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arXiv:quant-ph/0409170 · Physical Review A

Bounds on the entanglement attainable from unitary transformed thermal states in liquid-state nuclear magnetic resonance

Terri M. Yu, Kenneth R. Brown, Isaac L. Chuang

Published Mar 24, 2005• 11 citations

Abstract

The role of mixed-state entanglement in liquid-state nuclear magnetic resonance (NMR) quantum computation is not yet well understood. In particular, despite the success of quantum-information processing with NMR, recent work has shown that quantum states used in most of those experiments were not entangled. This is because these states, derived by unitary transforms from the thermal equilibrium state, were too close to the maximally mixed state. We are thus motivated to determine whether a given NMR state is entanglable---that is, does there exist a unitary transform that entangles the state? The boundary between entanglable and nonentanglable thermal states is a function of the spin system size $N$ and its temperature $T$. We provide bounds on the location of this boundary using analytical and numerical methods; our tightest bound scales as $N\ensuremath{\sim}T$, giving a lower bound requiring at least $N\ensuremath{\sim}22\phantom{\rule{0.2em}{0ex}}000$ proton spins to realize an entanglable thermal state at typical laboratory NMR magnetic fields. These bounds are tighter than known bounds on the entanglability of effective pure states.

DOI: 10.1103/physreva.71.032341

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The role of mixed-state entanglement in liquid-state nuclear magnetic resonance (NMR) quantum computation is not yet well understood. In particular, despite the success of quantum-information processing with NMR, recent work has shown that quantum states used in most of those experiments were not entangled. This is because these states, derived by unitary transforms from the thermal equilibrium state, were too close to the maximally mixed state. We are thus motivated to determine whether a given NMR state is entanglable---that is, does there exist a unitary transform that entangles the state? The boundary between entanglable and nonentanglable thermal states is a function of the spin system size

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and its temperature

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N\ensuremath{\sim}T

, giving a lower bound requiring at least

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proton spins to realize an entanglable thermal state at typical laboratory NMR magnetic fields. These bounds are tighter than known bounds on the entanglability of effective pure states.

Recently, Yu, Brown, and Chuang [Phys. Rev. A {\bf 71}, 032341 (2005)] investigated the entanglement attainable from unitary transformed thermal states in liquid-state nuclear magnetic resonance (NMR). Their research gave an insight into the role of the entanglement in a liquid-state NMR quantum computer. Moreover, they attempted to reveal the role of mixed-state entanglement in quantum computing. However, they assumed that the Zeeman energy of each nuclear spin which corresponds to a qubit takes a common value for all; there is no chemical shift. In this paper, we research a model with the chemical shifts and analytically derive the physical parameter region where unitary transformed thermal states are entangled, by the positive partial transposition (PPT) criterion with respect to any bipartition. We examine the effect of the chemical shifts on the boundary between the separability and the nonseparability, and find it is negligible.

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Abstract Decoherence of the bipartite entangled mixed state in a five‐qubit system has been studied experimentally with liquid‐state nuclear magnetic resonance (NMR). The experimental results indicate that when only two qubits of the five qubits are partially initialized, the decoherence process of the bipartite entangled mixed state is not quantized, as the entanglability is weak. In contrast, when all five qubits are initialized, the decoherence process is quantized, which is understood to indicate that the entanglability is actually strong. From these results, it is conceivable that the entropy of the deviation density matrix is a crucial factor in measuring the entanglement in the liquid‐state NMR quantum computer. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

While quantum computing offers the potential of using new quantum algorithms to tackle problems that are intractable for classical processors, its implementation requires the development of quantum devices, which are as yet unavailable. The most complex implementations of quantum algorithms to date have used techniques adapted from nuclear magnetic resonance (NMR) spectroscopy, but current liquid state NMR approaches cannot be extended to systems with many quantum bits, as it is not possible to prepare pure initial states by directly cooling the spin system into its ground state. Furthermore, it has been shown that current NMR experiments involve only separable states, and thus could in principle be described by local hidden variable models. Here we demonstrate how para-hydrogen can be used to prepare a two-spin system in an almost pure state and so overcome these two objections. To achieve polarisation comparable to our observed value of

\\epsilon = 0.916 \\pm 0.019

by thermal means would require cooling to 6.4 mK in the 9.6 T field used, or a magnetic field of 0.45 MT at room temperature.

Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10—30 qubits.