Wednesday, October 10, 2012

Quantum Teleportation

Alice and Bob make measurement choices concerning entangled photons A and B
One reason why medieval philosophers such as Thomas Aquinas never developed a sophisticated description of the material world might have been that the priorities of thinkers in the Middle Ages were different than our own. The aim of Thomas Aquinas and his colleagues was to discover the nature of God. Hence they treated the world not as a physical object with its own intrinsic laws but as the personal creation of a divine being. The facts of this world, unimportant in themselves, could tell us about the nature of God in much the same way as a painting or sculpture might inform us about the nature of the artist that created it. A bit of this same theological spirit surfaces in Einstein's famous statement: "I want to know the mind of God; all else is details."

Anyone seeking to know the nature of God by studying the physical universe will certainly be fascinated by quantum theory, our deepest and most successful theory of matter. Quantum theory is deeply paradoxical and seems to obey a distinctly non-human logic. One of the most peculiar feature of this theory is the way it seems to effortlessly embody seemingly contradictory aspects in the same phenomena. One of the most elegant examples of quantum theory's union of opposites is the recently discovered fact of quantum teleportation which unites in one system both a faster-than-light transmission of information plus a clear prohibition against humans using this undeniable FTL connection for sending FTL messages.

Quantum theory also embodies the unusual feature that the world we see depends on the questions that we pose. Hence the more sophisticated we become in asking questions of Nature, the more sophisticated will be Her replies.

Quantum teleportation is a special feature of quantum entanglement in which two photons emitted from a special source give up their individual identities and enter a collective state. The collective two-photon state has definite properties but the individual photons do not, until they are actually observed. For example, in the entangled state W(A,B,+), photons A and B will always be observed to have the same polarization; in the state X(A,B, -), photons A and B are always observed to have opposite polarization. These two entangled states W and X are part of a complete set of entangled two-photon states W, X, Y and Z, called the "Bell states" (after Irish physicist John Stewart Bell). Any two-photon state, whether entangled or not, can be expressed as a sum of the 4 Bell states. This fact is essential to the process of teleportation.

Alice obtains unknown photon "?" she wants to send to Bob

Alice acquires a photon "?" with an unknown polarization which she wants to teleport to Bob. This photon IS NOT ENTANGLED with Alice's photon A but Alice employs a clever trick--only possible in quantum theory. Alice expresses the quantum state of photon "?" and photon A as the sum of the four entangled Bell states W, X, Y and Z. She does this sum in such a way that all the entanglements cancel and the total quantum state of "?" and "A" is unentangled.

Alice's move reminds me of a string trick I learned as a kid in which you wrap a loop of string around your fingers in a complicated way so that it looks as though the fingers are entangled in the string. But upon pulling the string the fingers are freed--every loop of string was cancelled by an anti-loop. It's the same with the two photons--every seeming entanglement is cancelled somewhere by an anti-entanglement.

However because Bob's B photon is entangled with Alice's A photon, a kind of quantum magic occur in which the polarization "?" of Alice's unknown photon is transferred to Bob's photon B, although in a somewhat hidden form. To every term W, X, Y, Z in Alice's expression for her two states, there corresponds on Bob's side of things a quantum state that is either identical to "?" or differs from "?" only by a rotation R and/or a phase shift S. (R and S are fixed by the nature of the original AB entanglement and do not depend on "?".)

Given this setup, here's how quantum teleportation works.

Alice asks the question: which Bell state is my system in? This question can have one of four answers W, X, Y or Z. If the answer is W, then Bob's photon has the polarization "?". Teleportation is accomplished.

If the answer is X, Y or Z, the polarization of Bob's photon differs from "?" only by a rotation R, a phase shift S or a combination of both. So for 100% efficient teleportation all Bob has to know is what Alice's result was--W, X, Y or Z--a piece of knowledge that consists of only 2 bits of information. Without these two bits all that Bob sees is a random hash. With these two bits an infinite amount of information can be teleported. (The polarization of a photon can point anywhere on a sphere. The teleported information corresponds then to sending an unknown latitude and longitude on the surface of the Earth to a distant location faster than light. However this information cannot be decoded without the 2-bit key which must be sent by Alice to Bob at light speed or slower.) Thus a large quantity of quantum information can be teleported faster-than-light but this information is unrecognizable in the absence of a 2-bit code which can only be transferred over conventional channels.

Alice sends a 4-bit signal allowing Bob to decode an infinite-bit message

Quantum teleportation was discovered by a six-man team in 1993 and experimentally demonstrated a few years later. Teleportation is a particularly elegant example of quantum theory's subtle union of opposites--in this case the coexistence of a large FTL data transmission with the impossibility of sending signals faster-than-light.

Let's face it. We are only at the beginning of experiencing and appreciating the inhumanly beautiful mysteries of the quantum world.

QUANTUM REALITY

Shall I look at Her
Or shall I not?

Hard, small, separated
If I look;
Soft, spread-out, connected
If I don't.

Hard particle and soft wave: both?
Utterly random and perfectly predictable: both?
Small right-here and spread-out everywhere: both?
Deep connected yet lonely separate?

Honey
Some day You gotta show me
How You do that.

3 comments:

Wes Hansen said...

“Honey
Some day You gotta show me
How You do that.”

I’m with you on that . . !

“Alice's move reminds me of a string trick I learned as a kid in which you wrap a loop of string around your fingers in a complicated way so that it looks as though the fingers are entangled in the string. But upon pulling the string the fingers are freed--every loop of string was cancelled by an anti-loop. It's the same with the two photons--every seeming entanglement is cancelled somewhere by an anti-entanglement.”

It’s interesting that knot theory is finding potential application in quantum gravity considerations and that it (knot theory) has been demonstrated to map to G. Spencer Brown’s Law of Form. From http://www.lawsofform.org/lof.html:

Knots

Kauffman has shown a mapping between distinction laws and knot theory. In knot theory, string crossings draw a distinction.

Knot theory classifies equivalent knots with unravelling rules based on where these crossings occur. These rules are variations on calling, crossing, and distribution of the other forms.

Idempotency; Crossing; Distribution

Another coincidence, given your contribution to Esalen: http://www.lawsofform.org/aum/index.html.

The AUM conference link contains transcripts from Brown’s talk at Esalen in 1973.

nick herbert said...

Note that since Alice's state space is 4-dimensional, the 4 states she projects onto Bob's 2-D state space will necessarily be overcomplete (OC) and non-orthogonal (NO).

For those who believe that entangled OC and NO states offer a loophole for FTL signaling, this elementary teleportation scheme provides a simple test bed on which to explore this speculation.

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