses, except Hans Bethe told me that it doesn't) in Part 4. 

For the purpose of illustration, I'm going to describe the measurement of a position, but everything can be done just as well for discrete degrees of freedom, such as, you know, spins. In fact, I'll show you a bunch of spin measurements waaay later, like the Stern-Gerlach experiment, or the quantum eraser. But I'm getting ahead of myself.

Say our quantum system that we would like to measure is in state \(|Q\rangle=|x\rangle\). I'm going to use Q to stand in for quantum systems a lot. Measurement devices will be called "M", or sometimes "A" or "B". 

All right. How do you measure stuff to begin with?

In classical physics, we might imagine that a system is characterized by the position variable \(x\), [I'll write this as "(x)"] and to measure it, all we have to do is to transfer that label "x" to a measurement device. Say the measurement device (before measurement) points to a default location (for example '0') like this: (0). Then, we'll place that device next to the position we want to measure, and attempt to make the device "reflect" the position:

                                                         \((x)(0)\to (x)(x)\)

This is just what I want, because now I can read the position of the thing I want to measure off of my measurement device. 

I once in a while get the question: "Why do you have to have a measurement device? Can't you just read the position off of the system you want to measure directly?" The answer is no, no you can't. The thing is the thing: it stands there in the corner, say. If you measure something, you have to transfer the state of the thing to something you read off of. The variable that reflects the position can be very different from the thing you are measuring. For example, a temperature can be transferred to the height of a mercury column. In a measurement, you create a correlation between two systems. 

In a classical measurement, the operation that makes that possible is a copying operation. You copy the system's state onto the measurement device's state. The copy can be made out of a very different material (for example, a photograph is a copy of a 3D scene onto a two-dimensional surface, made out of whatever material you choose). But system and measurement refer to each other.

All right, so measuring really is copying. And reading this the sophisticated reader (yes, I mean you!) starts smelling a rat right away. Because you already know that copying is just fine in classical physics, but it really is against the law in quantum physics. That's right: there is a no-cloning (or no-xeroxing) theorem, in effect in quantum mechanics. You're not allowed to make exact copies. Ever. 

So how can quantum measurement work at all, if measurement is intrinsically copying?

That, dear reader, is indeed the question. And what I'll try to convince you of is now fairly obvious, namely that quantum measurement is really impossible in principle, unless you just happen to be in the "right basis". This "right basis", basically, is a basis where everything looks classical to begin with. (We'll get to this in more detail later). What I will try to convince you here is that quantum measurement is impossible, if you want a quantum measurement to do what you expect from a classical measurement, namely that your device reflects the state of the system. 

The no-cloning theorem makes that impossible. 

I could stop here, you know. "Stop worrying about quantum measurement", I could write, "because I just showed you that quantum measurement is impossible in principle!"

But I won't, because there is so much more to be said. For example, even though quantum measurements are impossible in principle, it's not like people haven't tried, right? So what is it that people are measuring? What are the measurement devices saying? 

I'll tell you, and I guarantee you that you will not like it one bit.

But first, I owe you this piece: to show you how quantum measurement works. So our quantum system \(Q\) is in state \(|x\rangle\). Our measurement device is conveniently already in its default state \(|0\rangle\). You can, by the way, think about what happens if the measurement device is not pointing to an agreed-upon direction (such as '0') before measurement, but Johnny vN has already done this for you on page 233 of his "Grundlagen". Here he is, by the way, discussing stuff with Ulam and Feynman, most likely in Los Alamos.

Left to right: Stanislaw Ulam, Richard Feynman, John von Neumann

To be a fly on the wall there! Note how JvN (to the right) is always better dressed than the people he hangs out with!

So investigating various possible initial states of the quantum measurement device does nothing for you, he finds, and of course he is correct. So we'll assume it points to \(|0\rangle\). 

So we start with \(|Q\rangle|M\rangle=|x\rangle|0\rangle\). What now? Well, the measurement operator, which of course has to be unitary (meaning it conserves probabilities, yada yada) must project the quantum state, then move the needle on the measurement device. For a position measurement, the unitary operator that does this is

                                                        \(U=e^{iX\otimes P}\)

where \(X\) is the operator whose eigenstate is \(|x\rangle\) (meaning \(X|x\rangle=x|x\rangle\)), and where \(P\) is the operator conjugate to \(X\). \(P\) (the "momentum operator") makes spatial translations. For example, \(e^{iaP}|x\rangle=|x+a\rangle\), that is, \(x\) was made into \(x+a\).  The \(\otimes\) reminds you that \(X\) acts on the first vector (the quantum system), and \(P\) acts on the second (the measurement device). 

So, what this means is that

                           \(U|x\rangle|0\rangle=e^{iX\otimes P}|x\rangle|0\rangle=e^{ix P}|x \rangle|0\rangle=|x\rangle|x\rangle .\)

Yay: the state of the quantum system was copied onto the measurement device! Except that you already can see what happens if you try to apply this operator to a superposition of states such as \(|x+y\rangle\):

\(U|x+y\rangle|0\rangle=e^{iX\otimes P}|x+y\rangle|0\rangle=e^{ix P}|x \rangle|0\rangle+e^{iy P}|y \rangle|0\rangle=|x\rangle|x\rangle + |y\rangle|y\rangle .\)

And that's not at all what you would have expected if measurement was like the classical case, where you would have gotten \((|x\rangle + |y\rangle)(|x\rangle + |y\rangle)\). And what I just showed you is really just the proof that cloning is impossible in quantum physics.

So there you have it: quantum measurement is impossible unless the state that you are measuring just happens to already be in an eigenstate of the measurement operator, that is, it is not in a quantum superposition. 

Whether or not a quantum system is in a superposition depends on the basis that you choose to perform your quantum measurement. I do realize that the concept of a "basis" is a bit technical: it is totally trivial to all of you who have been working in quantum mechanics for years, but less so for those of you who are just curious. In everyday life, it is akin to measuring temperature in Celsius or Fahrenheit, for example, or location in Euclidean as opposed to polar coordinates. But in quantum mechanics, the choice of a basis is much more fundamental, and I really don't know of a good way to make it more intuitive (meaning, without a lot more math). A typical distinction is to measure photon polarization either in terms of horizontal/vertical, or left/right circular. I know, I'm not helping. Let's just skip this part for now. I might get back to it later.

So what happens when you measure a quantum system, and your measurement device is not "perfectly aligned" (basis-wise) with the quantum system? As it in fact almost never will be, by the way, unless you use a classical device to measure a classical system. Because in classical physics, we are all in the same basis automatically.  (OK, I see that I'll have to clarify this to you but trust me here.)

Look forward to Part 4 instead. Where I will finally delve into "Stage 2" of the measurement process. That is the one that baffled von Neumann, because he could not understand where exactly the wavefunction collapses. And in hindsight, there was no way he could have figured this out, because the wavefunction never collapses. Ever. What I'll show you in Part 4 is how a measurement device can be perfectly (by which I mean intrinsically) consistent, yet tell you a story about what the quantum state is and lie to you at the same time. Lie to you, through its proverbial teeth, if it had any.  

But come on, cut the measurement device some slack. It is lying to you because it has no choice. You ask it to make a copy of the quantum state, and it really is not allowed to do so. What will happen (as I will show you), is that it will respond by displaying to you a random value, with a probability given by the square of some part of the amplitude of the quantum wavefunction. In other words, I'll show you how Born's rule comes about, quite naturally. In a world where no wavefunction collapses, of course.

Part 4 is here. 

On quantum measurement (Part 2: Some history, and John von Neumann is confused)

This is Part 2 of the "On quantum measurement" series. Part 1: (Hans Bethe, the oracle) is here.

Before we begin in earnest, I should warn you, (or ease your mind, whichever is your preference): this sequence has math in it. I'm not in it to dazzle you with math. It's just that I know no other way to convey my thoughts about quantum measurement in a more succinct manner. Math, you see, is a way for those of us who are not quite bright enough, to hold on to thoughts which, without math, would be too daunting to formulate, too ambitious to pursue. Math is for poor thinkers, such as myself. If you are one of those too, come join me. The rest of you: why are you still reading? Oh, you're not. OK. 

Hey, come back: this historical interlude turns out to be math-free after all. But I promise math in Part 3.

Before I offer to you my take on the issue of quantum measurement, we should spend some time reminiscing, about the history of the quantum measurement "problem". If you've read my posts (and why else would you read this one?), you'll know one thing about me: when the literature says there is a "problem", I get interested. 

This particular problem isn't that old. It arose through a discussion between Niels Bohr and Albert Einstein, who disagreed vehemently about measurement, and the nature of reality itself.  

Bohr and Einstein at Ehrenfest's house, in 1925. Source: Wikimedia

The "war" between Bohr and Einstein only broke out in 1935 (via dueling papers in the Physical Review), but the discussion had been brewing for 10 years at least.

Much has been written about the controversy (and a good summary albeit with a philosophical bent can be found in the Stanford Encyclopedia of Philosophy). Instead of going into that much detail, I'll just simplify it by saying:

Bohr believed the result of a measurement reflects a real objective quantity (the value of the property being measured).

Einstein believed that quantum systems have objective properties independent of their measurements, and that because quantum mechanics cannot properly describe them, the theory must necessarily be incomplete.

In my view, both views are wrong. Bohr's because his argument relies on a quantum wavefunction that collapses upon measurement (which as I'll show you is nonsense), and Einstein's because the idea that a quantum system has objective properties (described by one of the eigenstates of a measurement device) is wrong and that, as a consequence the notion that quantum mechanics must be incomplete is wrong as well. He was right, though, about the fact that quantum systems have properties independently of whether you measure them or not. It is just that we may not ever know what these properties are.

But enough of the preliminaries. I will begin to couch quantum measurement in terms of a formalism due to John von Neumann. If you think I'm obsessed by the guy because he seems to make an appearance in every second blog post of mine: don't blame me. He just ended up doing some very fundamental things in a number of different areas. So I'm sparing you the obligatory picture of his, because I assume you have seen his likeness enough. 

John von Neumann's seminal book on quantum mechanics is called "Mathematische Grundlagen der Quantenmechanik" (Mathematical foundations of quantum theory), and appeared in 1932, three years before the testy exchange of papers (1) between Bohr and Einstein. 

My copy of the "Grundlagen". This is the version issued by the U.S. Alien Property Custodian from 1943 by Dover Publications. It is the verbatim German book, issued in the US in war time. The original copyright is by J. Springer, 1932.

In this book, von Neumann made a model of the measurement process that had two stages, aptly called "first stage" and "second stage". [I want to note here that JvN actually called the first stage "Process 2" and the second stage "Process 1", which today would be confusing so I reversed it.]

The first stage is unitary, which means "probability conserving". JvN uses the word "causal" for this kind of dynamics. In today's language, we call that process an "entanglement operation" (I'll describe it in more details momentarily, which means "wait for Part 3"). Probability conservation is certainly a requisite for a causal process, and I actually like JvN's use of the word "causal". That word now seems to have acquired a somewhat different meaning. 

The second stage is the mysterious one. It is (according to JvN) acausal, because it involves the collapse of the wavefunction (or as Hans Bethe called it, the "reduction of the wavepacket"). It is clear that this stage is mysterious to Johnny, because he doesn't know where the collapse occurs. He is following "type one" processes in a typical measurement (in the book, he measures temperature as an example) from the thermal expansion of the mercury fluid column, to the light quanta that scatter off the mercury column and enter our eye, where the light is refracted in the lense and forms an image on the retina, which then stimulate nerves in the visual cortex, and ultimately creates the "subjective experience" of the measurement. 

According to JvN, the boundary between what is the quantum system and what is the measurement device can be moved in an arbitrary fashion. He understands perfectly that a division into a system to be measured and a measuring system is necessary and crucial (and we'll spend considerable time discussing this), but the undeniable fact—that it is not at all clear where to draw the boundary— is a mystery to him. He invokes the philosophical principle of "psychophysical parallelism"—which states that there can be no causal interaction between the mind and the body— to explain why the boundary is so fluid. But it is the sentence just following this assertion that puts the finger on what is puzzling him. He writes: 

"Because experience only ever makes statements like this: 'an observer has had a (subjective) perception', but never one like this: 'a physical quantity has taken on a particular value'."(2)

This is, excuse my referee's voice, very muddled. He says: We never have the experience "X takes on x", we always experience "X looks like it is in state x". But mathematically they should be the same. He makes a distinction that does not exist. We will see later why he feels he must make that distinction. But, in short, it is because he thinks that what we perceive must also be reality. If a physical object X is perceived to take on state x, then this must mean that objectively "X takes on x". In other words, he assumes that subjective experience must mirror objective fact.

Yet, this is provably dead wrong. 

That is what Nicolas and I discovered in the article in question, and that is undoubtedly what Hans Bethe immediately realized, but struggled to put into words. 

Quantum reality, in other words, is a whole different thing than classical reality. In fact, in the "worst case" (to be made precise as we go along) they may have nothing to do with each other, as Nicolas and I  argue in a completely obscure (that is, unknown) article entitled "What Information Theory Can Tell us About Quantum Reality" (3).

What you will discover when following this series of posts, is that if your measurement device claims "the quantum spin that you were measuring was in state up", then this may not actually tell you anything about the true quantum state. The way I put it colloquially is that "measurement devices tend to lie to you". They lie, because they give you an answer that is provably nonsense. 

In their (the device's) defense, they have no choice but to lie to you (I will make that statement precise when we do math). They lie because they are incapable of telling the truth. Because the truth is, in a precise information-theoretic way that I'll let you in on, bigger than they are. 

JvN tried to reconcile subjective experience with objective truth. Subjectively, the quantum state collapsed from a myriad of possibilities to a single truth. But in fact, nothing of the sort happens. Your subjective experience is not reflecting an objective truth. The truth is out there, but it won't show itself in our apparatus. The beauty of theoretical physics is that we can find out about how the wool is being pulled over our eyes—how classical measurement devices are conspiring to deceive us—when our senses would never allow us a glimpse of the underlying truth.

Math supporting all that talk will start in Part 3. 

(1) Einstein (with Podolsky and Rosen) wrote a paper entitled "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". It appeared in Phys. Rev. 47 (1935) 777-780. Four pages: nowadays it would be a PRL. I highly recommend reading it. Bohr was (according to historical records and the narrative in Zurek's great book about it all) incensed. Bohr reacted by writing a paper with the same exact title as Einstein's, that has (in my opinion) nothing in it. It is an astonishing paper because it is content-free, but was meant to serve as a statement that Bohr refutes Einstein, when in fact Bohr had nothing. 

(2) Denn die Erfahrung macht nur Aussagen von diesem Typus: ein Beobachter hat eine bestimmte (subjektive) Wahrnehmung gemacht, und nie eine solche: eine physikalische Größe hat einen bestimmten Wert. 

(3) C. Adami & N.J. Cerf, Lect. Notes in Comp. Sci. 1509 (1999) 258-268

Part 3 (No cloning allowed) continues here.