physics

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Three papers published

When it rains it pours. I had three papers published in the last week. One experimental paper and two papers about entanglement.

  1. Experimental violation of the Leggett–Garg inequality in a three-level system. A cool experimental project with IQC’s liquid state NMR group.    Check out the outreach article  about this experiment.
  2. Extrapolated quantum states, void states and a huge novel class of distillable entangled states. My first collaboration with Tal Mor and Michel Boyer and my first paper to appear in a bona fide CS journal (although the content is really mathematical physics). It took about 18 months to get the first referee reports.
  3. Entanglement and deterministic quantum computing with one qubit. This is a follow up to the paper above, although it appeared on arXiv  a few months earlier.
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Towards quantum supremacy

 Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.

Asher Peres

Being a theorist, it is easy to forget that physics is an empirical science.  This is especially true for those of us working on quantum information. Quantum theory has been so thoroughly tested, that we have gotten into the habit of assuming our theoretical predictions must correspond to physical reality. If an experiment deviates from the theory, we look for technical flaws (and usually find them) before seeking an explanation outside the standard theory. Luckily, we have experimentalists who insist on testing our prediction.

Quantum computers are an extreme prediction of quantum theory. Those of us who expect to see working quantum computers at some point in the future, expect the theory to hold for fairly large systems undergoing complex dynamics.  This is a reasonable expectation but it is not trivial.  Our only way to convince ourselves that quantum theory holds at fairly large scales, is through experiment. Conversely, the most reasonable way to convince ourselves that the theory breaks down at some scale, is through experiment. Either way, the consequences are immense,  either we build quantum computers or we make the most significant scientific discovery in decades.

Unfortunately, building quantum computers is very difficult.

There are many different routes towards  quantum computers.  The long and difficult roads, are those gearing towards universal quantum computers, i.e those that are at least as powerful as any other quantum  computer. The (hopefully) shorter and less difficult roads are those aimed at specialized (or semi or sub-universal) quantum computers. These should outperform classical computers for some specialized tasks and allow a demonstration of quantum supremacy; empirical evidence that quantum mechanics does not break down at a fairly high level of complexity.

One of the difficulties in building quantum computers is optimizing the control sequences. In many cases we end up dealing with catch-22. In order to optimize the sequence we need to simulate the system; in order to simulate the system we need a quantum computer; in order to build a quantum computer we need to optimize the control sequence…..

Recently Jun Li and collaborators found a loophole. The optimization algorithm requires a simulation of the quantum system under the imperfect pulses. This type of simulation can be done efficiently on the same quantum processor. We can generate the imperfect pulse `perfectly’, on our processor and it can obviously simulate itself.   In-fact, the task of optimizing pulses seems like a perfect candidate for demonstrating quantum supremacy.

I was lucky to be in the right place at the right time and be part of the group that implemented this idea on a 12-qubit processor. We showed that at the 12-qubit level, this method can outperform a fairly standard computer. It is not a demonstration of quantum supremacy yet, but it seems like a promising road towards this task. It is also a promising way to optimize control pulses.

As a theorist, I cannot see a good reason why quantum computers will not be a reality, but it is always nice to know that physical reality matches my expectations at least at the 12-qubit level.

P.S – A similar paper appeared on arXiv a few days after ours.

  1. Towards quantum supremacy: enhancing quantum control by bootstrapping a quantum processor – arXiv:1701.01198
  2. In situ upgrade of quantum simulators to universal computers – arXiv:1701.01723
  3. Realization of a Quantum Simulator Based Oracle Machine for Solving Quantum Optimal Control Problem – arXiv:1608.00677
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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

 

nl

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

 

 

Photons in curved space-time
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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

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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.

6. The reply

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.

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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.

 

click

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.

TB

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.

vN

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’.

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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.

 

 

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Discord and completely positive maps

After over two years of work  we finally published our results showing that the connection between discord and complete positivity is quite weak, and probably has no operational significance. But let me start at the beginning:

In the beginning… and then there was a big discussion/argument about possible maps describing the evolution of a quantum system interacting with the environment. In the case where they are initially correlated this discussion is still not settled. In 2007 came a paper by Cesar Rodriguez-Rosario Kavan Modi, Aik-meng Kuah, Anil Shaji and ECG Sudarshan titled “Completely positive maps and classical correlations“, I call it Cesar and Kavan’s paper. They examined a situation where some initial family of system-environement, $$mathcal{SE}$$ states is classically correlated (has zero discord). It turns out that such a family of states is in the consistency domain of  a completely positive assignment map. In slightly less technical language: given a family of classically correlated $$mathcal{SE}$$ states it is possible to describe the evolution of the system using a completely positive map. Without going into details this comes with some caveats. Cesar, Kavan and Alán Aspuru-Guzik explained these caveats in another paper.

About a year after Cesar and Kavan’s paper Alireza Shabani and Daniel Lidar published a paper titled “Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps” This result was published in PRL, I will call it the SL paper. It made a lot of waves and has since been cited around 150 times. Unfortunately no one really understands it. I don’t know who should be blamed here, the authors for writing an unreadable paper (I assume they can read it), the editor for accepting an unreadable paper, or the referees who thought the paper was readable. But as it stands this paper was accepted, and since it was published in a prestigious journal and has such a bombastic title, people love to cite it. Especially to justify their research on discord. I guess I could start a rant but it’s nothing new so let us return to the story.

In September 2010 I had the extreme pleasure of attending the “Quantum Coherenece and Decoherence” workshop in Benasque where I met Cesar and Animesh Datta. After a short conversation about discord and interesting results in the field we discovered that although we have all cited SL we don’t know what they actually claim. We all assumed it was the “necessary” part of Cesar and Kavan’s “sufficient” result for completely positive maps but none of us could really explain the bottom line. After spending a few days in trying to understand the paper together we finally gave up, and instead came up with a counterexample. That is, we found a family of discordant states which is consistent with a completely positive assignment map.

A few weeks later I met Kavan in Singapore and we discussed this result further…

[missing reel]

.. and finally  Ángel Rivas joined our jolly group. The work was very slow, mostly due to us being on 4 different continents. By the time time I was at IQC we had a draft. When Kavan came to visit we finalized the paper.

The final version is much more then a comment on SL’s result. We showed that the problem of finding the map that correctly describes the evolution is a matter of how the problem is stated. More to the point, we showed that in at least thee sensible frameworks for approaching this problem there is at best a one way connection between positivity of the map and discord. Presumably there might be a framework where zero discord is both necressary and sufficient for completely positive maps. Unfortunately we were unable to identify this framework.

 

Vanishing quantum discord is not necessary for completely-positive maps
Aharon Brodutch, Animesh Datta, Kavan Modi, Ángel Rivas, César A. Rodríguez-Rosario arXiv:1212.4387, Phys. Rev. A 87, 042301

 

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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.

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…

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Faster than light quantum information

A cool new paper appeared in Nature Physics this week and made a lot of waves (at least on my facebook feed).

In this paper titled “Quantum non-locality based on finite-speed causal influences leads to superluminal signalling” the authors show a method for testing if the “weird” effects of entanglement such as the violation of Bell inequalities can be explained using a faster then light source for exchanging information.

The main problem with these kinds of theories is that the usual Bell experiments can at best put a lower bound on the speed of exchanging this kind of information. The way to overcome this problem is to look at Bell violations in a 4 party system. The result of the paper is that one can perform a test that requires the speed of these causal influences to be infinite.

http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys2460.html