verse order, as in this case, is the real essence of the Heisenberg uncertainty principle. Besides its fundamental importance, the experimental implementation of such a sequence of basic quantum operations is an essential tool for the full-scale engineering of a quantum light state optimized for a multitude of different tasks (15), including robust quantum communication. As any quantum operation, including non-Gaussian operations, is composed of photon additions and subtractions (i.e, it can be expressed as f ð% a; % a † Þ), our experimental results constitute a step toward the full quantum control of a field and the generation of highly entangled states (16).

We present a protocol to evaluate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems, and use it to experimentally demonstrate-under an assumption of fair sampling-the violation of an inequality that is satisfied by any non-contextual hidden-variables (NCHV) theory. The experiment is performed on an ensemble of molecular nuclear spins in the solid state, using established Nuclear Magnetic Resonance (NMR) techniques for quantum information processing (QIP).The Bell-Kochen-Specker (BKS) theorem [1,2,3,4] states that no noncontextual hidden-variables (NCHV) theory can reproduce the predictions of quantum mechanics for correlations between measurement outcomes of some sets of observables. Any such set of observables constitutes a proof of the theorem. Recently, Cabello [5] and others [6] used BKS proofs to derive a set of inequalities that are satisfied by any NCHV theory but are violated by quantum mechanics for any quantum state. These inequalities bound certain linear combinations of ensemble averages of correlations between measurement outcomes of compatible observables; thus creating a separation between the predicted outcomes of quantum mechanics, and the bound that is satisfied by NCHV theories.This provides an opportunity to test noncontextuality with finite-precision experiments -which has been the subject of contention for many years [7,8,9]-and without the need for the creation of special quantum states [10,11,12]. Already, two experiments, on a pair of trapped 40 Ca + ions [13], and with single photons [14], have demonstrated this state-independent conflict with noncontextuality. In this letter, we examine testing contextuality on quantum ensembles. This manuscript is organized as follows. First, we sketch the arguments leading to one of the inequalities derived in [5]. Then we present an algorithm to estimate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems. And lastly, we report and discuss the result of experimentally implementing the algorithm on a 3-qubit ensemble of molecular nuclear spins in the solid state.Inequality -For a quantum system prepared according to some state, ρ, one can assign simultaneous outcomes {ν(S k )} of measurements of a set {S k } of coobservables (i.e. comeasureable; mutually compatible; commuting). In this case, the correlation between the measurement outcomes is given byirrespective of the product ordering. Repeating the preparation and measurement many times, and averaging over the outcomes, one obtains an estimate of the ensemble average of the correlation π {S k } ρ = k ν(S k ) ρ . For the case where the coobservables {S k } are also dichotomic, with possible outcomes {ν(S k ) = ±1}, the correlation (1) also takes on the possible values ±1, and the ensemble average satisfies −1 ≤ π {S k } ρ ≤ +1. Note, that in this case, these operators are Hermitian and unitary (also known as Quantum Boolean Functions).Consider any set of observables with possible outcomes ±1 arranged in ...

The counter-intuitive properties of quantum mechanics have the potential to revolutionize information processing by enabling efficient algorithms with no known classical counterparts [1,2]. Harnessing this power requires developing a set of building blocks [3], one of which is a method to initialize the set of quantum bits (qubits) to a known state. Additionally, fresh ancillary qubits must be available during the course of computation to achieve fault tolerance [4,5,6,7]. In any physical system used to implement quantum computation, one must therefore be able to selectively and dynamically remove entropy from the part of the system that is to be mapped to qubits. One such method is an "open-system" cooling protocol in which a subset of qubits can be brought into contact with an external large heat-capacity system. Theoretical efforts [8,9,10] have led to an implementation-independent cooling procedure, namely heat-bath algorithmic cooling (HBAC). These efforts have culminated with the proposal of an optimal algorithm, the partner-pairing algorithm (PPA), which was used to compute the physical limits of HBAC [11]. We report here the first experimental realization of multi-step cooling of a quantum system via HBAC. The experiment was carried out using nuclear magnetic resonance (NMR) of a solid-state ensemble three-qubit system. It demonstrates the repeated repolarization of a particular qubit to an effective spin-bath temperature and alternating logical operations within the three-qubit subspace to ultimately cool a second qubit below this temperature. Demonstration of the control necessary for these operations is an important milestone in the control of solid-state NMR qubits and toward fault-tolerant quantum computing.