effects If the measurement registers the i-th outcome, the overall state collapses to The tensor factor in is while that of is . Bob then applies a corresponding local operation Ψi on system 3. On the combined system, this is described by where Id is the identity map on the composite system . Therefore, the channel Φ is defined by Notice Φ satisfies the definition of LOCC. As stated above, the teleportation is said to be successful if, for all observable O on Bob's system, the equality holds. The left hand side of the equation is: where Ψi* is the adjoint of Ψi in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes The success criterion for teleportation has the expression Local explanation of the phenomenon A local explanation of quantum teleportation is put forward by David Deutsch and Patrick Hayden, with respect to the many-worlds interpretation of quantum mechanics. Their paper asserts that the two bits that Alice sends Bob contain "locally inaccessible information" resulting in the teleportation of the quantum state. "The ability of quantum information to flow through a classical channel […], surviving decoherence, is […] the basis of quantum teleportation." Recent developments While quantum teleportation is in an infancy stage, there are many aspects pertaining to teleportation that scientists are working to better understand or improve the process that include: Higher dimensions Quantum teleportation can improve the errors associated with fault tolerant quantum computation via an arrangement of logic gates. Experiments by D. Gottesman and I. L. Chuang have determined that a "Clifford hierarchy" gate arrangement which acts to enhance protection against environmental errors. Overall, a higher threshold of error is allowed with the Clifford hierarchy as the sequence of gates requires less resources that are needed for computation. While the more gates that are used in a quantum computer create more noise, the gates arrangement and use of teleportation in logic transfer can reduce this noise as it calls for less "traffic" that is compiled in these quantum networks. The more qubits used for a quantum computer, the more levels are added to a gate arrangement, with the diagonalization of gate arrangement varying in degree. Higher dimension analysis involves the higher level gate arrangement of the Clifford hierarchy. Information quality Considering the previously mentioned requirement of an intermediate entangled state for quantum teleportation, there needs to be consideration placed on to the purity