User:Baileyruesch/sandbox

In quantum information theory, superdense coding is a quantum communication protocol to transmit two classical bits of information (i.e., either 00, 01, 10 or 11) from a sender (often called Alice) to a receiver (often called Bob), by sending only one qubit from Alice to Bob, under the assumption of Alice and Bob pre-sharing an entangled state. If Alice and Bob do not already share entanglement before the protocol begins, then it is impossible to send two classical bits using 1 qubit, as this would violate Holevo's theorem.

It can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair.

Contents

 * 1Overview
 * 2The protocol
 * 2.1Preparation
 * 2.2Sharing
 * 2.3Encoding
 * 2.4Sending
 * 2.5Decoding
 * 2.5.1Example
 * 3Eavesdropper
 * 4General dense coding scheme
 * 5References
 * 6External links

Overview[edit]
Suppose Alice wants to send two classical bits of information (00, 01, 10, or 11) to Bob using qubits (instead of classical bits). To do this, an entangled state (e.g. a Bell state) is prepared using a Bell circuit or gate by Charlie, a third person. Charlie then sends one of these qubits (in the Bell state) to Alice and the other to Bob. Once Alice obtains her qubit in the entangled state, she applies a certain quantum gate to her qubit depending on which two-bit message (00, 01, 10 or 11) she wants to send to Bob. Her entangled qubit is then sent to Bob who, after applying the appropriate quantum gate and making a measurement, can retrieve the classical two-bit message. Alice must tell Bob which gate to apply after he receives the entangled qubit in order to obtain the correct classical bits.

The protocol[edit]
The protocol can be split into different steps (as described below).

Preparation[edit]
The protocol starts with the preparation of an entangled state, which is then shared between Alice and Bob. Suppose the following Bell state

where  denotes the tensor product, is prepared. Note: we can omit the tensor product symbol and write the Bell state as



Sharing[edit]
After the preparation of the Bell state, the qubit denoted by subscript A is sent to Alice and the qubit denoted by subscript B is sent to Bob (note: this is the reason these states have subscripts). At this point, Alice and Bob may be in completely different locations, which can be very distant from each other.

There may be a long period of time between the preparation and sharing of the entangled state  and the rest of the steps in the procedure.

Encoding[edit]
By applying a quantum gate to her qubit locally, Alice can transform the entangled state into any one of the 4 Bell states (including, of course, ). Note that this process cannot "break" the entanglement between the two qubits.

Let's now describe which operations Alice needs to perform on her entangled qubit, depending on which classical two-bit message she wants to send to Bob. We'll later see why these specific operations are performed. There are four cases, which correspond to the four possible two-bit strings that Alice may want to send.

1. If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate,, to her entangled qubit, so that her it remains unchanged. The resultant entangled state is then



In other words, the entangled state shared between Alice and Bob has not changed, i.e. it is still. The notation  is also used to remind us of the fact that Alice wants to send the two-bit string 00.

2. If Alice wants to send the classical two-bit string 01 to Bob, then she applies the quantum NOT (or bit-flip) gate,, to her entangled qubit, so that the resultant entangled quantum state becomes



3. If Alice wants to send the classical two-bit string 10 to Bob, then she applies the quantum phase-flip gate  to her entangled qubit, so the resultant entangled state becomes



4. If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate  to her entangled qubit, so that the resultant entangled state is



The matrices  and  are two of the Pauli matrices. The quantum states, ,  and  (or, respectively,  and ) are the Bell states.

Sending[edit]
After having performed one of the operations described above, Alice can send her entangled qubit to Bob using a quantum network through some conventional physical medium.

Decoding[edit]
In order for Bob to find out which classical bits Alice sent, he will perform the CNOT unitary operation with A as the control qubit and B as the target qubit. Then, he will perform  unitary operation on the entangled qubit A. In other words, the Hadamard quantum gate H is only applied to A (see the figure above).


 * If the resultant entangled state was  then after the application of the above unitary operations the entangled state will become
 * If the resultant entangled state was  then after the application of the above unitary operations the entangled state will become
 * If the resultant entangled state was  then after the application of the above unitary operations the entangled state will become
 * If the resultant entangled state was  then after the application of the above unitary operations the entangled state will become

These operations performed by Bob can be seen as a measurement which projects the entangled state onto one of the four two-qubit basis vectors  or  (as you can see from the outcomes and the example below).

Example[edit]
For example, if the resultant entangled state is, then a CNOT with A as control bit and B as target bit will change  to become. Now, the Hadamard gate is applied only to A, to obtain

For simplicity, let's get rid of the subscripts, so we have

Now, Bob has the basis state, so he knows that Alice wanted to send the (classical) two-bit string 01.

Eavesdropper[edit]
If an eavesdropper, which we can call Eve, intercepts Alice's qubit en route to Bob, all that is obtained by Eve is part of an entangled state. Therefore, no useful information whatsoever is gained by Eve, unless she also has access to Bob's qubit.

General dense coding scheme[edit]
General dense coding schemes can be formulated in the language used to describe quantum channels. Alice and Bob share a maximally entangled state ω. Let the subsystems initially possessed by Alice and Bob be labeled 1 and 2, respectively. To transmit the message x, Alice applies an appropriate channel



on subsystem 1. On the combined system, this is effected by



where I denotes the identity map on subsystem 2. Alice then sends her subsystem to Bob, who performs a measurement on the combined system to recover the message. Let the effects of Bob's measurement be Fy. The probability that Bob's measuring apparatus registers the message y is



Therefore, to achieve the desired transmission, we require that



where δxy is the Kronecker delta.

*** Plan to add to this article with more recent information and expand upon existing statements and examples using the following sources:

https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.118.050501

https://www.cs.mcgill.ca/~yli252/files/quantum.pdf

https://www.cs.ubc.ca/~condon/cpsc506/lectures/lec16.pdf