On quantum entanglement and teleportation (part II)

(The introductory text can be read here.)

Apparatus

Imagine two devices separated by a large distance (any distance large enough to nullify quantum mechanical effects). These are devices that receive inputs and yield results. There are two modes in which an input may be received: classical and quantum-mechanical. Moreover, the input is generated by the same source and is delivered simultaneously.



Procedure

1. A set of inputs is generated at the source.


2. Each input may instruct the device to yield a result ‘x’ or ‘y’.


3. Device A reads the instructions and yields a result, A’


4. Device B reads the instructions and yields a result, B’

(A’ and B’ are now the states of the instructions after they have been measured by A and B, respectively)

Observations

1. If A’ and B’ are in the same state – i.e., if they are the result {x, x} or {y, y} – then they may be said to be entangled. To have achieved this, A and B must have communicated in some way to yield the same results arising from the input or must be in possession of some information that enabled them to yield the same result.


2. If it so happened that A and B communicated instantaneously – i.e., at faster than the speed of light – then they may be said to be quantum entangled. (Note that, in a quantum mechanical context, the results are found to be identical only upon observation, which means the act of observing the result is also a participant in the measurement process.)

Inferences

[caption id="attachment_23229" align="alignleft" width="115"] John Stewart Bell[/caption]

#1 If we assume the observations are being made on particles instead of some arbitrary “inputs”, then Heisenberg’s uncertainty principle kicks in. When we make the measurement, we are changing the value of some state variable of the particle, and it is the final state that we end up observing. John Stewart Bell, a Scottish physicist, was the first to make this observation, and added that the act of observation was somehow tied in with quantum entanglement (i.e., that the results were quantum-entangled in some way owed itself in part to the act of observing).

#2 The act of observing is a classical phenomenon because the devices A and B that enable the measurement are classical devices. That said, J.S. Bell argued that this is where the line between classical mechanics and quantum mechanics blurred.

Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and results in our classical domain. (…) Now nobody knows just where the boundary between the classical and the quantum domain is situated. (…) More plausible to me is that we will find that there is no boundary. The wave functions would prove to be a provisional or incomplete description of the quantum-mechanical part. It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called "hidden variable" possibility.

- J.S. Bell

#3 The EPR (Einstein-Podolsky-Rosen) paradox is aimed at refuting quantum mechanics by leveling itself against the possibility of quantum entanglement. Since entanglement occurred only on conjugate entities – particles that are somehow but definitely paired – then the measurement of one of the A’-state variables should render indeterminate that state variable in B’ (Heisenberg uncertainty prin.). However, entanglement has already been observed. This means that either the two particles should have communicated or that the information necessary to generate the same outcome was already present in the two particles.

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The physicists preferred the latter explanation, asserting that some “hidden local variable” was responsible for controlling the outcome of the act of observing. Of course, they made two assumptions in reaching this conclusion: locality and realism. (Note that this is a classical explanation of a perceived quantum mechanical effect.)

#4 In 1964, Bell came out with his famous theorem that refuted EPR’s preferred explanation. He observed that any local realist theories are incompatible with quantum mechanics. Essentially, this means that since a great number of experiments agree with the predictions of quantum mechanical theory, and since many of the results are greater than to be explicable by local hidden variables, either locality or realism is in conflict with quantum mechanics.

The principle of locality states that an object is affected directly only by its immediate surroundings, not by an event that is occurring simultaneously a large distance away. Realism, or counterfactual definiteness (CFD), is the ability to assume the existence of objects and parameters even when they have not been observed – or, to believe that constitutionally present particles shape the properties of the object at all levels. Bell posited that locality had been violated, and that superluminal communication was happening.

[caption id="attachment_23242" align="alignleft" width="114"] David Bohm[/caption]

#5 Bell’s hypothesis was based on the de Broglie-Bohm theory, which interpreted quantum mechanical effects as being caused by an encoding function (called the wave function) that did not let the particles that it guided feedback into itself. The interpretation also assumed that the velocities of the particles depended solely on the wave function, and that the wave function depended on the whole configuration of the universe. That's a contradiction of locality right there.

What does this have to do with teleportation?

Teleportation, by definition, is an instance of non-locality because it implies instantaneous communication. If two particles can communicate faster than at the speed of light to replicate quantum mechanical effects, then perhaps complex objects can someday be replicated instantaneously across large distances by simultaneously reproducing the quantum states of the particles associated with the object.

Of course, such a possibility is hinged on Bell’s theorem being true, on the EPR paradox’s implied existence of locality being false. To date, numerous experiments have been conducted that have neither conclusively validated nor invalidated Bell’s theorem. The difficulty lies in what Bell's theorem implied for the real world: it made quantum mechanics and local realism mutually exclusive. Either quantum mechanics fell short of explaining some physical parameters, or superluminal information communication was occurring (see Lorentz covariance).

[youtube http://www.youtube.com/watch?v=V0zQHNmz0gU]

Disentangling quantum entanglement

Very few scientific concepts enjoy the popularity of teleportation: the idea is equally awe-inspiring among scientists and laymen. To the most inspired, so to speak, what is fascinating is not how it’s accomplished as much as the possibility of “leaving” one space and “arriving” at another, but traversing the interim distance instantaneously. The implications of such travel are significant even at first sight. For example, imagine being able to teleport an object from earth into interstellar space, hundreds of thousands of kilometers away, without having to bother with rockets that might take years to span the same distance.

The sole field of physics seemingly capable of tackling the problems associated with such an esoteric and fragile system—quantum mechanics—still has leaps and bounds to go, however, before it can realize the teleportation of objects. For starters, it hasn’t figured out what really happens during the teleportation of a few photons even though it has accomplished just that with an 80-per-cent accuracy over a distance of 97 km.

In a paper submitted to arXiv on May 9, 2012, Jian-Wei Pan, et al, demonstrate how they used an exotic phenomenon called quantum entanglement to achieve teleportation across Qinghai Lake in western China. Using an ultraviolet laser aimed at a barium crystal, Pan’s team generated pairs of quantum-entangled photons. Then, each photon of a pair was transmitted using a telescope to two parties on either sides of the lake, A and B.

[caption id="attachment_23171" align="aligncenter" width="460"] The experimental setup[/caption]

Making a measurement of the photons yields a good description of the “state” the photons are collectively in. Therefore, A’s and B’s goals are to see if a third party interacting with these photons ends up in a state similar to the control group even when separated by 97 km of free-space. Here, the state of the system refers to the values of a few fixed variables: if the variables hold a particular set of values, then the system is said to be in a particular state (states are usually independent of extrinsic properties such as mass, etc).

To measure this change, the researchers at Site A let photons generated at locally to interact with the incoming modified photons. Simply put, the foreigners would leave an imprint on the locals upon interaction. This imprinted state is then measured and compared with the state of the photons at Site B. In Pan’s experiment, the between the two agreement was 8 on 10.

The aspect that makes such an outcome wonderful is that the particles didn’t have to end up with the same state. Further, 80 per cent is a value large enough to rule out any coincidence (but small enough to rule out a complete success). If this long-distance communication between nanoscopic particles is further investigated, it becomes evident that their pre-travel entanglement provided for a form of durability and predictability of state that let the particles behave similarly in two very different measurement experiments. At the same time, it is the nature of this entanglement that baffles most scientists.

The formal definition of entanglement is very fundamental in the sense that quantum mechanics deals with it in terms of probabilities. When two groups of photons are said to be quantum-entangled, it means that the states that the groups are in are related to each other by means of a variable. If the variable changes, then the properties of the photons change, too. However, the groups’ relationship with each other does not, like siblings who remain siblings despite how old they get or when they each die.

The existence of this variable is not as much disputed as it is hoped into existence (Little wonder then that it’s handled as a product of probabilities?). Because it remains outside the realm of human control, experiments with teleportation tend to leave the hidden variable alone and instead focus on how much the measurement sites can be separated by, how efficiently large molecules can be entangled, etc., i.e., testing the limits of its practicability.

In order to do so, the photons are subjected to a simplified treatment—one conceptualised so as to make the fewest assumptions as well as not introduce new sources of error. Instead of groups of photons, two are addressed, and each is “allowed” to exist in one of two states. Schrodinger’s cat takes off here and asserts that the particles may exist in this state, that state, or the counter-intuitive superposition of both, and that revelation can come only with observation.

Let’s say the two particles are ‘a’ and ‘b’, the states ‘0’ and ‘1’. The four possible combinations of states, then, are:

{0, 0}
{0, 1}
{1, 0}
{1, 1}

Entanglement is said to have occurred when b is in a particular state when a is in a particular state. That is, if b is 1 every time a is 0, then a and b are entangled. Since this property is commutative, a will be 0 every time b is 1. Further, the change occurs instantaneously irrespective of the distance between the two particles (giving the impression that they're "communicating" at a speed faster than light's). The presence of such an order coupled with the four possible outcomes makes each outcome a particular state of the system, called a Bell state.

To find out what the Bell state is, a Bell measurement is made. Because of Heisenberg’s uncertainty principle, however, the act of making the measurement changes the state of the system. Even so, this alteration doesn’t matter as long as the pre-measurement state is discovered. In Pan’s experiment, with six initial states, the Bell measurement was made using the imprinting mechanism to reveal that the entangled photons had interacted with other particles to yield a final state that resembled the initial.

[caption id="attachment_23175" align="aligncenter" width="729"] xkcd #824[/caption]

Earlier, another experiment had been conducted that demonstrated the teleportation of quantum information across 16 km. The principal shortcoming of that experiment was that the photons to be teleported—the “locals” —had been specially generated within the lab under careful conditions. Practically, this is a highly ideal condition that can seldom be met: if this blogger is to be teleported, he cannot be carefully “prepared” in a lab. Pan and his colleagues eliminated this necessity by generating local photons with random quantum states.

At the same time, they have failed to discount a possible source of error: the entangled photons and the to-be-teleported photons were generated by the same source. Even though this limitation doesn’t interfere significantly, it is a limitation nonetheless. (What is the confidence with which it may be asserted that the “local” photons are created in a mixed state? If it’s being assumed that this is not a source of error, what were the considerations made? Et cetera.)

I must concede that looking behind the teleportation curtain kills a lot of the fantasy. Even if entanglement continues to elude understanding, simplifying something so enigmatic to probabilistic proportions and then to linear algebra can be a bit of a buzzkill—disregarding that that is the purpose of scientific endeavour, of course. With their paper, Jian-Wei Pan and his team currently sit pretty at the forefront of quantum mechanical teleportation. Even if we still have a long way go, the knowledge of Pan’s experiment gives us the best shot at ultimately achieving the teleportation of complex objects.