We show that any unital qubit channel can be implemented by letting the input system interact unitarily with a four-dimensional environment in the maximally mixed state and then tracing out the ...environment. We also provide an example where the dimension of such an environment has to be at least 3.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
It is known that every two-qubit unitary operation has Schmidt rank one, two or four, and the construction of three-qubit unitary gates in terms of Schmidt rank remains an open problem. We explicitly ...construct the gates of Schmidt rank from one to seven. It turns out that the three-qubit Toffoli and Fredkin gate, respectively, have Schmidt rank two and four. As an application, we implement the gates using quantum circuits of CNOT gates and local Hadamard and flip gates. In particular, the collective use of three CNOT gates can generate a three-qubit unitary gate of Schmidt rank seven in terms of the known Strassen tensor from multiplicative complexity.
Unitary operations are physically implementable. We further the understanding of such operators by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite ...unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a dA×dB bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank min{dA2,dB} when dA≤dB. We further show that any multipartite unitary operator U of Schmidt rank three can be controlled by one system or collectively controlled by two systems, regardless of the number of systems of U. In the scenario of n-qubit, we construct non-controlled U for any odd n≥5, and prove that U is a controlled unitary for any even n≥4.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
The Schmidt number of a state of an infinite-dimensional composite quantum system is defined and several properties of the corresponding Schmidt classes are considered. It is shown that there are ...states with given Schmidt number such that any of their countable convex decompositions does not contain pure states of finite Schmidt rank. The classes of infinite-dimensional partially entanglement-breaking channels are considered, and generalizations of several properties of such channels, which were obtained earlier in the finite-dimensional case, are proved. At the same time, it is shown that there are partially entanglement-breaking channels (in particular, entanglement-breaking channels) such that all the operators in any of their Kraus representations are of infinite rank.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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