Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems
We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a spin lattice. For a quantum state on a D -dimensional lattice that has correlation length σ and has aver...
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| Format: | Article |
| Language: | English |
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IOP Publishing
2016-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/18/8/083011 |
| _version_ | 1797750952834367488 |
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| author | Anurag Anshu |
| author_facet | Anurag Anshu |
| author_sort | Anurag Anshu |
| collection | DOAJ |
| description | We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a spin lattice. For a quantum state on a D -dimensional lattice that has correlation length σ and has average energy e with respect to a given local Hamiltonian (with n local terms, each of which has norm at most 1), we show that the overlap of this state with eigenspace of energy f is at most $\exp {(-({(e-f)}^{2}\sigma )}^{\tfrac{1}{D+1}}/{n}^{\tfrac{1}{D+1}}D\sigma )$ . This bound holds whenever $| e-f| \gt {2}^{D}\sqrt{n\sigma }$ . Thus, on a one-dimensional lattice, the tail of the energy distribution decays exponentially with the energy. For product states, we improve above result to obtain a Gaussian decay in energy, even for quantum spin systems without an underlying lattice structure. Given a product state on a collection of spins which has average energy e with respect to a local Hamiltonian (with n local terms and each local term overlapping with at most m other local terms), we show that the overlap of this state with eigenspace of energy f is at most $\exp (-{(e-f)}^{2}/{{nm}}^{2})$ . This bound holds whenever $| e-f| \gt m\sqrt{n}$ . |
| first_indexed | 2024-03-12T16:41:16Z |
| format | Article |
| id | doaj.art-31abe59da7f845c2a056eb1f76f6d38d |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:41:16Z |
| publishDate | 2016-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-31abe59da7f845c2a056eb1f76f6d38d2023-08-08T14:30:37ZengIOP PublishingNew Journal of Physics1367-26302016-01-0118808301110.1088/1367-2630/18/8/083011Concentration bounds for quantum states with finite correlation length on quantum spin lattice systemsAnurag Anshu0Centre for Quantum Technologies, National University of Singapore , SingaporeWe consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a spin lattice. For a quantum state on a D -dimensional lattice that has correlation length σ and has average energy e with respect to a given local Hamiltonian (with n local terms, each of which has norm at most 1), we show that the overlap of this state with eigenspace of energy f is at most $\exp {(-({(e-f)}^{2}\sigma )}^{\tfrac{1}{D+1}}/{n}^{\tfrac{1}{D+1}}D\sigma )$ . This bound holds whenever $| e-f| \gt {2}^{D}\sqrt{n\sigma }$ . Thus, on a one-dimensional lattice, the tail of the energy distribution decays exponentially with the energy. For product states, we improve above result to obtain a Gaussian decay in energy, even for quantum spin systems without an underlying lattice structure. Given a product state on a collection of spins which has average energy e with respect to a local Hamiltonian (with n local terms and each local term overlapping with at most m other local terms), we show that the overlap of this state with eigenspace of energy f is at most $\exp (-{(e-f)}^{2}/{{nm}}^{2})$ . This bound holds whenever $| e-f| \gt m\sqrt{n}$ .https://doi.org/10.1088/1367-2630/18/8/083011energy distributionlocal Hamiltonian systemChernoff boundquantum spin lattice system |
| spellingShingle | Anurag Anshu Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems New Journal of Physics energy distribution local Hamiltonian system Chernoff bound quantum spin lattice system |
| title | Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| title_full | Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| title_fullStr | Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| title_full_unstemmed | Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| title_short | Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| title_sort | concentration bounds for quantum states with finite correlation length on quantum spin lattice systems |
| topic | energy distribution local Hamiltonian system Chernoff bound quantum spin lattice system |
| url | https://doi.org/10.1088/1367-2630/18/8/083011 |
| work_keys_str_mv | AT anuraganshu concentrationboundsforquantumstateswithfinitecorrelationlengthonquantumspinlatticesystems |