dorsal/arxiv
View SchemaThe Conal representation of Quantum States and Non Trace-Preserving Quantum Operations
| Authors | Pablo Arrighi, Christophe Patricot |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0212062 |
| URL | https://arxiv.org/abs/quant-ph/0212062 |
| DOI | 10.1103/PhysRevA.68.042310 |
| Journal | Phys. Rev. A 68, 042310 (2003) |
Abstract
We represent generalized density matrices of a $d$-complex dimensional quantum system as a subcone of a real pointed cone of revolution in $\mathbb{R}^{d^2}$, or indeed a Minkowskian cone in $\mathbb{E}^{1,d^2-1}$. Generalized pure states correspond to certain future-directed light-like vectors of $\mathbb{E}^{1,d^2-1}$. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particluar to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real linear transform. We give detailed formulae for the one qubit case and express the post-measurement states in terms of the initial state vectors and measurement vectors. We apply these results in order to find the information gain versus disturbance tradeoff in the case of two equiprobable pure states. Thus we recover Fuchs and Peres' formula in an elegant manner.
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"abstract": "We represent generalized density matrices of a $d$-complex dimensional\nquantum system as a subcone of a real pointed cone of revolution in\n$\\mathbb{R}^{d^2}$, or indeed a Minkowskian cone in $\\mathbb{E}^{1,d^2-1}$.\nGeneralized pure states correspond to certain future-directed light-like\nvectors of $\\mathbb{E}^{1,d^2-1}$. This extension of the Generalized Bloch\nSphere enables us to cater for non-trace-preserving quantum operations, and in\nparticluar to view the per-outcome effects of generalized measurements. We show\nthat these consist of the product of an orthogonal transform about the axis of\nthe cone of revolution and a positive real linear transform. We give detailed\nformulae for the one qubit case and express the post-measurement states in\nterms of the initial state vectors and measurement vectors. We apply these\nresults in order to find the information gain versus disturbance tradeoff in\nthe case of two equiprobable pure states. Thus we recover Fuchs and Peres\u0027\nformula in an elegant manner.",
"arxiv_id": "quant-ph/0212062",
"authors": [
"Pablo Arrighi",
"Christophe Patricot"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.68.042310",
"journal_ref": "Phys. Rev. A 68, 042310 (2003)",
"title": "The Conal representation of Quantum States and Non Trace-Preserving Quantum Operations",
"url": "https://arxiv.org/abs/quant-ph/0212062"
},
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