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===Universal Resource=== | ===Universal Resource=== | ||
A set of <math>|+_\theta\rangle </math>states on which applying Clifford operations is enough for universal quantum computation. | A set of <math>|+_\theta\rangle </math>states on which applying Clifford operations is enough for universal quantum computation. | ||
===Superposition=== | |||
Superposition is a fundamental concept in quantum mechanics which states if two states <math>|\psi\rangle</math> and <math>|\phi\rangle</math> are representing two valid state in a Hilbert space, their linear combination exist in the same Hilbert space as well and refer to another valid states. We call state <math>\alpha|\psi\rangle + \beta|\phi\rangle</math> a superposition of the two states. This property leads to most of the non-classical properties of quantum mechanics such as entanglement. | |||
===Entangled States=== | |||
An entangled state is the quantum state of a group of particled (or a two party or multiparty system) that cannot be described as the independent states of these particles (or subsystems). The subsystems of such a quantum state, have quantum correlation even over a long distance. Mathematically, a multiparty entangled state cannot be written in following way:<br/> | |||
<math>|\psi_{1-N}\rangle = |\psi_1\rangle \otimes ... \otimes |\psi_N\rangle</math><br/> | |||
The states that can be written in the above from, are called separable states. | |||
===EPR Pairs=== | |||
EPR pairs refer to the pairs of particles with a conjugate physical property such as angular momentum. This concept has been introduced for the first time by the EPR (Einstein–Podolsky–Rosen) paradox which is a thought experiment challenging the explanation of physical reality provided by Quantum Mechanics. | |||
The particles that have been used in the EPR paradox had perfect correlation in such a way that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance. A two party quantum state with above property can be described with the following state:<br/> | |||
<math>|\Phi^+\rangle = \frac{1}{sqrt{2}} (|00\rangle + |11\rangle)</math><br/> | |||
This is one of the Bell states. | |||
===Bell states or Two-qubit Maximally Entangled States=== | |||
Bell states are maximally-entangled two-qubit states. These are the states that violate the Bell's inequality with maximal value of <math>2\sqrt{2}</math>. These states make a compelete basis for the two-qubit (4 dimensional) Hilbert space:<br/> | |||
<div style='text-align: center;'> | |||
<math>|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes |0\rangle_{B}+|1\rangle_{A}\otimes |1\rangle_{B}) </math></br> | |||
<math>|\Phi ^{-}\rangle =\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes |0\rangle_{B}-|1\rangle_{A}\otimes |1\rangle_{B}) </math></br> | |||
<math>|\Psi ^{+}\rangle =\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes |1\rangle_{B}+|1\rangle_{A}\otimes |0\rangle_{B}) </math></br> | |||
<math>|\Psi ^{-}\rangle =\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes |1\rangle_{B}-|1\rangle_{A}\otimes |0\rangle_{B}) </math><br/></div> | |||
===Fidelity=== | |||
The Fidelity is a quantum distance measure between two quantum states. For two general state $\rho$ and $\sigma$ it is defined as followes:<br/><br/> | |||
<math>F(\rho, \sigma) = [Tr(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})]^2</math><br/> | |||
The definition reduces to the squared overlap between the pure states <math>|\psi_{\rho}\rangle$ and $|\psi_{\sigma}\rangle</math>:<br/> | |||
<math>F(|\psi_{\rho}\rangle, |\psi_{\sigma}\rangle) = |\langle\psi_{\rho}|\psi_{\sigma}\rangle|^2</math> | |||
===Density Matrix, Pure and Mixed states=== | |||
A density matrix is the matrix representation of a statistical state in quantum mechanics. This is a useful representation for mixed states. The mixed states are the states which cannot be described with a single vector $|\psi\rangle$ in Hilbert space. Instead, they are statistical mixture of several pure states. These states are also useful for describing the quantum state of a subsystem of a multi-party or larger quantum sysytem where the overall state can be shown as pure states. a density matrix in general can be shown as:<br/><br/> | |||
<math>\rho = \sum_{i} p_i |\psi_i\rangle\langle\psi_i|</math><br/> | |||
where <math>|\psi_i\rangle</math> are pure states and <math>p_i</math> are the relative probabilities. For a pure quantum state <math>|\psi\rangle</math> the density matrix representation will be:<br/> | |||
<math>\rho = |\psi\rangle\langle\psi|</math> | |||
===Unitary transformation=== | |||
A unitary transformation is an isomorphism between two Hilbert spaces, These transformation preserve the inner products of the vector states and can be shown as matrices where <math>UU^{\dagger} = U^{\dagger}U = I</math>. | |||
===Monogomy of entanglement=== | |||
Monogamy is one of the most fundamental properties of entanglement and can, in its extremal form, be expressed as follows: If two qubits A and B are maximally quantumly correlated they cannot be correlated at all with a third qubit C. In general, there is a trade-off between the amount of entanglement between qubits A and B and the same qubit A and qubit C. This is mathematically expressed as:<br/> | |||
<math>E(A|B1) + E(A|B2) \leq E(A|B1B2)</math><br/> | |||
where <math>E()</math> is a measure for entanglement. | |||
===Ancilla or Ancillary states=== | |||
Ancillary states are extra states used in some quantum algorithms and are usually measured or discarded at the end of the procedure or they represent the state of an extra quantum system that is used for computation or etc. | |||
===Bloch Sphere=== | |||
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit). The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors <math>|0\rangle</math>, <math>|1\rangle</math> respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. | |||
===Gate Teleportation=== | ===Gate Teleportation=== | ||