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The fidelity between two quantum states and , expressed as density matrices, is commonly defined as: [1] [2] (,) = ().The square roots in this expression are well-defined because both and are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem.
In general, a quantum code for a quantum channel is a subspace , where is the state Hilbert space, such that there exists another quantum channel with () = =, where is the orthogonal projection onto .
Its logical operators are ¯ = and ¯ =. [3] Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors.
Rotations about the x, y, z axes of the Bloch sphere are represented by the rotation operator gates. Quantum logic gates are represented by unitary matrices . A gate that acts on n {\displaystyle n} qubits (a register ) is represented by a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} unitary matrix, and the set of all such gates with the group ...
The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x 2 + y 3 + z 5) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x 2 + y 3 + z 7) at the prime ideal (x, y, z) the local ring is a UFD but ...
The canonical desingularization of the ideal with these generators would blow up the center C 0 given by x=y=z=w=0. The transform of the ideal in the x-chart if generated by x-y 2 and y 2 (y 2 +z 2-w 3). The next center of blowing up C 1 is given by x=y=0. However, the strict transform of X is X 1, which is generated by x-y 2 and y 2 +z 2-w 3.
The mass of X(3872) is close to the mass of DD* and makes it a candidate as a meson molecule or a possible tetraquark. In 2005 the BaBar collaboration found Y(4260) from Initial state radiation as well in π+ π- J/ψ production. Again a charmonium-like particle with a large coupling to final states without open charm mesons.
is acted on by the order 5 symmetry , where is a fifth root of 1. There are two solutions invariant under this transformation, one with a pole of order 2 at 0, and the other with a zero of order 3 at 0.