dorsal/arxiv
View SchemaExtremal Quantum States in Coupled Systems
| Authors | K. R. Parthasarathy |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0307182 |
| URL | https://arxiv.org/abs/quant-ph/0307182 |
Abstract
Let ${\cal H}_1,$ ${\cal H}_2$ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose $\rho_i$ is a state in ${\cal H}_i, i=1,2.$ Let ${\cal C} (\rho_1, \rho_2)$ be the convex set of all states $\rho$ in ${\cal H} = {\cal H}_1 \otimes {\cal H}_2$ whose marginal states in ${\cal H}_1$ and ${\cal H}_2$ are $\rho_1$ and $\rho_2$ respectively. Here we present a necessary and sufficient criterion for a $\rho$ in ${\cal C} (\rho_1, \rho_2)$ to be an extreme point. Such a condition implies, in particular, that for a state $\rho$ to be an extreme point of ${\cal C} (\rho_1, \rho_2)$ it is necessary that the rank of $\rho$ does not exceed $(d_1^2 + d_2^2 - 1)^{{1/2}},$ where $d_i = \dim {\cal H}_i, i=1,2.$ When ${\cal H}_1$ and ${\cal H}_2$ coincide with the 1-qubit Hilbert space $\mathbb{C}^2$ with its standard orthonormal basis $\{|0 >, |1> \}$ and $\rho_1 = \rho_2 = {1/2} I$ it turns out that a state $\rho \in {\cal C} ({1/2}I, {1/2}I)$ is extremal if and only if $\rho$ is of the form $|\Omega>< \Omega|$ where $| \Omega > = \frac{1}{\sqrt{2}} (|0> | \psi_0 > + |1 > | \psi_1 >),$ $\{| \psi_0 >, | \psi_1> \}$ being an arbitrary orthonormal basis of $\mathbb{C}^2.$ In particular, the extremal states are the maximally entangled states.
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"abstract": "Let ${\\cal H}_1,$ ${\\cal H}_2$ be finite dimensional complex Hilbert spaces\ndescribing the states of two finite level quantum systems. Suppose $\\rho_i$ is\na state in ${\\cal H}_i, i=1,2.$ Let ${\\cal C} (\\rho_1, \\rho_2)$ be the convex\nset of all states $\\rho$ in ${\\cal H} = {\\cal H}_1 \\otimes {\\cal H}_2$ whose\nmarginal states in ${\\cal H}_1$ and ${\\cal H}_2$ are $\\rho_1$ and $\\rho_2$\nrespectively. Here we present a necessary and sufficient criterion for a $\\rho$\nin ${\\cal C} (\\rho_1, \\rho_2)$ to be an extreme point. Such a condition\nimplies, in particular, that for a state $\\rho$ to be an extreme point of\n${\\cal C} (\\rho_1, \\rho_2)$ it is necessary that the rank of $\\rho$ does not\nexceed $(d_1^2 + d_2^2 - 1)^{{1/2}},$ where $d_i = \\dim {\\cal H}_i, i=1,2.$\nWhen ${\\cal H}_1$ and ${\\cal H}_2$ coincide with the 1-qubit Hilbert space\n$\\mathbb{C}^2$ with its standard orthonormal basis $\\{|0 \u003e, |1\u003e \\}$ and $\\rho_1\n= \\rho_2 = {1/2} I$ it turns out that a state $\\rho \\in {\\cal C} ({1/2}I,\n{1/2}I)$ is extremal if and only if $\\rho$ is of the form $|\\Omega\u003e\u003c \\Omega|$\nwhere $| \\Omega \u003e = \\frac{1}{\\sqrt{2}} (|0\u003e | \\psi_0 \u003e + |1 \u003e | \\psi_1 \u003e),$\n$\\{| \\psi_0 \u003e, | \\psi_1\u003e \\}$ being an arbitrary orthonormal basis of\n$\\mathbb{C}^2.$ In particular, the extremal states are the maximally entangled\nstates.",
"arxiv_id": "quant-ph/0307182",
"authors": [
"K. R. Parthasarathy"
],
"categories": [
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"title": "Extremal Quantum States in Coupled Systems",
"url": "https://arxiv.org/abs/quant-ph/0307182"
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