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arXiv:quant-ph/0509036 · Physical Review A

Sharing polarization within quantum subspaces

Muhammad Sabieh Anwar, Jonathan A. Jones, Simon B. Duckett

Published Feb 13, 2006• 2 citations
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Abstract

Given an ensemble of $n$ spins, at least some of which are partially polarized, we investigate the sharing of this polarization within a subspace of $k$ spins. We assume that the sharing results in a pseudopure state, characterized by a single purity parameter which we call the bias. As a concrete example we consider ensembles of spin-$1∕2$ nuclei in liquid-state nuclear magnetic resonance systems. The shared bias levels are compared with some current entanglement bounds to determine whether the reduced subspaces can give rise to entangled states.

DOI
10.1103/physreva.73.022322

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Given an ensemble of nnn spins, at least some of which are partially polarized, we investigate the sharing of this polarization within a subspace of kkk spins. We assume that the sharing results in a pseudopure state, characterized by a single purity parameter which we call the bias. As a concrete example we consider ensembles of spin-1∕21∕21∕2 nuclei in liquid-state nuclear magnetic resonance systems. The shared bias levels are compared with some current entanglement bounds to determine whether the reduced subspaces can give rise to entangled states.
We established important relationships between entanglement measures and the order parameter (spin polarization) in nuclear spin systems controlled by the nuclear magnetic resonance (NMR) technique. Since spin polarization can be easily manipulated by the NMR technique, experimentalists are presented with an opportunity to study the dynamic properties of entanglement, i.e., the creation and evolution of entangled states. Our approach may constitute the basis for researching the relations between the entanglement measures and measurable parameters of order in other quantum systems.
We experimentally and theoretically demonstrate the purity (polarization) control of qubits entangled with multiple spins, using induced dephasing in nuclear magnetic resonance setups to simulate repeated quantum measurements. We show that one may steer the qubit ensemble towards a quasiequilibrium state of a certain purity by choosing suitable time intervals between dephasing operations. These results demonstrate that repeated dephasing at intervals associated with the anti-Zeno regime leads to ensemble purification, whereas those associated with the Zeno regime lead to ensemble mixing.
The premise of nuclear magnetic resonance (NMR) quantum computing is that pseudopure quantum states, for which the populations of all energy levels but one are equal, can be used instead of pure quantum states. However, for systems with a large number of quantum bits (qubits), it would require elaborate pulse sequences with many steps to prepare these pseudopure states. To overcome this problem, a method using pairs of pseudopure states (POPS) instead of individual pseudopure states as the basis for NMR quantum computing is introduced here. The POPS approach has a significant advantage over the existing methods that, regardless of the number of qubits, only a single step is needed to prepare each pair of pseudopure states. A detailed analysis of a 16-level spin system is given, and the results of applying a c2{c}^{2}c2-NOT quantum logic gate (a controlled-NOT gate conditional upon the state of two other qubits) to all the POPS in the system are presented.
The classical analogues of quantum correlations have been the subject of considerable interest in the past few years; however, all of these investigations consider the classical analogue of only pure quantum mechanical states. In this work, we study mixed classical light states and derive relations between the polarization coherence of the field and the (classical) entanglement between its degrees of freedom. We show, for a specific model of mixed states, where the purity is determined by a single parameter, how the coherence shared between polarization and entanglement shrinks as the level of purity decreases. The sum of the square of polarization and entanglement remains constant, a behavior consistent with previous results [Phys. Rev. Lett.117, 153901 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.153901] of pure states, even though the constant for the mixed-state case is now smaller in value.
The use of nuclear magnetic resonance (NMR) to carry out quantum information processing (QIP) often requires the preparation, transformation, and detection of pseudopure states. In our previous work, it was shown that the use of pairs of pseudopure states (POPS) as a basis for QIP is very convenient because of the simplicity in experimental execution. It is now further demonstrated that the product of the NMR spectra corresponding to two sets of POPS that share a common pseudopure state has the same peak frequencies as those of the common (single) pseudopure state. Examples of applying two different quantum logic gates to a 5-qubit system are given.
We report the implementation of an entanglement transfer on a four-qubit liquid-state nuclear-magnetic resonance quantum information processor. This consists of creating an (pseudopure) entangled state among two directly coupled spins, and then transferring this two-spin state to another pair of spins whose direct interactions are negligible. Such transfers are expected to be an essential operation in scalable quantum computer architectures, and provide a useful benchmark for the coherent control available in specific implementations.