What is "teleportation"?

We probably all have watched Star Trek episodes.  Often crew members enter a transporter and are beamed from the starship Enterprise to the surface of a planet, and later back to the starship.  The crew members de-materialize in one location and then re-materialize in another location.  The details of the process are not explained. 

The theory of special relativity does not allow instantaneous de/re-materialization.  Something (matter, information) must flow from location 1 to location 2 with a speed no greater than the speed of light.  But does the transporter physically disassemble a crew member while making measurements on each particle, move the particles from location 1 to location 2, and then reassemble them using the results of the measurements?  Or does it scan all the information about the crew member's physical state and transmit that information to the new location, where it is used to construct a new crew member out of raw materials found at that location.  In either case, the transporter needs to have complete information about the physical state of the crew member in order to reconstruct him/her.  Quantum Mechanics, however, tells us that not all the measurements needed to obtain complete information are compatible measurements.  This is often expressed in terms of the uncertainty principle, which in this case implies that it is impossible to obtain complete information about a crew member.  Can the transporter therefore do no better than make an approximate copy?   We will see that quantum mechanics provides us with two ways of making exact copies without ever having complete information about the original.  But it also requires that in these processes the original is destroyed.  This is called the no cloning theorem.

Let us investigate two procedure that will allow the teleportation of the quantum state of a spin ½ particle from “Dark Sciences” laboratory to  “Globalux” laboratory.  The laboratories are located far away from each other.

Motivation

Dark Sciences” is a laboratory doing fundamental physics research.  It is the leading research laboratory of the “Axis Empire”, and scientists working at "Dark Sciences" are not sharing their results with scientists from the “Allied Nations”.  Bob works at “Globalux”, the leading research laboratory of the “Allied Nations”.  His partner Alice has been able to infiltrate “Dark Sciences” and is trying to send information about one of the latest “Dark Sciences” experiments to Bob.  Something unexpected is happening in this experiment.  Very slow muons are created.  The probability of creation is very low, but they are definitely being produced.  Muons are spin ½ particles which resemble electrons, but they are approximately 200 times more massive than electrons.  And they are unstable with a mean lifetime of about 2.2 microseconds.  The scientists at “Dark Sciences” are using these muons in a follow up experiment, which Bob would like to duplicate at “Globalux”.  Alice suspects that the spin state of the slow muons is the key to the success of the follow up experiment.  Bob needs muons in exactly the same spin state.

Bob can produce muons at “Globalux” and put them into an arbitrary spin state.  So if Alice could measure the spin state of a muon produced at "Dark Sciences", then she could send this information to Bob in some encrypted form, and, upon receiving this information, Bob could recreate the spin state in his laboratory.

The spin state of any spin 1/2 particle can be expressed as |ψ> = a|+> + b|->.  Here |+> and |-> denote spin up or down with respect to some chosen axis, and |a|² and |b|² are the probabilities that a measurement will yield the respective result. 

Alice can choose any axis and measure the spin of the particle with respect to that axis.  If Alice finds the particle in the spin up state, then she knows the spin state of the particle after the measurement, but she does not know the original spin state.  Furthermore, she cannot make any more measurements on the original spin state, because it has been destroyed.  After her first measurement the particle is in the |+> spin state.  So measuring the spin state and sending the information to Bob is impossible.  This is one of the consequences of the uncertainty principle.  If the muon were stable, Alice could try to put it in a trap, smuggle it out of the “Dark Sciences” laboratory, and ship it to Bob.  But the short mean lifetime of the muon makes this impossible.

Procedure 1

Even though Alice cannot measure the spin state of the muon, she can manipulate it.  The basic operations she can do if she has the appropriate equipment are described below.  Performing some operations in sequence, Alice can transfer the unknown quantum state of the muon to an electron.  The electron is a stable particle, and it is possible to ship it to Bob.  Using the same type of equipment Bob can then perform the same sequence of operations to transfer the electron’s spin state to one of his muons, and use this muon to duplicate the “Dark Sciences” experiment.  We label this procedure "classical" teleportation".  It does not require entangled states.

Procedure 2

Shipping the electron to “Globalux “ is risky.  Alice is under cover and the probability is high that her cover will be blown if she ships something from “Dark Sciences” to “Globalux”.  Shipping also takes a very long time.  But there is another way for Bob to get Alice's muon.  It is possible to recreate an unknown quantum state in a different physical location through the process of quantum teleportation.  It allows a quantum state to be recreated by exchanging only 2 bits of classical information.  Quantum teleportation was proposed by Charles Bennett et al in 1993, and in 1997 Anton Zeilinger et al, at University of Innsbruck in Austria, performed the first successful teleportation experiment, teleporting the polarization state of a photon.  In 2004 the quantum state of an atom was teleported for the first time.  To teleport a quantum state, a pair of particles in an entangled state is needed.  But for this process to work, Alice and Bob have to have made preparations and Alice needs an additional experimental apparatus, called a Bell-state analyzer.

To later use quantum teleportation, Alice and Bob must have created pairs of electrons before Alice left to go undercover.  The process that they used to create the pairs conserves angular momentum, and in that process angular momentum conservation requires that the spin angular momentum of the two electrons adds up to zero.  The square of the magnitude of the total spin is zero and the component of the total spin measured along any axis is zero.  The two electrons are entangled.  Entanglement means as far as spin is concerned neither particle has properties of its own, they only have common properties.  Alice and Bob put these entangled electrons into individual traps and Alice took one of the electrons from each pair with her to "Dark Sciences", while Bob kept the other one.  To teleport the first muon from "Dark Sciences" laboratory to "Globalux" laboratory, Alice now chooses her entangled electron from the first pair and puts it in her Bell-state analyzer, together with the muon.  The Bell-state analyzer makes a measurement on the muon-electron system.  The muon-electron system is a quantum-mechanical system and the analyzer can obtain one of four different results, which we label result 0 through result 3.  Each result is obtained with equal probability, independent of the exact quantum state of the muon.  Alice's measurement destroys her muon spin state, but it put Bob's entangled electron into a definite quantum state that is related to the original muon spin state.  Alice notes the result of the Bell-state analyzer and transmits the result number (00 or 01 or 10 or 11 in binary form) to Bob at approximately the speed of light, using a radio transmission, a satellite phone, etc.  Given this result Bob knows exactly what to do to put his electron into the original muon spin state, and then transfer this state to one of his muons.  The muon has been teleported and Bob can perform the experiment.

"Classical" teleportation

Let us first concentrate on classical teleportation.  What equipment is needed and how does it work?

The equipment needed by Alice and Bob are a set of quantum gates.  Quantum gates allow us to manipulate quantum states without measuring them.  The unknown spin states of electrons and muons can be manipulated in well defined ways using quantum gates.  The quantum gates needed by Alice and Bob are a phase shifter, a bit flipper, and some CNOT gates. 

What do these gate do?

The spin state of a spin 1/2 particle or the polarization state of a photon are representations of a quantum bit or qbit (or qubit).  A qbit is any quantum system with exactly two degrees of freedom.  For the spin state we denote these degrees of freedom by |+> and |->.  For a photon we can denote them by |V> and |H>.

The phase shifter and bit flipper perform the following operations on a single qbit:

A controlled NOT or CNOT gate operates on two qbits.

This gate uses one of the qbits a a control qbit and changes the state of the other qbit depending on the state of control qbit.

To examine the results of the various quantum gates, open the linked spreadsheet.  You can examine the state of two spin 1/2 particles before and after quantum gates were applied.  The graphs show the orientation of the "up" direction of a spin filter that would pass a spin 1/2 particle in this state with probability 1.

What is the physics behind these gates?
The spin of a muon or electron is associated with a magnetic moment.  The particles therefore interact with external magnetic fields.  If two particles are brought close enough together, they interact with each other.  One can use these magnetic interactions to change the spin state of one or more particles in well defined ways without knowing the initial spin state.  This can be done in a time interval much shorter than the mean lifetime of a muon.  Polarization states of a photon can also be manipulated without measuring them.  Here anisotropic and non-linear transparent materials are used.

What does Alice have to do?

Assume Alice and Bob did not prepare entangled electrons or Alice does not have access to her entangled electrons or the Bell state analyzer, but she can use her quantum gates.  She has to transfer the muon spin state to an electron and ship the electron to Bob.  She prepares an electron in the spin up (|+>_e) state and puts it into a trap together with the muon in the unknown quantum state a|+>_μ + b|->_μ.  The quantum state of the two particles in the trap is

(a|+>_μ + b|->_μ)|+>_e  = a|+>_μ|+>_e + b|->_μ|+>_e.

Using quantum gates, i. e. phase shift, bit flip, or CNOT operations (first or second qbit as control bit), Alice develops a procedure that allows her to transfer the quantum state of the muon to the electron.

Her goal:  (a|+>_μ + b|->_μ)|+>_e  -->   |+>_μ(a|+>_e + b|->_e)

Task 1:

Using CNOT operations (first or second qbit as control bit) develop a procedure that allows Alice to transfer the quantum state of the muon to the electron.
Start with a CNOT₁ operation.  Add the other operations needed and the appropriate algebraic expressions to the table below.

This sequence of operations leaves the electron in the state a|+>_e + b|->_e.  The spin state of the muon has been transferred to the electron.  Alice can now ship the electron to Bob.  (Note: the state has not been cloned, the original muon state has been destroyed.  It has been proved by Wootters and Zurek that it is impossible to clone, or duplicate, an unknown quantum state.)

Question 1:

If Bob receives the electron, which procedure can he use to transfer the quantum state back to a muon?

Quantum Teleportation

Alice has a muon, which she wants to teleport to Bob.  Alice and Bob have made preparations to accomplish this quantum teleportation.  Before leaving to go undercover, Alice made sure that a Bell-state analyzer was part of the equipment available to her at "Dark Sciences" laboratory.  Then Alice and Bob created pairs of electrons.  The process that they used to create the pairs conserves angular momentum, and in that process angular momentum conservation requires that the spin angular momentum of the two electrons adds up to zero.  The two electrons are entangled.  Alice and Bob put these entangled electrons into individual traps and Alice took one of the electrons from each pair with her to "Dark Sciences", while Bob kept the other one.  To teleport the first muon from "Dark Sciences" laboratory to "Globalux" laboratory, Alice now chooses her entangled electron from the first pair and puts it in her Bell-state analyzer, together with the muon. 

The Bell-state analyzer makes a measurement whose outcome depends on the spin state of the two-particle system.  This measurement can have 4 different outcomes (result 0 – result 3) and after the measurement the spin state of the two-particle system is one of four different states, commonly known as Bell states.

For a muon and an electron the four different Bell states are listed below.

Alice’s electron has no property of its own, as far as spin is concerned.  So even so the other electron of the entangled pair is kept at a far away location, it must be included in the description of the spin state before the measurement.  The notation for this spin-0 entangled state of the two electrons is  

N(|+>_eA|->_eB  -  |->_eA|+>_eB),

with the normalization constant N = 2^–½.  The subscripts A and B refer to the owners of the electron, Alice and Bob.

The muon is in the unknown state (a|+>_μ + b|->_μ), so the state of the electron-muon system in the Bell-state analyzer before the measurement is

(a|+>_μ + b|->_μ) N(|+>_eA|->_eB  -  |->_eA|+>_eB)
= N(a|+>_μ|+>_eA|->_eB  -  a|+>_μ|->_eA|+>_eB 
+  b|->_μ|+>_eA|-> _eB - b|->_μ|->_eA|+>_eB).

The rightmost electron belongs to Bob.

We can rewrite this expression in terms of the Bell states.
(Click here if you are interested in the algebraic details.)

N (a|+>_μ|+>_eA|->_eB  -  a|+>_μ|->_eA|+>_eB 
+  b|->_μ|+>_eA|->_eB  -  b|->_μ|->_eA|+>_eB)
=
- ½|ψ+>(a|+>_eB - b|->_eB)
- ½|ψ->(a|+>_eB + b|->_eB)
+ ½|Φ+>(a|->_eB - b|+>_eB)
+ ½|Φ->(a|->_eB + b|+>_eB)

When Alice makes a measurement with the Bell state analyzer, every outcome (|ψ+>, |ψ->, |Φ+>, |Φ->) is equally likely.  Each outcome has probability (½)² = ¼.  Once Alice has made a measurement, Bob’s electron has been put into a particular state.  Alice notes the result of the Bell-state analyzer and transmits the result number (00 or 01 or 10 or 11 in binary form) to Bob at approximately the speed of light, using a radio transmission, a satellite phone, etc.  When Bob receives the result of Alice's measurement over a classical channel, he knows which of the states listed below is the spin state of his electron, even though he does not know the coefficients a and b.

Given this result Bob knows exactly what to do to put his electron into the original muon spin state, and then transfer this state to one of his muons.  Alice's muon has been teleported.

Task 2:

Develop a procedure for Bob to produce a muon in the original spin state a|+> + b|-> after he receives the result of Alice's measurement.  Bob can use phase shift, bit flip, and CNOT operations.
List the appropriate operations in the table below.

Question 2:

At the instant Alice obtains the result of her Bell state measurement, the quantum state of Bob's electron is determined.  Does this mean we can teleport quantum states faster than the speed of light?  What is your take on that?

Question 3:

According to the no-cloning theorem, an arbitrary quantum state cannot be copied.  But does Alice not copy her qbit onto Bob's qbit?  Explain?

Summary:

The result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts.  This is known as "non-local behavior".

Quantum teleportation using the above outlined scheme was accomplished for the first time in 1997.  The quantum state of a photon was teleported.