dorsal/arxiv
View SchemaCanonical Decompositions of n-qubit Quantum Computations and Concurrence
| Authors | Stephen S. Bullock, Gavin K. Brennen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0309104 |
| URL | https://arxiv.org/abs/quant-ph/0309104 |
| DOI | 10.1063/1.1723701 |
| Journal | Journal of Mathematical Physics, vol 45, issue 6, pp. 2447-2467 (2004) |
Abstract
The two-qubit canonical decomposition SU(4) = [SU(2) \otimes SU(2)] Delta [SU(2) \otimes SU(2)] writes any two-qubit quantum computation as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an n-qubit decomposition, the concurrence canonical decomposition (C.C.D.) SU(2^n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any computation in K preserves the tangle |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 for n even. Thus, the C.C.D. shows that any n-qubit quantum computation is a composition of a computation preserving this n-tangle, a computation in A which applies relative phases to a set of GHZ states, and a second computation which preserves it. As an application, we study the extent to which a large, random unitary may change concurrence. The result states that for a randomly chosen a in A within SU(2^{2p}), the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approaches infinity. Any v=k_1 a k_2 for such an a \in A has the same property. Finally, although |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 vanishes identically when the number of qubits is odd, we show that a more complicated C.C.D. still exists in which K is a symplectic group.
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"abstract": "The two-qubit canonical decomposition SU(4) = [SU(2) \\otimes SU(2)] Delta\n[SU(2) \\otimes SU(2)] writes any two-qubit quantum computation as a composition\nof a local unitary, a relative phasing of Bell states, and a second local\nunitary. Using Lie theory, we generalize this to an n-qubit decomposition, the\nconcurrence canonical decomposition (C.C.D.) SU(2^n)=KAK. The group K fixes a\nbilinear form related to the concurrence, and in particular any computation in\nK preserves the tangle |\u003cphi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi\u003e|^2 for n\neven. Thus, the C.C.D. shows that any n-qubit quantum computation is a\ncomposition of a computation preserving this n-tangle, a computation in A which\napplies relative phases to a set of GHZ states, and a second computation which\npreserves it.\n As an application, we study the extent to which a large, random unitary may\nchange concurrence. The result states that for a randomly chosen a in A within\nSU(2^{2p}), the probability that a carries a state of tangle 0 to a state of\nmaximum tangle approaches 1 as the even number of qubits approaches infinity.\nAny v=k_1 a k_2 for such an a \\in A has the same property. Finally, although\n|\u003cphi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi\u003e|^2 vanishes identically when the\nnumber of qubits is odd, we show that a more complicated C.C.D. still exists in\nwhich K is a symplectic group.",
"arxiv_id": "quant-ph/0309104",
"authors": [
"Stephen S. Bullock",
"Gavin K. Brennen"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1723701",
"journal_ref": "Journal of Mathematical Physics, vol 45, issue 6, pp. 2447-2467\n (2004)",
"title": "Canonical Decompositions of n-qubit Quantum Computations and Concurrence",
"url": "https://arxiv.org/abs/quant-ph/0309104"
},
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