Recently, there has been much interest in a new kind of ``unspeakable'' quantum information that stands to regular quantum information in the same way that a direction in space or a moment in time stands to a classical bit string: the former can only be encoded using particular degrees of freedom while the latter are indifferent to the physical nature of the information carriers. The problem of correlating distant reference frames, of which aligning Cartesian axes and synchronizing clocks are important instances, is an example of a task that requires the exchange of unspeakable information and for which it is interesting to determine the fundamental quantum limit of efficiency. There have also been many investigations into the information theory that is appropriate for parties that lack reference frames or that lack correlation between their reference frames, restrictions that result in global and local superselection rules. In the presence of these, quantum unspeakable information becomes a new kind of resource that can be manipulated, depleted, quantified, etcetera. Methods have also been developed to contend with these restrictions using relational encodings, particularly in the context of computation, cryptography, communication, and the manipulation of entanglement. This article reviews the role of reference frames and superselection rules in the theory of quantum information processing.Comment: 55 pages, published versio

Measurement underpins all quantitative science. A key example is the measurement of optical phase, used in length metrology and many other applications. Advances in precision measurement have consistently led to important scientific discoveries. At the fundamental level, measurement precision is limited by the number N of quantum resources (such as photons) that are used. Standard measurement schemes, using each resource independently, lead to a phase uncertainty that scales as 1/ √ N -known as the standard quantum limit. However, it has long been conjectured [1,2] that it should be possible to achieve a precision limited only by the Heisenberg uncertainty principle, dramatically improving the scaling to 1/N [3]. It is commonly thought that achieving this improvement requires the use of exotic quantum entangled states, such as the NOON state [4,5]. These states are extremely difficult to generate. Measurement schemes with counted photons or ions have been performed with N ≤ 6 [6, 7, 8, 9, 10, 11, 12, 13, 14, 15], but few have surpassed the standard quantum limit [12,14] and none have shown Heisenberg-limited scaling. Here we demonstrate experimentally a Heisenberg-limited phase estimation procedure. We replace entangled input states with multiple applications of the phase shift on unentangled single-photon states. We generalize Kitaev's phase estimation algorithm [16] using adaptive measurement theory [17,18,19,20] to achieve a standard deviation scaling at the Heisenberg limit. For the largest number of resources used (N = 378), we estimate an unknown phase with a variance more than 10 dB below the standard quantum limit; achieving this variance would require more than 4,000 resources using standard interferometry. Our results represent a drastic reduction in the complexity of achieving quantumenhanced measurement precision.Phase estimation is a ubiquitous measurement primitive, used for precision measurement of length, displacement, speed, optical properties, and much more. Recent work in quantum interferometry has focused on nphoton NOON states [5,6,7,8,9,10,11,12,21], (|n |0 + |0 |n ) / √ 2, expressed in terms of number states of the two arms of the interferometer. With this state, an improved phase sensitivity results from a decrease in the phase period from 2π to 2π/n. We achieve improved phase sensitivity more simply using an insight from quantum computing. We apply Kitaev's phase estimation algorithm [16,22] to quantum interferometry, wherein the entangled input state is replaced by multiple passes through the phase shift. The idea of using multi-pass protocols to gain a quantum advantage was proposed for the problem of aligning spatial reference frames [23], and further developed in relation to clock synchronization [24] and phase estimation [25,26].The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. 1a. The algorithm yields, with K + 1 bits of precision, an estimate φ est of a classical phase parameter φ, where e iφ is an eigenvalue of a unitary operator U . It requires us t...

Quantum state tomography-deducing quantum states from measured data-is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. one of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.

Quantum-dot spin qubits characteristically use oscillating magnetic or electric fields, or quasi-static Zeeman field gradients, to realize full qubit control. For the case of three confined electrons, exchange interaction between two pairs allows qubit rotation around two axes, hence full control, using only electrostatic gates. Here, we report initialization, full control, and single-shot readout of a three-electron exchange-driven spin qubit. Control via the exchange interaction is fast, yielding a demonstrated 75 qubit rotations in less than 2 ns. Measurement and state tomography are performed using a maximum-likelihood estimator method, allowing decoherence, leakage out of the qubit state space, and measurement fidelity to be quantified. The methods developed here are generally applicable to systems with state leakage, noisy measurements and non-orthogonal control axes.

We analyze the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the "quantum coin toss" in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.

We produce and holographically measure entangled qudits encoded in transverse spatial modes of single photons. With the novel use of a quantum state tomography method that only requires two-state superpositions, we achieve the most complete characterisation of entangled qutrits to date. Ideally, entangled qutrits provide better security than qubits in quantum bit-commitment: we model the sensitivity of this to mixture and show experimentally and theoretically that qutrits with even a small amount of decoherence cannot offer increased security over qubits. PACS numbers: 42.50.Dv, 03.65.Wj, 03.67.Dd, 03.67.Mn Many two-level quantum systems, or qubits, have been used to encode information [1]; using d-level systems, or qudits, enables access to larger Hilbert spaces, which can provide significant improvements over qubits such as increased channel capacity in quantum communication [2]. When entangled, qutrits (d=3) provide the best known levels of security in quantum bit-commitment and coinflipping protocols, which cannot be matched using qubitbased systems [3]. The ability to completely characterise entangled qudits is critical for applications. This is only possible using quantum state tomography [4,5].Entangled qudits have been realised in few physical systems, and only indirect measurements have been made of the quantum states of these systems. Qutrit entanglement has been generated between the arrival times of correlated photon pairs, where fringe measurements were used to infer features such as fidelities with specific entangled states and to estimate a potential Bell violation [6]. It is also possible to encode qudits in the transverse spatial modes of a photon, Fig. 1. There have been measurements demonstrating, but again not quantifying, spatial mode entanglement in parametric downconversion [7], including fringe measurements [8,9] and the violation of a two-qutrit Bell inequality [10,11].Here, we use quantum state tomography to completely characterise entangled, photonic qudits (both d = 2 and 3) encoded in transverse spatial modes, measuring the amount of entanglement and the degree of mixture. We show how to use the qutrit system in a quantum bitcommitment protocol and investigate the experimental requirements for achieving the best known security [3]. To illustrate these results, we first introduce and demonstrate two conceptually distinct ways of encoding information in transverse spatial modes, which differ in the behaviour of superposition states. This work constitutes the most complete characterisation of spatially-encoded qubits and qutrits and the first quantitative measurement of entangled qutrit states.The Gaussian spatial modes are a complete basis for describing the paraxial propagation of light [13]. Two orthonormal mode families are shown in Fig. 1(a): the Hermite-Gauss (HG rs ) and Laguerre-Gauss-Vortex (LGV pl ). These modes are self-similar under propagation; modes of the same order experience the same propagation-dependent phase shift, the Gouy phase shift. We define degenerate qudits...

We present a method for estimating the probabilities of outcomes of a quantum circuit using Monte Carlo sampling techniques applied to a quasiprobability representation. Our estimate converges to the true quantum probability at a rate determined by the total negativity in the circuit, using a measure of negativity based on the 1-norm of the quasiprobability. If the negativity grows at most polynomially in the size of the circuit, our estimator converges efficiently. These results highlight the role of negativity as a measure of nonclassical resources in quantum computation.