dorsal/arxiv
View SchemaNonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
| Authors | Yuriy Makhlin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0002045 |
| URL | https://arxiv.org/abs/quant-ph/0002045 |
| DOI | 10.1023/A:1022144002391 |
| Journal | Quant. Info. Proc. 1, 243-252 (2002) |
Abstract
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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"abstract": "Entanglement of two parts of a quantum system is a non-local property\nunaffected by local manipulations of these parts. It is described by quantities\ninvariant under local unitary transformations. Here we present, for a system of\ntwo qubits, a set of invariants which provides a complete description of\nnon-local properties. The set contains 18 real polynomials of the entries of\nthe density matrix. We prove that one of two mixed states can be transformed\ninto the other by single-bit operations if and only if these states have equal\nvalues of all 18 invariants. Corresponding local operations can be found\nefficiently. Without any of these 18 invariants the set is incomplete.\n Similarly, non-local, entangling properties of two-qubit unitary gates are\ninvariant under single-bit operations. We present a complete set of 3 real\npolynomial invariants of unitary gates. Our results are useful for optimization\nof quantum computations since they provide an effective tool to verify if and\nhow a given two-qubit operation can be performed using exactly one elementary\ntwo-qubit gate, implemented by a basic physical manipulation (and arbitrarily\nmany single-bit gates).",
"arxiv_id": "quant-ph/0002045",
"authors": [
"Yuriy Makhlin"
],
"categories": [
"quant-ph"
],
"doi": "10.1023/A:1022144002391",
"journal_ref": "Quant. Info. Proc. 1, 243-252 (2002)",
"title": "Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations",
"url": "https://arxiv.org/abs/quant-ph/0002045"
},
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