## Entangled particles, faster than light communications and the no-cloning theorem

In 1982, a paper entitled “FLASH – A Superluminal Communicator Based Upon a New Kind of Quantum Measurement” was published by Nick Herbert, an American physicist who meant to prove that by using EPR entangled pairs and quantum effects, a superluminial communicator could be built to transmit information at speeds faster than the speed of light . Such a statement caused quite some uproar in the scientific community, because, as you may know, there are no signals than can travel faster than light.

The story of Herbert’s paper is not a story of an experimental proposal with a fundamental error, but the story of scientific progress and how, by studying the possibility of a superluminial communicator, the no-cloning theorem came to be. We will see in this article how the resolution of this problem improved our understanding of physics by showing that we cannot copy an unknown quantum state.

Now, before proceeding to review Herbert’s paper, we should first answer the question:

# Could there be superluminial velocities?

The answer to this questions is no… and yes… (of course it is, isn’t it?). Let’s first try to understand why it is that the answer to this question should be no.

## It’s all about causality!

It’s a very common misconception to believe that special relativity somehow forbids superluminial communications. Einstein’s theory states that 

“… light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body”.

So, if the principle of invariant light speed imposes a condition on the speed of light in all inertial frames of reference, how come we usually believe that it directly implies a speed limit on communications?

If we would like to transmit information using massive particles, we know that as the velocity $v$ of the particle (of mass $m$) approaches the speed of light it’s momentum tends to infinity: $p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}$

so we know that we cannot use them to transmit information at velocities faster than $c$. However, this inertia constraint doesn’t apply to massless particles such as photons. Hence, it’s not a question of whether something can, or cannot, travel faster than the speed of light but a question of what would happen to causality if we had superluminial communications.

Consider the following scenario : Alice and Bob are time-like separated, let’s say in two distant planets. Their Minkowski diagram is shown in Fig 1. Figure 1: Alice and Bob’s Minkowski Diagram. Time increases from bottom to top and the distance that separates them is given by the horizontal axis.

Alice sends a signal to Bob using photons (green dashed line), if we could indeed have superluminial communications, Alice could also simultaneously send a message A to Bob letting him know that he will receive a light signal. Furthermore, Alice could also send a signal B that Bob would receive BEFORE she actually sent it. By means of superluminial signals we could then “send signals into the past”! When receiving this signal (B), Bob can then immediately proceed to send a message C to Alice that she would receive even before sending message B. Now, here we have a real brain teaser. Alice then can inform herself of something that she’s going to do in the future. Thus, is we could have faster than light communications then either special relativity or causality are false.

However, superluminal group velocities have been oberved. So… what’s going on?!

## Superluminial velocities

The key aspect to have in mind is that we shouldn’t be able to transmit energy and information faster than the speed of light. Nevertheless, there are certain velocities which are not bounded by this speed, i.e. waves’s phase velocities,  group velocities in packets or quantum tunneling velocities . In wave phenomena, special relativity sets that front velocities can’t be faster than the speed of light, and it’s this velocity which is associated with the information carried by the wave or the wave packet. Thus, even if we can find faster than light velocities, we can’t use them to transfer information.

# Flash: First Laser-Amplified Superluminial Hookup

Now that we know why it is that Herbert’s Superluminial communicator shouldn’t work, let’s review his proposal. In his paper, he states that by using entangled EPR pairs (two entangled photons) we could use the fact that a measurement over one of the photons has an immediate effect on the other, and if this connection could be observed, then it could permit faster than light signaling. The device he proposed is shown in Fig. 2.

A pair of entangled photons are emitted in the Two-photon Source and are sent in two opposite directions to two different observers (Alice and Bob). The two-photon state is given by the rotationally invariant state $\frac{|HH\rangle + |VV\rangle}{\sqrt{2}}$        (1)

where $H$ and $V$ respectively stand for horizontal and vertical polarization. We may also rewrite this maximally entangled state in the circular polarization basis $\frac{|RR\rangle + |LL\rangle}{\sqrt{2}}$        (2) $R$ stands for right polarization and $L$ for left polarization. Once she receives her photon, Alice can choose to either measure the plane polarization, or the circular polarization.  As a consequence of her measurement, Bob’s photon will be projected into either a plane or circular polarized state. Bob can now use his photon (whose polarization depends on Alice’s measurement) to stimulate a Laser gain tube, which will emit $4n$ photons (with $n$ large), with the same polarization as the input photon. These $4n$ photons are then split into 4 beams, each consisting of $n$ photons,. Finally Bob measures each beam with a detector devised to measure a specific polarization: H, V, R, L. The following table shows the probabilities for the outcomes of the measurements in each of the devices given the different possible beam polarizations:

Measurement Device H polarization V polarization R polarization L polarization
H 1 0 .5 .5
V 0 1 .5 .5
R .5 .5 1 0
L .5 .5 0 1

Thus, if the outcome of Alice’s measurement was the plane polarization $H$, we know that Bob’s photon was projected into a $|H\rangle$ state, and therefore his devices are gonna measure

H: $n$ photons

V: $0$ photons

R: $n/2$ photons

L: $n/2$ photons.

All that Bob has to do now is see which of his devices didn’t measure anything and that will let him know that Alice had chosen to measure plane polarization. Following the same reasoning it’s easy to see that if Alice had chosen to measure circular polarization, then either R or L wouldn’t have detected anything, and Bob would have known Alice’s choice.

By means of this experiment Bob and Alice now have a way to perform superluminial communications!

# The no-cloning theorem

At the time the paper was published, it was well known that the photons emitted by a laser gain tube have the same polarization as the input photons. However, Herbert’s mistake was not to realize that if the input photon is in the state $|H\rangle$, the output will not be $4n$ photons in the $|H\rangle$ state. This is due to the fact that $|H\rangle$ can be written as a superposition of $|R\rangle$ and $|L\rangle$ states: $|H\rangle=\frac{|R\rangle + |L\rangle}{\sqrt{2}}$.       (3)

Therefore,  the laser gain tube would amplify $|R\rangle$ states and $|L\rangle$ states each with a probability of 50%. The output of the laser gain tube is not $4n$ photons in the $|H\rangle$ state but rather $2n$ photons in $|R\rangle$ and $2n$ photons in $|L\rangle$. Thus, the outcome of Herbert’s measurements will be

H: $n/2$ photons

V: $n/2$ photons

R: $n/2$ photons

L: $n/2$ photons,

and this will be true regardless of Alice’s choice of measurement!

A. Peres, one of the referee’s of Herbert’s paper wrote :

It was obvious to me that the paper could not be correct, because it violated the special theory of relativity. However I was sure this was also obvious to the author. […] I recommended to the editor of Foundations of Physics that this paper be published. I wrote that it was obviously wrong, but I expected that it would elicit considerable interest and that finding the error would lead to significant progress in our understanding of physics.

Just as he predicted, there were many physicists who took it upon themselves to show that Herbert’s proposal was wrong. GianCarlo Ghirardi , Tullio Weber, Wojciech Zurek, Bill Wootters and Dennis Dieks  showed (independently) that there was a fundamental error in Herbert’s device, and they all agreed that Herbert’s mistake was to suppose that we can copy a random quantum state. Thus was born the no-cloning theorem:

No quantum amplifier can duplicate accurately two or more non-orthogonal quantum states.

The no-cloning theorem is a very profound and powerful statement about quantum mechanics, and ever since it’s discovery it has had practical implications for quantum cryptography, quantum encryption, quantum disentanglement and has led to more general theorems, like the proof of no-cloning of impure density matrices.

The moral of this story is that sometimes we can learn more from mistakes than from perfect reasoning, and scientific progress thrives just as much from errors than it does from correct theories. In word’s of A Peres: “Nick Herbert’s erroneous paper was a spark that generated immense progress”.

All text copyright © Marco Vinicio Sebastian Cerezo de la Roca. Entangled particles, faster than light communications and the no-cloning theorem by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

#### . GianCarlo Ghirardi “Entanglement, Nonlocality, Superluminal Signaling and Cloning”. 