Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing
Open accessAbstract
We give a constructive proof that all mixed states of $N$ qubits in a sufficiently small neighborhood of the maximally mixed state are separable (unentangled). The construction provides an explicit representation of any such state as a mixture of product states. We give upper and lower bounds on the size of the neighborhood, which show that its extent decreases exponentially with the number of qubits. The bounds show that no entanglement appears in the physical states at any stage of present NMR experiments. Though this result raises questions about NMR quantum computation, further analysis would be necessary to assess the power of the general unitary transformations, which are indeed implemented in these experiments, in their action on separable states.
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In the liquid NMR demonstration, because the qubits are at all times in a highly mixed state and the dynamics can be fully modelled classically, neither the entanglement nor the coherent control at the core of Shor's algorithm can be implemented or verified.
“However, since the qubits are at all times in a highly mixed state [15], and the dynamics can be fully modelled classically [16], neither the entanglement nor the coherent control at the core of Shor's algorithm can be implemented or verified.”
From Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement (2007) · cites this paper as [15]
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- Authors
- Braunstein, S. L., Caves, C. M., Jozsa, R., Linden, N., Popescu, S., & Schack, R.
- Affiliations
- Bangor University (GB)
- University of New Mexico (US)
- University of Plymouth (GB)
- Isaac Newton Institute for Mathematical Sciences (GB)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1999-08-02
- Volume
- 83
- Issue
- 5
- Pages
- 1054-1057
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2040921410
- Cited by
- 566
- References
- 23
- Citations by year
- 2026: 2 · 2025: 7 · 2024: 9 · 2023: 19 · 2022: 12 · 2021: 20 · 2020: 11 · 2019: 15 · 2018: 11 · 2017: 12 · 2016: 19 · 2015: 13 · 2014: 18 · 2013: 24 · 2012: 43
- FWCI
- 45.78
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Bristol Research (University of Bristol) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- maximally mixed state, product states, entanglement, NMR quantum computing, qubits, constructive proof
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