Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations

Details Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations

2000-02-15

arxiv.org/abs/quant-ph/0002045v2

Quant. Info. Proc. 1, 243-252 (2002)

Physics

Quantum Physics

4 pages; minor changes; relation of our invariants and those of quant-ph/9712040 clarified

Scientific paper

10.1023/A:1022144002391

Entanglement of two parts of a quantum system is a non-local property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of non-local properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-bit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete. Similarly, non-local, entangling properties of two-qubit unitary gates are invariant under single-bit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-bit gates).

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