dorsal/arxiv
View SchemaGibbs States and the Consistency of Local Density Matrices
| Authors | Yi-Kai Liu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0603012 |
| URL | https://arxiv.org/abs/quant-ph/0603012 |
Abstract
Suppose we have an n-qubit system, and we are given a collection of local density matrices rho_1,...,rho_m, where each rho_i describes some subset of the qubits. We say that rho_1,...,rho_m are "consistent" if there exists a global state sigma (on all n qubits) whose reduced density matrices match rho_1,...,rho_m. We prove the following result: if rho_1,...,rho_m are consistent with some state sigma > 0, then they are also consistent with a state sigma' of the form sigma' = (1/Z) exp(M_1+...+M_m), where each M_i is a Hermitian matrix acting on the same qubits as rho_i, and Z is a normalizing factor. (This is known as a Gibbs state.) Actually, we show a more general result, on the consistency of a set of expectation values <T_1>,...,<T_r>, where the observables T_1,...,T_r need not commute. This result was previously proved by Jaynes (1957) in the context of the maximum-entropy principle; here we provide a somewhat different proof, using properties of the partition function.
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"abstract": "Suppose we have an n-qubit system, and we are given a collection of local\ndensity matrices rho_1,...,rho_m, where each rho_i describes some subset of the\nqubits. We say that rho_1,...,rho_m are \"consistent\" if there exists a global\nstate sigma (on all n qubits) whose reduced density matrices match\nrho_1,...,rho_m.\n We prove the following result: if rho_1,...,rho_m are consistent with some\nstate sigma \u003e 0, then they are also consistent with a state sigma\u0027 of the form\nsigma\u0027 = (1/Z) exp(M_1+...+M_m), where each M_i is a Hermitian matrix acting on\nthe same qubits as rho_i, and Z is a normalizing factor. (This is known as a\nGibbs state.) Actually, we show a more general result, on the consistency of a\nset of expectation values \u003cT_1\u003e,...,\u003cT_r\u003e, where the observables T_1,...,T_r\nneed not commute. This result was previously proved by Jaynes (1957) in the\ncontext of the maximum-entropy principle; here we provide a somewhat different\nproof, using properties of the partition function.",
"arxiv_id": "quant-ph/0603012",
"authors": [
"Yi-Kai Liu"
],
"categories": [
"quant-ph"
],
"title": "Gibbs States and the Consistency of Local Density Matrices",
"url": "https://arxiv.org/abs/quant-ph/0603012"
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