# 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

# 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

# Spring

Its officially spring!!!! We had 23 degrees on Thursday. Actually spring started with freezing rain

and is continuing with snow today, hopefully the last of the year.

Apart from the freezing rain the gallery is a short ode to winter , mostly taken with my phone on the few days I walked to the old IQC building (RAC).

In the past year I had the pleasure of reading two popular science books about topics that are very close to my heart. Lee Smolin’s “The trouble with physics” and Vlatko Vedral’s “Decoding reality”.  Both are excellent scientists and science communicators and the books are highly recommended.

If there is one criticism it is that both authors try to convince the reader  of  their own ideas about how to do ground braking science and how to answer the most fundamental questions in science and philosophy.  In the case of Smolin it is “background independence” and a complete overhaul of the academic evaluation and hiring process. In the case of Vedral it is quantum information theory as the basis of everything. I agree with both on most points (at least those I understand) but feel that when one author tries to undersell string theory and the other tries to oversell information theory some objectivity is lost.

Now you have to ask yourselves a , “Why would anyone recommend a book on science  if it is somewhat subjective”. Well… if I could only recommend truly objective books I would probably be limited to textbooks in mathematics.

Scientists are (for the most part) human and tend to loose objectivity about as fast as anyone else. Ok, true overwhelming evidence will usually make a scientist change his mind and admit he is wrong, but in the absence of such OVERWHELMING  evidence (if the evidence was overwhelming it is probably not pushing the boundaries ) scientist tend to stick to what they believe.  Especially if they have been working in this direction for years and years.

This is indeed the trouble with science communicators. People believe in their own research programs and will try to convince you that these research programs are the best path to the given goal.  One very evident example is research on anthropogenic  global warming. Seemingly there are two sides fighting it out, the “believers” and the “skeptics”.  We, the general public were probably most influenced by our first impression on this debate. It depends on the newspaper article you read first or the first t.v show  you saw on the subject. These are usually written by a non-scientist and include some (often carefully chosen) pieces of interviews with the people who are actually doing the research. The researchers will obviously try to sell their own findings and present them as the state of the art.  And lets’ face it, most of these researchers got into their field with their own prejudice.

So, what is the solution? Don’t try communicate science to the public? Clearly that would miss  the point, the public is funding your research and you better show them something for it. Moreover you should educate and use science to benefit everyone.  One thing we should stop doing is selling scientists as the model of objectivity, people should realize that scientists too have OPINIONS and that every scientific theory should be questioned. The evidence is very seldom overwhelming  at the forefront of science and debates always exist.  Even quantum mechanics, the most successful physical theory so far, may turn out to be incomplete. A number of scientists, even some very good ones, are actively looking for possible flaws in the theory and alternative explanations for known phenomena.
So yes be skeptical about everything.  That being said, don’t outright reject good science just for the sake of skepticism.  There are smart guys out there and they are probably closer to being right then wrong, even if they are somewhat subjective.

# Quantum discord

After a long an eventful month that included a visit by Kavan Modi to IQC and my visit to Israel (I’m posting from Israel), it’s time I got back to writing something. This time I’ll say something about my work for the past four years (as promised). One of the main subjects of my research is quantum correlations, and their role in defining the difference between quantum and classical (not quantum) systems.
Imagine a piece of information shared between two people Alice and Bob. Now think of a way to quantify the correlations between them. One way to quantify correlations is to ask what can Alice know about Bob’s part of the information by looking at her own part.
For example lets say Alice and Bob are each given a queen from a chess board. Alice then looks at her queen and sees it is white. She now knows Bob’s queen must be black. Alice and Bob are strongly correlated, since Alice always knows Bob’s piece by looking at her own.
For the second example Alice and Bob are each given a queen, but this time from a Deck of cards. If Alice sees a red queen she can say that it is more likely that Bob has a black queen, but she has no certainty. Correlations are lower in this case then in the chess example.
There is another way to account for correlations. We can ask about difference between the information in Alice and Bob’s hands individually and the information in their hands together. In the chess example Alice and Bob can each get one of two types of queens: black or white Together they also have two options Black White or White Black.
It turns out that both options for quantifying correlations are the same. To see this in the example we need to quantify the information in bits. Since Bob has two options in his hand “black queen” or “white queen” he has one bit of information. The amount of information Alice can discover about Bob is precisely this one bit. So they have one bit of correlations. Alternatively we can say that Alice has one bit of information: “black queen” or “white queen”; Bob has one bit of information: “black queen” or “white queen” and together they also have one bit “black white” or “white black”. The difference (1+1)-1 is again 1 bit so there is one bit of correlations.
Since i’m avoiding maths you will have to take my word that both methods give the same result in all cases… in the classical world. In the quantum world things are a bit different.
There are two essential (and related) aspects of quantum theory that make these two ideas about how to to quantify correlations give different results. 1) Measurements affect the system. If Alice wants to know the color of her queen, she needs to make a measurement, this measurement can change the state of the system; and 2) Quantum systems can be correlated in a much stronger way then classical systems, a phenomenon known as entanglement.
Before discussing the first aspect in detail, I will say a bit about entanglement. Entanglement was a term coined by Schrodinger in his famous “cat” paper, this paper was inspired by the earlier “paradox” of Einstein Podolski and Rosen (EPR). They showed that quantum mechanics predicts a situation where a system shared by two parties is in a well defined state although locally it is not defined. A system is in a well defined state if making a measurement on this system will give some result with certainty. So if I give Alice and Bob an entangled system I can predict the result of a measurement made on the whole system, but I cannot predict the result of a measurement made by Alice and Bob separately.
Entanglement is the most remarkable prediction of quantum mechanics, and in one way of another it is the driving force behind most of the really cool quantum phenomena. From quantum computers to Schrodinger’s cat. Nevertheless entanglement does not account for all the non-classical features of the theory. At least not directly. When discussing correlations, measurements and their effects on the system play a crucial role in describing non-classicality. To explain quantum measurements we can imagine a quantum system as an arrow pointing to some direction, X, in the simplest case we can think of this problem in two dimensions.
A quantum measurement is a question regarding the direction of the arrow. Is the arrow pointing in direction A? This has one of two results either yes or no. The probability is given by the angle between the “actual” direction and the direction in question. The effect of the measurement is that the arrow will now point in the same direction as the result. If the answer is yes it will point in direction A if the answer is no, it will point in the opposite direction.

A quantum measurement will “collapse” the state X into A or Not A.

Ok so what does this have to do with correlations? Well lets go back to the two definitions for correlations. The first was “What information can Alice find about Bob by looking at her own system”. In the quantum case this is no a clear question, we need to also say what measurement Alice is making. Different measurements will reveal different information about Bob. The second definition for correlations is what is the difference between the information in Alice’s hands plus the information in Bob’s hands and the information in their joint system” This is not directly related to measurement, so clearly it is not the same as the first definition. The difference between these definitions in the quantum case is the quantum discord. It is a measure of the “quantumness” of correlations.
As it turns out discord can be found in many interesting quantum systems and paradigms, but it is not yet clear what this means…

# I’m an expert

Two expected, but long awaited events happened on my birthday. One: I found out that my Phd was approved, so no more bureaucratic shit regarding that. Two: The review paper on discord and similar quantities was finally published. This sums up about one and a half years’ work spent on reading, writing and rewriting this review with my collaborators.  Two versions have been posted on  arXiv  since the end of last year. The latest one, posted in August, is pretty much the published version.

I will soon post something longer about discord and non-classical correlations, for now it is enough to say that quantum theory allows more general correlations then a classical theory.  Entanglement is the best example of these types of correlations, but as it turns out there are unentangled systems with non-classical correlations. Quantum discord captures entanglement and more (but not everything).

Since the beginning of the century (i.e 12 years ago) people started studying these kinds of correlations “beyond” entanglement in various forms and physical scenarios. The area exploded about 5 years ago and discord became a “hot” topic. The review includes almost all the work done on the subject until the end of 2011. discord was studied in so many different scenarios like quantum information, thermodynamics, many body systems, relativistic quantum information and others which made work on this review so much fun, on the one hand, but a lot of work on the other.

# Resurrection!

In good time for my 33rd birthday The Wiki Religion was resurrected. After being martyred by the vast evil community of spammers, the wiki rose from the ashes of no-host and no-domain back to the land of the online.

All should drink and rejoice to our great prophet  ‎Kaer.fyzarc may his booze ever remain strong.

Proph. Fwiffo