Bell state
[1]: 25 The Bell's states are a form of entangled and normalized basis vectors.
[2] Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability.
[3] These mechanisms cannot transmit information faster than the speed of light, a result known as the no-communication theorem.
Their entanglement means the following: The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1.
Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated.
This perfect correlation at a distance is special: maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.
Hence, following Albert Einstein, Boris Podolsky, and Nathan Rosen in their famous 1935 "EPR paper", there is something missing in the description of the qubit pair given above – namely this "agreement", called more formally a hidden variable.
In his famous paper of 1964, John S. Bell showed by simple probability theory arguments that these correlations (the one for the 0, 1 basis and the one for the +, − basis) cannot both be made perfect by the use of any "pre-agreement" stored in some hidden variables – but that quantum mechanics predicts perfect correlations.
In a more refined formulation known as the Bell–CHSH inequality, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of local "hidden-variable" theory (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as
Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables".
They are known as the four maximally entangled two-qubit Bell states and form a maximally entangled basis, known as the Bell basis, of the four-dimensional Hilbert space for two qubits:[1] Although there are many possible ways to create entangled Bell states through quantum circuits, the simplest takes a computational basis as the input, and contains a Hadamard gate and a CNOT gate (see picture).
As an example, the pictured quantum circuit takes the two qubit input
In addition, the Bell states form an orthonormal basis and can therefore be defined with an appropriate measurement.
The CNOT gate performs the act of un-entangling the two previously entangled qubits.
The result of a Bell state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.
Such devices can be constructed from, for example: mirrors, beam splitters, and wave plates – and are attractive from an experimental perspective because they are easy to use and have a high measurement cross-section.
For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques.
This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as teleportation.
If a Bell state is measured from this ambiguous class, the teleportation event fails.
Entangling particles in multiple qubit variables, such as (for photonic systems) polarization and a two-element subset of orbital angular momentum states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other.
It also has advantages for other protocols such as superdense coding, in which hyper-entanglement increases the channel capacity.
Bell state using the same basis, the qubits would appear positively correlated when measuring in the
correlations can be understood by measuring both qubits in the same basis and observing perfectly anti-correlated results.
Superdense coding allows two individuals to communicate two bits of classical information by only sending a single qubit.
gate to her qubit; and finally, if Alice wanted to send the two bit message
The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob.
This process has become a fundamental research topic for quantum communication and computing.
More recently, scientists have been testing its applications in information transfer through optical fibers.
[7] The process of quantum teleportation is defined as the following: Alice and Bob share an EPR pair and each took one qubit before they became separated.
It enables two parties to produce a shared random secret key that can be used to encrypt messages.
