User:Krisch53/sandboxQuantumTeleportation

This is the sandbox for Aryeh Krischer and Breannah Lemieux on Quantum Teleportation for Dr. Kaplan's class.

Quantum teleportation is the article we are working on. What follows is a general list of areas we want to update and additions we want to include.

- General writing, grammar, removing excess words to promote clarity, etc...

- The non-technical introduction is too technical at a number of points. We would like to provide additional clarity.

- The article sticks strictly to discussion of measurement in the Bell basis. It would be quite helpful to mention explicitly (and even show) that the measurements can be done in the computational basis as well.

- Place the results and records section after the formal presentation section?

- Results and records could benefit from a great deal of fleshing out and defining terms.

- Formal presentation could use a fair amount of clarifying.

- New (useful) diagrams needed in Entanglement Swapping

- d-dimension systems has a poorly handled reference.

Original
In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, the two-state system. In classical information, this is a bit, commonly represented using one or zero (or true or false). The quantum analog of a bit is a quantum bit, or qubit. Qubits encode a type of information, called quantum information, which differs sharply from "classical" information. For example, quantum information can be neither copied (the no-cloning theorem) nor destroyed (the no-deleting theorem).

Edits
In matters relating to quantum or classical information theory it is convenient to work with two-state systems, the simplest possible units of information. For classical information these two-state systems are called bits. A simple bit is a switch with the possible states on and off. 0 and 1, or true and false, are other common names for the states of a bit. The quantum analog of a bit is a quantum bit, also called a qubit. Qubits encode quantum information, which differs sharply from "classical" information. For example, unlike classical information, quantum information can be neither copied (the no-cloning theorem) nor destroyed (the no-deleting theorem).

Original
The movement of qubits does not require the movement of "things" any more than communication over the internet does: no quantum object needs to be transported, but it is necessary to communicate two classical bits per teleported qubit from the sender to the receiver. The actual teleportation protocol requires that an entangled quantum state or Bell state be created, and its two parts shared between two locations (the source and destination, or Alice and Bob). In essence, a certain kind of quantum channel between two sites must be established first, before a qubit can be moved. Teleportation also requires a classical information channel to be established, as two classical bits must be transmitted to accompany each qubit. The reason for this is that the results of the measurements must be communicated between the source and destination so as to reconstruct the qubit, or else the state of the destination qubit would not be known to the source, and any attempt to reconstruct the state would be random; this must be done over ordinary classical communication channels. The need for such classical channels may, at first, seem disappointing, and this explains why teleportation is limited to the speed of transfer of information, i.e., the speed of light. The main advantages is that Bell states can be shared using photons from lasers, and so teleportation is achievable through open space, i.e., without the need to send information through cables or optical fibers.

Edits
Counterintuitively, the movement of qubits, and thus quantum information, does not require the transport of a quantum object. Instead, two classical bits are communicated from the sender to the receiver to accomplish the teleportation. Although only classical information is exchanged in the teleportation, the protocol requires that source and destination (often known as Alice and Bob) each posses one particle from a pair that is in an entangled quantum state or Bell state. How the Bell pair came to be shared is irrelevant and could even be prepared and distributed by a third party. In other words, teleportation can even be effected between parties who have never previously communicated. The actual information exchanged between the sender and receiver and used to reconstruct the quantum state is a pair of classical bits sent over ordinary classical communication channels. The need for such classical channels guarantees that teleportation is limited to the speed of transfer of information, i.e., the speed of light. One advantage of the protocol is that Bell states can be shared using photons from lasers, and so teleportation is achievable through open space, i.e., without the need to send information through cables or optical fibers.

Original
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments. Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn't yet had time to travel the distance; a conclusion seemingly at odds with special relativity (EPR paradox). However such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem. Thus, teleportation, as a whole, can never be superluminal, as a qubit cannot be reconstructed until the accompanying classical information arrives.

Edits
''Just get rid of this whole paragraph. It is highly technical and does not add anything that has not already been said.''

Original
Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces and projection matrixes. A qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below. A working knowledge of quantum mechanics is not absolutely required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious.

Edits
Complete understanding of quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces, and projection matrixes. In particular, a qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which is the primary basis for the formal manipulations given below. While the mathematics of quantum teleportation may be understood without a working knowledge of quantum mechanics, in lieu of such acquaintance the deeper meaning of the equations may remain quite mysterious.

Original
The prerequisites for quantum teleportation are a qubit that is to be teleported, a conventional communication channel capable of transmitting two classical bits (i.e., one of four states), and means of generating an entangled EPR pair of qubits, transporting each of these to two different locations, A and B, performing a Bell measurement on one of the EPR pair qubits, and manipulating the quantum state of the other pair. The protocol is then as follows:


 * 1) An EPR pair is generated, one qubit sent to location A, the other to B.
 * 2) At location A, a Bell measurement of the EPR pair qubit and the qubit to be teleported (the quantum state $$|\phi \rangle$$) is performed, yielding one of four measurement outcomes, which can be encoded in two classical bits of information. Both qubits at location A are then discarded.
 * 3) Using the classical channel, the two bits are sent from A to B. (This is the only potentially time-consuming step after step 1, due to speed-of-light considerations.)
 * 4) As a result of the measurement performed at location A, the EPR pair qubit at location B is in one of four possible states. Of these four possible states, one is identical to the original quantum state $$|\phi \rangle$$, and the other three are closely related. Which of these four possibilities actually obtained, is encoded in the two classical bits. Knowing this, the EPR pair qubit at location B is modified in one of three ways, or not at all, to result in a qubit identical to $$|\phi \rangle$$, the qubit that was chosen for teleportation.

It is worth to notice that the above protocol assumes that the qubits are individually addressable, that means the qubits are distinguishable and physically labeled. However, there can be situations where two identical qubits are indistinguishable due to the spatial overlap of their wave functions. Under this condition, the qubits cannot be individually controlled or measured. Nevertheless, a teleportation protocol analogous to that described above can still be (conditionally) implemented by exploiting two independently prepared qubits, with no need of an initial EPR pair. This can be made by addressing the internal degrees of freedom of the qubits (e.g., spins or polarizations) by spatially localized measurements performed in separated regions A and B shared by the wave functions of the two indistinguishable qubits.

Edits
The resources required for quantum teleportation are a communication channel capable of transmitting two classical bits, a means of generating an entangled EPR pair of qubits and distributing to two different locations, performing a Bell measurement on one of the EPR pair qubits, and manipulating the quantum state of the other qubit from the pair. Of course, there must also be some input qubit (in the quantum state $$|\phi \rangle$$) to be teleported. The protocol is then as follows:


 * 1) An EPR pair is generated with one qubit sent to location A and the other sent to location B.
 * 2) At location A, a Bell measurement of the EPR pair qubit with the qubit to be teleported ($$|\phi \rangle$$) is performed, yielding one of four measurement outcomes, which can be encoded in two classical bits of information. Both qubits at location A are then discarded.
 * 3) Using the classical channel, the two bits are sent from A to B. (This is the only potentially time-consuming step after step 1, due to speed-of-light considerations.)
 * 4) As a result of the measurement performed at location A, the EPR pair qubit at location B is in one of four possible states. Of these four possible states, one is identical to the original quantum state $$|\phi \rangle$$, and the other three are closely related. Which of these four possibilities actually obtained is encoded in the two classical bits. The EPR pair qubit at location B is then modified in one of three ways, or not at all, to result in a qubit identical to $$|\phi \rangle$$, the qubit that was chosen for teleportation.

It is worth noticing that the above protocol assumes that the qubits are individually addressable, meaning that the qubits are distinguishable and physically labeled. However, there can be situations where two identical qubits are indistinguishable due to the spatial overlap of their wave functions. Under this condition, the qubits cannot be individually controlled or measured. Nevertheless, a teleportation protocol analogous to that described above can still be (conditionally) implemented by exploiting two independently prepared qubits, with no need of an initial EPR pair. This can be made by addressing the internal degrees of freedom of the qubits (e.g., spins or polarizations) by spatially localized measurements performed in separated regions A and B shared by the wave functions of the two indistinguishable qubits.

Original
One applies these identities with A and C subscripts. The total three particle state, of A, B and C together, thus becomes the following four-term superposition:



\begin{align} \frac{1}{2} \Big \lbrack \ & |\Phi^+\rangle_{CA} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B) \ + \ |\Phi^-\rangle_{CA} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B) \\ \ + \ & |\Psi^+\rangle_{CA} \otimes (\alpha |1\rangle_B + \beta|0\rangle_B) \ + \ |\Psi^-\rangle_{CA} \otimes (\alpha |1\rangle_B - \beta|0\rangle_B) \Big \rbrack. \\ \end{align} $$
 * & \psi\rangle_C \otimes\ |\Phi^+\rangle_{AB} \  = \\

The above is just a change of basis on Alice's part of the system. No operation has been performed and the three particles are still in the same total state. The actual teleportation occurs when Alice measures her two qubits A,C, in the Bell basis


 * $$|\Phi^+\rangle_{CA}, |\Phi^-\rangle_{CA}, |\Psi^+\rangle_{CA}, |\Psi^-\rangle_{CA}$$.

Edits
After expanding the expression for $ \begin{align} \end{align} $, one applies these identities to the quibits with A and C subscripts. In particular, $$ \alpha \frac{1}{\sqrt{2}} |0\rangle_C \otimes |0\rangle_A \otimes |0\rangle_B = \alpha \frac{1}{2}(|\Phi^+\rangle_{CA} + |\Phi^-\rangle_{CA}) \otimes |0\rangle_B $$ and the other terms follow similarly. Combining similar terms, the total three particle state of A, B and C together becomes the following four-term superposition:
 * & \psi\rangle_C \otimes\ |\Phi^+\rangle_{AB}



\begin{align} \frac{1}{2} \Big \lbrack \ & |\Phi^+\rangle_{CA} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B) \ + \ |\Phi^-\rangle_{CA} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B) \\ \ + \ & |\Psi^+\rangle_{CA} \otimes (\alpha |1\rangle_B + \beta|0\rangle_B) \ + \ |\Psi^-\rangle_{CA} \otimes (\alpha |1\rangle_B - \beta|0\rangle_B) \Big \rbrack. \\ \end{align} $$
 * & \psi\rangle_C \otimes\ |\Phi^+\rangle_{AB} \  = \\

Note that all three particles are still in the same total state since no operations have been performed. Rather, the above is just a change of basis on Alice's part of the system. The actual teleportation occurs when Alice measures her two qubits A,C, in the Bell basis


 * $$|\Phi^+\rangle_{CA}, |\Phi^-\rangle_{CA}, |\Psi^+\rangle_{CA}, |\Psi^-\rangle_{CA}$$.

Equivalently, the measurement may be done in the computational basis, $$ \{|0\rangle,|1\rangle\} $$, by mapping each Bell state uniquely to one of $$ \{|0\rangle \otimes|0\rangle, |0\rangle \otimes|1\rangle, |1\rangle \otimes|0\rangle, |1\rangle \otimes|1\rangle\} $$ with the quantum circuit shown below: [Add circuit]

Original
Alice's state in qubit 2 is transferred to Bob's qubit 0 using a priorly entangled pair of qubits between Alice and Bob, qubits 1 and 0.

Edits
Remove this, it provides no useful addition in context and the implied quibit labeling is inconsistent with the rest of the section.