# Nonlocal Measurements

My paper Nonlocal Measurements Via Quantum Erasure has  finally been published in PRL.  There is a short news story on the work on the IQC website. I also recently wrote a related blog post on nonlocal measurements for the IQC blog.

#### IQC blog post: Tomaytos, Tomahtos and Non-local Measurements

Nonlocal measurements via quantum erasure, A. Brodutch and E. Cohen  Phys. Rev. Lett 116 (2016)

# Misconceptions about weak measurements: 2. Weak, not noisy

It’s about time that I continue writing about misunderstandings surrounding weak measurements and weak values. This time I will try to explain the difference between weak measurements and noisy measurements.

## Why weak?

One of the first things we learn about quantum mechanics is that  the measurement process causes an unavoidable back-action on the measured system. As a consequence, some measurements are incompatible, i.e. the result of a measurement on observable  A  can change significantly if a different observable,  B, is measured before A.  A well known example is the measurements of position and momentum where the back-action leads to the Heisenberg uncertainty relation.

The measurement back-action can create  some seemingly paradoxical situations when we make counterfactual arguments such as

We measured A and got the result a, but had we measured B we would have go the result b which is incompatible with a.

These situations appear very often  when we consider  systems both past and future boundary conditions. In these cases they are known as pre and post selection (PPS) paradoxes. In PPS paradoxex the measurement back-action is important even when A and B commute.  An example is the three box paradox that I explain without mathematical detail:

A single particle is placed in one of three boxes A,B,C (actually in a superposition) at time t0 and is later found to be in some other superposition state at t1.  At time t0< t m < t1 one box is opened. The initial t0 and final t1 states of the particle are chosen in such a way that the following happens:

If box A is opened, the particle will be discovered with certainty. If box B is opened, the particle will also be found with certainty. If box C is opened the particle will be found with some probability. The situation seems paradoxical:

If the ball is found with certainty in box A, then it must have been in box A to begin with. But if it is also found with certainty in box B, so it must have been there …

One way to solve this apparent paradox is to note that the measurements are incompatible. i.e opening box A and not B,C is incompatible with opening box B and not A,C etc.

These are the types of questions that Aharonov Albert and Vaidman were investigating  1980s1 . Weak measurements were studied as a way to minimize the measurement back-action. These measurements then  provided a picture that arguably gives a solid (if somewhat strange) foundation to statements like the one above.

The motivation of weak measurements is therefore an attempt to derive a consistent picture where all observables are mutually compatible in a way which is similar to classical physics. In quantum mechanics this comes at a cost. The classical information gained by reading out the result of a single weak measurement is usually indistinguishable from noise. In other words  weak measurements are noisy measurements.

## Weak, not noisy

Part of the confusion around weak measurements lies in the fact that the statement above is not a sufficient condition for a weak measurement. One may argue noise is not even a necessary requirement, it is rather, a consequence of quantum mechanics. Weak measurements may be noisy, but noisy measurements are, in most cases, not weak. To understand this fact it is good to examine  both a classical and quantum scenario.

#### The classical scenario

Walking on the beach you see a person drowning. Being  a good swimmer you go in and try to save this person. As you get back to the beach you see that he is not responsive and decide to to find if he is alive. You are now faced with the choice of how to perform the measurement.

A weak measurement – You try to get a pulse – The measurement is somewhat noisy since the pulse may be too weak to notice. It is also a weak measurement since it is unlikely to change this person’s state.

A noisy measurement – You start screaming for help. There is some small chance that the guy will wake up and tell you to shut up.

A noisy, strong measurement – You start kicking the guy in the head, hoping that he regains conciseness. This is a strong measurement, but it is also noisy. The person might be alive and you still won’t notice after kicking his head, moreover the kick in the head might kill him.

#### The quantum scenario

You want to find the $\sigma_z$ component of a spin 1/2 particle.

A weak measurement – Perform the usual von Neumann measurement with weak coupling. There is still some back-action but if the coupling is sufficiently weak you can ignore it. The down side is that you will get very little information.

A noisy measurement – Perform the weak measurement as above, but follow it with a unitary rotation and some dephasing.

A noisy, strong measurement – Perform a standard projective measurement, but then add extra noise at the readout stage. This could, for example,  be the result of a defective amplifier.

While all of the measurements above are noisy, only the weak measurements follow the original motivation of making a measurement with a weak back-action.

## An extreme example

One neat example of a measurement which is noisy but not weak involves a wave function with a probability distribution that has no tails.

Take the measurement of a Pauli observable that has results $\pm1$ and imagine that after the readout we get the following probability distributions: If the system was initially in the state corresponding to +1 we get a flat distribution between -9 and 11, if the result is -1 we get a flat distribution between -11 and 9. The measurement is noisy, in fact any result between -9 and +9 will give us no information about the system. However it is not weak since any result outside this range will cause the state of the system to collapse into an eigenstate.

A pointer with no tails: The probability density function for the result of a  dichotomic measurement. A +1 state will produce the blue distribution while a   -1 state will produce the orange one. Although a result between -9 and +9 will provide no information, the measurement is still not weak.

It is not surprising that this type of measurement will not produce a weak value as the expectation value of a given set of measurements on a pre and post selected system.  While this is is obviously an extreme case,  any situation where the probability density function for the readout probabilities has no tails will not be weak for the same reason. The same is usually true in cases where the derivatives of the probability density function are very large. In less technical terms – noise is not a sufficient condition for a weak measurement.

1. To get a partial historic account of what AAV were thinking see David Albert’s remarks in Howard Wiseman’s QTWOIII talk on weak measurements (around minute 25-29)

# Photons in curved space-time

The IQC blog “Our quantum world” is finally open. The first blog post is about possible experiments that can detect the effect of Earth’s gravity on photons, check it out.

You can also check out my papers with Daniel Terno and other collaborators on the subject.

Polarization rotation, reference frames, and Mach’s principle

Photon polarization and geometric phase in general relativity

Post-Newtonian gravitational effects in optical interferometry

# Writing a PRL comment.

Getting a comment published  in PRL is far from trivial. I recently succeeded in publishing my Comment on: How the result of a single coin toss can turn out to be 100 heads.  Despite the fact that I pointed out a major  error in a published paper, the road to publication was long and difficult.  The odds were against me (5 other comments on the same paper were rejected despite being correct) and I had to fight the authors of the original paper on top of a biased editor and a referee.   I hope this post will help anyone  thinking about writing a comment (or a reply to a comment).  This was my second attempt at a PRL comment, the first one did not get published as a comment but eventually became a very well received paper.

I attempted to be as general as possible and minimize the specific technical details of  my comment. For anyone interested in the content, the arXiv version is slightly more complete.  It contains a one paragraph response to the published reply. This short response is a very good summary of the comment.

# Publishing a comment in PRL

## 1. A PRL with errors

So you think PRL XXX has mistakes and makes false claims, and you think you should let the world know. What better way than submitting a comment in PRL?  Well… You’re in for a treat.

Generally PRL will only publish a comment if it identifies a central error. This has  following implications: If the paper in question is completely meaningless it cannot contain an error and is therefore comment-proof. If the discussion of the results is speculative to the point where it is not supported by the results of the paper, it is also safe, unless you can convince the editor that the discussion is a central point. etc..

Example 1: The paper I commented on made absurd claims such as:

Our results provide evidence that weak values are not inherently quantum but rather a purely statistical feature of pre- and postselection with disturbance

such claims, although unjustified, are unfortunately safe from comments since it is almost impossible to demonstrate they are the central point of the paper.

My guess is that the majority of the five or six comments submitted on this specific paper were rejected for this reason. Each comment (e.g 1,2,3,4) showed that central features of weak measurements were missing in the supposed classical analogue’. The relation to weak values is so weak that all subsequent conclusions about the nature of weak values are speculative at best.  But wild speculation is not a central mistake and the claim is safe from criticism in PRL.

Technical note: I commented on a paper that supposedly provided a classical analogue for anomalous weak values. The central result of the paper was a measurement scheme that provides strange results which they claimed are classical weak values. Most of the other comments showed that the scheme is not a weak measurement in one way or another. My comment was different in that I simply showed that they are using a nonsensical method to calculate the result of the measurement.

## 2. You will probably end up fighting with both the referee and authors of the original paper

Before submitting your comment, you may want to contact the authors of the original paper to see their reaction. I guess that in some cases scientists will put science ahead of their ego, but in many (most?) cases they will not. The result is that they will try to fight the comment. You should be aware of this.

The editor also has an ego. He accepted the paper and he does not like to admit he accepted a paper with a major mistake (remember PRL only accepts comments that point out central errors).

After you submit the comment, the editor will usually send it directly to the original authors (the other option is a quick reject). They will then write a report claiming that your comment is not worth publishing.

Here you are at a disadvantage. The editor will probably side with the authors. The likely outcome is a reject or (as in my case) a “we cannot accept” unless you make a good rebuttal.

Example 2: A few years ago we tried to write a comment on a paper titled “Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps“. We provided a counterexample to the statement in the title. The main objections of the authors were that we were working under different assumptions. WTF? There were no explicit assumptions in the paper (or in the preceding literature) that contradicted our counterexample, moreover they never event attempted to point out what the different assumptions were. Nevertheless the editor agreed with the authors. The counterexample never made it as a comment (despite going through phase 3-4 below), but did eventually turn into a very nice paper.

## 3. The rebuttal

If things went well, this is the point where the editor asks for a rebuttal of the claims in the informal reply. Now, here is the most important piece of advice. Do not make any changes to your comment at this stage! Just as the authors try to show that your comment is a piece of junk, you should at this point show that their reply is a joke. Remember, the authors are not your friends, they are not trying to improve your comment, they want it out of their way.

So you need to plan ahead. When submitting the comment you must anticipate all possible replies by the authors. If your argument is correct, and furthermore if you hit a mistake that may be considered critical, the rebuttal should not be a major task. This is what I learned from my previous experience and it paid off.

One issue that appeared in all author responses (to this comment and the previous one) was misdirection. On top of the attempt to show that the claims are not a central issue, the authors try to misdirect the editor/referee and move the argument in a different direction.  The best advice is to try and ignore those issues that are tangential to your central claim.

Example 3:  In both the informal and published reply, the authors claimed that there are other papers with classical analogues of weak measurements. The issue has little to do with my comment. My comment was: There is a mathematical error in this paper. I don’t really care if some other guys have similar results. I never responded to that criticism.

## 4. If you were successful with your rebuttal, your paper will be sent to a referee

The referee will give you comments on your manuscript. Treat them like any other referee reports. In my case they made very good points and asked for clarifications and changes before making a decision. I made the changes and the paper is much much better as a result.

It is perhaps important to remember that the referee will get some of the editor’s bias, in-fact in my previous comment we got the following negative report “I agree with the editor”.

## 5. Back to the authors

And again they get to reply, and again they can make it as long as they want.

I was reasonably lucky at this stage because the authors are not experts in the field and it was very easy to point out mistakes in their reply. As before, I did not make any changes based on the author’s reply, and only responded to real criticism.

The paper went back to the referee and he made some suggestions on possible changes but otherwise recommended publication.

At this point the authors had to give their one page reply. This reply did not go to referees. It was sent to me, but the editor explained that it will be published regardless of what I say. My only reply was to point out two technical errors. The authors quoted some results that do not exist and are plain wrong (in fact they base their argument on an inequality where the units don’t match).  The editor was nice enough to let me add a (very brief) note to the comment pointing out these errors. He was also nice to the original paper’s authors and allowed them to make the same type of change (but nothing else), i.e add a note at the end of their paper. Their note contained another mistake (actually the same mistake again) and subsequently they now have a reply with three major technical errors.

This is a lesson for anyone writing a reply. It might not be refereed so make fucking sure it is all correct. One way to get some feedback is to use similar arguments to the ones used to fight the comment. If you make up new things at the last moment you are risking it.

Since there was no option to react to their reply, I added a short response on the arXiv version. Actually this short response is better than my comment and it is a shame it would not get published, but such is life.

## 7. Have fun

If like me, you like a good argument, a comment is an extreme challenge with a referee to help decide the winner. At the end of the day (actually months) the back and forth can be enjoyable. On top having to make concise and precise arguments I  had to read the literature presented in the counter-arguments, this forced me to read some  nice papers that I had been putting off for a while, or missed.  Moreover, each time  it was great to learn how well I anticipated the  attempts at refuting my comment (I did not anticipate the  mathematical mistakes that appeared in  the published reply).

## 8. Conclusion

Writing a comment can be a very rewarding experience, especially if it gets published. On the other hand, comments are a lot more work than you would expect. The process is ugly and biased against the person making a comment. As a rule, PRL editor try to fight off comments and make to road to publication tough. The upside is that getting it published is extremely gratifying.

# Misconceptions about weak measurements: 1. Weak ‘measurements’.

It seems that I am somehow drawn to controversial subjects. Maybe it’s my nature as an Israeli. Much of my Phd research has been around the controversial topic of ‘quantum discord‘.  Now I find myself working hard on the subject of my Master’s thesis, ‘weak measurements’, yet another topic which is both controversial and misunderstood.  Unlike quantum discord which became controversial mainly due to it’s popularity (aka the discord bubble),  weak measurements were controversial from day one.   This controversy  is, at leas in part, due to both misunderstandings, different interpretations, and choice of words; in particular the word measurement’.

I first realized that the use of the word measurement may  cause of confusion after watching a recorded lecture by Anthony Leggett.  But the problem really sank in after many discussion with Marco Piani who helped me clarify my thoughts about the subject.  At one point Marco’s reaction reminded me of the phrase “You keep using that word. I do not think it means what you think it means.“.

Let me explain.

In a discussion of measurements, Asher Peres, one of the main critics of weak measurements (and my academic grandfather’) wrote The “detector clicks” are the only real thing we have to consider. Their observed relative frequencies are objective data.’ This is the usual sense we think of measurements in quantum information, a measurement  is a channel that takes a quantum state as input and gives probabilities (the relative frequencies) as an output.  Let us call this the quantum information approach. In more technical language a measurement is completely specified by the POVM elements. Roughly speaking any set of POVM elements that sum up to the identity can describe a measurement i.e probabilities for the various detectors clicking. However, a weak measurement cannot be described in this way, the POVM elements are at best, only part of the picture.

The quantum information approach. Quantum in – classical out

The first time I encountered the term POVM was during the time between my B.Sc and M.Sc when I started learning the basics of quantum information. As  an undergad, I was taught about a different type of quantum measurement. The quantum world, I was told, is made up of quantities that are observable; these correspond to Hermitian operators. The results of the measurements are eigenvalues and after the measurement the measured system will change its state to the relevant eigenvector, the so-called collapse of the wave function. I will call this the textbook approach. The big difference between this and the quantum information approach’ (above) is that the channel has a classical input and a classical (eigenvalue) and quantum (eigenvector) output. This is closer in spirit to the measurement’ in a weak measurement. However this framework does not have any variable strength.

The textbook approach. Quantum in- classical [eigenvalue] and quantum [eigenvector] out.

The textbook approach’ is unsatisfactory in two ways. First it allows a limited class of measurements that do not necessarily correspond to realistic situations. Second it does not include a dynamical picture: measurements simply happen. Although a full dynamical picture is still an (if not the) open problem, von Neumann gave the a reasonable dynamical picture for the measurement which is know as the von Neumann scheme. The measurement is described as a coherent interaction between the measured system and a (quantum) meter initially in a state |0>.  The interaction Hamiltonian is set up so that: if the system is in an eigenstate a of the desired observable, the meter will shift accordingly i.e it will end up in the state |a>.  Generally the system-meter state will be entangled after the measurement. With the right choice of interaction Hamiltonian the local picture will be  a mixed state that gives the right statistics for the textbook measurement. A slightly more elaborate picture can be used to describe more general measurements.

The von Neumann scheme. Quantum [product system-meter] in- quantum [entangled system-meter] out.

A weak measurement is a measurement in this sense, i.e it is a channel that has a system-meter (quantum) input and a system-meter (quantum) output. The measurement can be followed by a readout stage where a single detector clicks’, but this part simply tells us something about the meter and only indirectly about the system whose state has changed.To complete the transition from the von Neumann scheme to a weak measurement we simply need to make the interaction Hamiltonian weak. It should be so weak that, after the measurement, the shifts corresponding to different eigenvalues will strongly overlap. The first  advantage of this  method is that the system state is virtually unchanged by the measurement process. Other, surprising  advantages follow, especially when one considers the fact that this measurement process is symmetric with respect to time.

I hope I convinced you that the term measurement’ means different things to different people; While quantum information theorists say measurement and mean a quantum to classical channel’ the weak measurement community think of a quantum to quantum channel’. I believe this is major source of confusion that leads to controversy around weak measurements. My advice to people in the quantum information community is: either stop thinking about weak measurements as measurements, or read the literature and try to convince yourself that this channel represents the closest thing we have to a measurement in quantum theory.  Either way stop trying to understand weak measurements simply in terms of POVM elements.

In upcoming posts I will try to clarify some other misconceptions including the difference between noisy measurements’ and weak measurements’, and an explanation of what is anomalous about ‘anomalous weak values’.

# Weak measurements and post selection in NMR.

Last summer  I started thinking about my master’s thesis on weak measurements. I’ve been keeping an eye out  for interesting weak measurement papers for a while and have had the opportunity to referee a few  papers on the subject that forced me to keep up to date.     I started playing around with some weak measurement ideas  when Raymond Laflamme (one of my current supervisors)  suggested I  give a short introduction at the next group meeting. The biggest question at the end of this short introduction was “can we do this in (liquid state)  NMR?”. My first response was an outright NO’, because any interesting weak measurement experiment would require post-selection (see below), a very difficult task in an ensemble system like NMR.  After some serious thought I realized that the solution was actually very simple.  What I found amazing was that the experimentalists  were able to perform the experiment immediately, in-fact these guys can perform any small quantum circuit without too much trouble. The result was the first weak measurement experiment that did not involve any optics. The paper was published in NJP (open access) and a video-abstract is available on the NJP website and youtube.

This month I also taught a short four lecture module on weak measurements and the two state vector formalism as part of QIC 890. But I will keep the discussion of weak measurements to another post. For now I will explain the trick used in the NMR experiment. That will require me to first explain some issues regarding ensemble quantum computers.

## Ensemble quantum computing

Today we don’t know what a quantum computer will look like. We don’t know what it will be made of and we don’t know how it will work. While from a computer science perspective all architectures are the same, that is they can solve the same problems, from a practical perspective they are quite different.  Nevertheless in most cases we like to think of an abstract quantum processor in a similar way to a standard processor, in terms of circuits.

The circuit accepts a classical input, a series of zeros and ones, encoded in quantum bits. The circuit itself is a sequence of operations on those quantum bits. These operations are reversible (unitary) but otherwise they can be quite general. At the end, some of the quantum bits are measured in a specific way and a classical output (a series of zeros and ones) is produced. This output is usually not deterministic so the program can produce different outputs for the same input. Although this seems like a flaw it is not, as long as the probability for an unwanted result is low.

In liquid state NMR the quantum bits are the nuclear spin degree of freedom of single atoms on a molecule. The molecule is the processor and the natural electromagnetic interactions inside the molecules are supplemented with controlled external fields to produce the dynamics (i.e the gates).   Control in this system is very good but there are a number of downsides. The main downside is that the signal is very noisy. To overcome the issue of noise a large number of molecules are used. This means that a large number of identical processors are running in parallel.

One of the drawbacks of running the computation on an ensemble of identical processors is in the readout stage.  The final measurement is an ensemble measurement and the result is a statistical average.  Why is this bad? Let us say for example that we are running a classical computation on two bits with two possible results. Half the time the result is 0,0 and half the time it is 1,1. Now if we read the average on each bit we get that each bit is 1 half the time and 0 half the time so on average it is 1/2. But this average result 1/2,1/2 is also consistent with an output which is 0,1 half the time and 1,0 half the time.  So we can’t distinguish between these results.

Liquid state NMR is not the only system where this kind of ensemble paradigm applies and it is quite possible that ensemble quantum processors will be the way to go for quantum computing, at least in the short term. Liquid state NMR is also the current record system, with good control of 12 qubits.  It is therefore not a surprise that people have come up with methods for circumventing the shortcomings of ensemble readouts.  Going back to the example above it is possible to have a third bit register set to   1 if the first two are equal and 0 if they are not. This will distinguish between the first and second scenario above. In the first case we will have 0,0,1 half the time and 1,1,1 the other half while in the second case we will have 1,0,0 half the time and 0,1,0 the other half.

## Post selection and weak measurements.

In the case of post selection we want to read the average result of the first (quantum) bit but only in the case where the second one is in a specific state (say 0).  So if we have 0,1 one third of the time 0,0 one third and 1,0 one third we should to read out 1/2, the average of the first bit only in the two cases where the second was 0.  A similar situation exists when we want to get an interesting result for a weak measurement. The reading on the measuring device  must be post-selected according to the  state of the measured system.

To perform the post-selection we used a (seemingly) non reversible operation. Sticking to the example of classical bits above our algorithm worked in the following way.  We want to post select on the cases where the second bit is 0. To achieve this we perform an operation that randomizes the first bit if the second bit is 1. When we get the averages at the end we know how many times we got a random result (by measuring the second bit) and how many times we got a real result.  Using this information we can get the statistical average of the post-selected states.

The quantum case is a little  more involved but the basic idea is the same.  This trick allowed us to perform the weak measurement experiment with post selection and get strange results such as complex values and values in outside the normal range. The method we used for post-selection goes beyond weak measurements. We are currently thinking about other weak measurement experiments as well as other experiments that involve post-selection. The advantage is that we can control bigger systems than anyone else (by we I mean the experimentalists, I can’t control anything).

This was also my first collaboration with experimentalists. I’m looking forward to more.

Experimental realization of post-selected weak measurements on an NMR quantum processor,

Dawei Lu, Aharon Brodutch, Jun Li, Hang Li, Raymond Laflamme,  NJP 2014.

# Riddles

As everyone knows I like riddles. At Macquarie I had a weekly blog post on riddles. Unfortunately it seems that this blog is down so: no link for you!

That would be a very unsatisfactory end to this blog post. Fortunately I was looking at the IBM website a short while back. IBM has an amazing group of researchers doing very basic research on quantum information. These guys, among other things, discovered quantum teleportation.

What I found on the IBM website was a fantastic page full of nice puzzles. http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/pages/index.html

Enjoy