Physicists measure complementary properties using quantum cloning

[.Scott]

The title of this thread is the title to this Phys.org article posted today:
https://phys.org/news/2017-08-physicists-complementary-properties-quantum-clones.html

First, correct me if I'm wrong, but the problem is not that complementary states cannot be both measured, but that the there is a limit ($\hslash/2$) describing how measuring one results in a trade-off in what can be measured in the other.

Here is their claim:

"In our daily lives, information is often copied, such as when we photocopy a document, or when DNA is replicated in our bodies," Thekkadath explained. "However, at a quantum level, information cannot be copied without introducing some noise or imperfections. We know this because of a mathematical result known as the no-cloning theorem. This has not stopped physicists from trying. They developed strategies, known as optimal cloning, that minimize the amount of noise introduced by the copying process. In our work, we go one step further. We showed that it is possible to eliminate this noise from our measurements on the copies using a clever trick that was theoretically proposed by Holger Hofmann in 2012. Our results do not violate the no-cloning theorem since we never physically produce perfect copies: we only replicate the measurement results one would get with perfect copies."

Why am I so thoroughly skeptical of this?

From any normal view of QM, complementary states cannot be measured because they don't both exist - there simply isn't that much information in the system to measure. So I'm putting this article into the same category as perpetual motion machines.

Of course, if I'm wrong, I'm sure I'll hear about it.

 

[jerromyjon]

Why am I so thoroughly skeptical of this?

From any normal view of QM, complementary states cannot be measured because they don't both exist - there simply isn't that much information in the system to measure. So I'm putting this article into the same category as perpetual motion machines.

All I have is instinctual skepticism. I saw the article earlier and thought the same thing, Heisenburg wasn't just an enthusiast like me so I'm sure the limit (ħ/2) will hold...

 

[vanhees71]

Already the 1st sentence in the above linked phys.org article is wrong. I wonder, why they didn't simply copy the correct 2nd sentence in the abstract by the authors of the PRL article:

In quantum mechanics, measurement-induced disturbance is largest for
complementary properties and, hence, limits the precision with which such properties can be determined
simultaneously.

The emphasis is mine, and I'd rather write "...can be prepared simultaneously" to make the meaning of the uncertainty relation as clear as possible. It says that for any state of a system the standard deviations of two incompatible observables are subject to the inequality
$$\Delta A \Delta B \geq \frac{1}{2} |\langle [\hat{A},\hat{B} \rangle|.$$
This does not imply that I cannot measure either of the observable more accurately than given by the state. This is also obvious since the accuracy doesn't depend on the measured system but on the measurement device. Usually you can only measure one of the observables on one system, but to verify the uncertainty relation you have to measure anyway the observables on a sufficiently large ensemble of equally prepared systems, and the measurement of each observable must be much more accurate than the standard deviations of this observable anyway.

It's as you learn in the 1st session of your 1st introductory lab about error analysis in experiments!

 

[bhobba]

Without even reading it rest assured it's wrong - look into the no cloning theorem.

Heuristically the reason for the no cloning theorem is you would then be able to measure at the same time with no uncertainty things QM says you cant. So - you can't clone systems.

There is also the issue of exactly what the indeterminacy relations of QM actually say - they put no limit on the precision of measurement as Vanhees correctly points out- but if you want to go further into it, it really is another thread.

Similar articles have been discussed on this forum before - they nearly always are a misunderstanding of what are called weak measurements - you can look them up and if anything is unclear fire away with a new thread.

Thanks
Bill

 

[vanhees71]

Heuristically the reason for the no cloning theorem is you would then be able to measure at the same time with no uncertainty things QM says you cant. So - you can't clone systems.

This should read
"Heuristically the reason for the no cloning theorem is you would then be able to determine (prepare) at the same time with no uncertainty things QM says you cant. So - you can't clone systems."

As I emphasized above, you can measure any observable with any accuracy you like, provided you have the technical means to do so. There is no fundamental limitation to the accuracy of measurement. You can measure position as accurately as you want and momentum as accurately as you want on a system, no matter in which state it is prepared. However, the uncertainty relation implies that you cannot prepare the system in a state such that both position and momentum are both more accurate than allowed by the uncertainty relation, i.e., for the standard deviations you always have $\Delta x \Delta p \geq \hbar/2$.

 

[Strilanc]

Maybe I can shed some light on this.

First of all, ignore the article and go straight to the paper (or rather the pre-print "Determining complementary properties with quantum clones" on the arXiv). In the paper they discuss various ways to approximately clone a qubit:

1. By randomly swapping between two qubits (e.g. controlled by a coin flip). This doesn't introduce any interesting entanglement between the two output:

The above is an animation of random-swap cloning in my simulator Quirk. The top qubit is the value to clone and set to various states along the ZY plane. The second qubit is the clone target and set to a mixed state by entangling it with the otherwise-unused third qubit. The fourth qubit is the coin-flip for whether to swap or not.

The green displays show the individual qubit values mapped onto the bloch sphere. The light-blue displays are showing how the qubits relate to each other. In particular, they show what the first qubit's state would be given for a nice covering of possible measurement axes / outcomes on the second qubit.

Notice that the right-most display is always showing a line. That's how you know the two qubits are only classically correlated: no matter where you measure one qubit, the other ends up along a single ray. Entanglement would look like a ball.

2. By doing a square-root-of-swap operation:

A bit better. There's entanglement now. But really we don't want entanglement, we want total independence. The closer we can get to that the better. Which leads to the known-optimal cloning method...

3. Projecting onto the symmetric state (and not counting the run if the projection fails):

Notice that there's still entanglement. However, the ball is smaller than before. We're closer to independence (i.e. perfect cloning). In fact, we're as close as we can get; this is the optimal approximate cloning method.

For reference, this is what's hidden inside the box:

What's happening: mix the clone-target qubit, count the input bits to figure out which symmetric basis state we should be in, un-prepare that basis state so that the top two qubits should be in the 0 state (unless we're not in the symmetric subspace), assert that we succeeded by throwing out runs where the top two qubits aren't zero, then re-prepare then and uncount.

The rest of the paper is just talking about properties of the approximately cloned state and what they measured on it.

 

[jerromyjon]

The rest of the paper is just talking about properties of the approximately cloned state and what they measured on it.

That's going to take some deep concentration for me to get into, thanks for the info! But in laymen's terms, is that kind of like a "weak" cloning along the lines of a weak measurement?

 

[bhobba]

This should read "Heuristically the reason for the no cloning theorem is you would then be able to determine (prepare) at the same time with no uncertainty things QM says you cant. So - you can't clone systems."

Of course - my bad

Thanks
Bill