Quantum algorithms in group theory
We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.
quant-ph↗
arXiv subjects
Samuel L. Braunstein
Publications and source records attributed to Samuel L. Braunstein.
Towards photostatistics from photon-number discriminating detectors
We study the properties of a photodetector that has a number-resolving capability. In the absence of dark counts, due to its finite quantum efficiency, photodetection with such a detector can only eliminate the possibility that the incident field corresponds to a number of photons less than the detected photon number. We show that such a {\em non-photon} number-discriminating detector, however, provides a useful tool in the reconstruction of the photon number distribution of the incident field even in the presence of dark counts.
quant-ph↗
On Strong Superadditivity of the Entanglement of Formation
We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation $E_F$ and the conjugate function $E^*$ of the entanglement function $E(ρ)=S(\trace_Aρ)$. We then consider the conjectured strong superadditivity of the entanglement of formation $E_F(ρ) \ge E_F(ρ_I)+E_F(ρ_{II})$, where $ρ_I$ and $ρ_{II}$ are the reductions of $ρ$ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of $E^*$. As an application, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for all non-trace-preserving quantum channels, when purity is measured by Schatten $p$-norms for $p$ tending to 1.
quant-ph↗
Quantum coherence in the presence of unobservable quantities
State representations summarize our knowledge about a system. When unobservable quantities are introduced the state representation is typically no longer unique. However, this non-uniqueness does not affect subsequent inferences based on any observable data. We demonstrate that the inference-free subspace may be extracted whenever the quantity's unobservability is guaranteed by a global conservation law. This result can generalize even without such a guarantee. In particular, we examine the coherent-state representation of a laser where the absolute phase of the electromagnetic field is believed to be unobservable. We show that experimental coherent states may be separated from the inference-free subspaces induced by this unobservable phase. These physical states may then be approximated by coherent states in a relative-phase Hilbert space.
quant-ph↗
Quantum deleting and Signalling
It is known that if we can clone an arbitrary state we can send signal faster than light. Here, we show that deletion of unknown quantum state against a copy can lead to superluminal signalling. But erasure of unknown quantum state does not imply faster than light signalling.
quant-ph↗
Quantum mechanical Universal constructor
Arbitrary quantum states cannot be copied. In fact, to make a copy we must provide complete information about the system. However, can a quantum system self-replicate? This is not answered by the no-cloning theorem. In the classical context, Von Neumann showed that a `universal constructor' can exist which can self-replicate an arbitrary system, provided that it had access to instructions for making copy of the system. We question the existence of a universal constructor that may allow for the self-replication of an arbitrary quantum system. We prove that there is no deterministic universal quantum constructor which can operate with finite resources. Further, we delineate conditions under which such a universal constructor can be designed to operate dterministically and probabilistically.
quant-ph↗
The Partition Ensemble Fallacy Fallacy
The Partition Ensemble Fallacy was recently applied to claim no quantum coherence exists in coherent states produced by lasers. We show that this claim relies on an untestable belief of a particular prior distribution of absolute phase. One's choice for the prior distribution for an unobservable quantity is a matter of `religion'. We call this principle the Partition Ensemble Fallacy Fallacy. Further, we show an alternative approach to construct a relative-quantity Hilbert subspace where unobservability of certain quantities is guaranteed by global conservation laws. This approach is applied to coherent states and constructs an approximate relative-phase Hilbert subspace.
quant-ph↗
Deutsch-Jozsa algorithm for continuous variables
We present an idealized quantum continuous variable analog of the Deutsch-Jozsa algorithm which can be implemented on a perfect continuous variable quantum computer. Using the Fourier transformation and XOR gate appropriate for continuous spectra we show that under ideal operation to infinite precision that there is an infinite reduction in number of query calls in this scheme.
quant-ph↗
Speed-up and entanglement in quantum searching
We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `charged for' then in existing machines speed-up without entanglement is not possible.
quant-ph↗
Distributed entanglement as a probe for the quantum structure of spacetime
Simultaneity is a well-defined notion in special relativity once a Minkowski metric structure is fixed on the spacetime continuum (manifold) of events. In quantum gravity, however, the metric is not expected to be a fixed, classical structure, but a fluctuating quantum operator which may assume a coherent superposition of two classically-distinguishable values. A natural question to ask is what happens to the notion of simultaneity and synchronization when the metric is in a quantum superposition. Here we show that the resource of distributed entanglement of the same kind as used by Jozsa et al. [Phys. Rev. Lett. 85, 2010 (2000)] gives rise to an experimental probe that is sensitive to coherent quantum fluctuations in the spacetime metric.
quant-ph↗
Multipartite entanglement
First, we show how the quantum circuits for generating and measuring multi-party entanglement of qubits can be translated to continuous quantum variables. We derive sufficient inseparability criteria for $N$-party continuous-variable states and discuss their applicability. Then, we consider a family of multipartite entangled states (multi-party multi-mode states with one mode per party) described by continuous quantum variables and analyze their properties. These states can be efficiently generated using squeezed light and linear optics.
quant-ph↗
Efficient Classical Simulation of Continuous Variable Quantum Information Processes
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.
quant-ph↗
Quantum interferometric optical lithography:towards arbitrary two-dimensional patterns
As demonstrated by Boto et al. [Phys. Rev. Lett. 85, 2733 (2000)], quantum lithography offers an increase in resolution below the diffraction limit. Here, we generalize this procedure in order to create patterns in one and two dimensions. This renders quantum lithography a potentially useful tool in nanotechnology.
quant-ph↗
Detection devices in entanglement-based optical state preparation
We study the use of detection devices in entanglement-based state preparation. In particular we consider optical detection devices such as single-photon sensitivity detectors, single-photon resolution detectors and detector cascades (with an emphasis on the performance of realistic detectors). We develop an extensive theory for the use of these devices. In entanglement-based state preparation we perform measurements on subsystems, and we therefore need precise bounds on the distinguishability of these measurements (this is fundamentally different from, e.g., tomography, where an ensemble of identical states is used to determine probability distributions, etc.). To this end, we introduce the confidence of preparation, which may also be used to quantify the performance of detection devices in entanglement-based preparation. We give a general expression for detector cascades of arbitrary size for the detection up to two photons. We show that, contrary to the general belief, cascading does not give a practical advantage over detectors with single-photon resolution in entanglement-based state preparation.
quant-ph↗
Telecloning and multiuser quantum channels for continuous variables
We propose entangled (M+1)-mode quantum states as a multiuser quantum channel for continuous-variable communication. Arbitrary quantum states can be sent via this channel simultaneously to M remote and separated locations with equal minimum excess noise in each output mode. For a set of coherent-state inputs, the channel realizes optimum symmetric 1-to-M cloning at a distance (``telecloning''). It also provides the optimal cloning of coherent states without the need of amplifying the state of interest. The generation of the multiuser quantum channel requires no more than two 10\log_{10}[(\sqrt{M}-1)/(\sqrt{M}+1)] dB squeezed states and M beam splitters.
quant-ph↗
Quantum versus Classical Domains for Teleportation with Continuous Variables
Fidelity F{classical} = 1/2 has been established as setting the boundary between classical and quantum domains in the teleportation of coherent states of the electromagnetic field (S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt. 47, 267 (2000)). Two recent papers by P. Grangier and F. Grosshans (quant-ph/0009079 and quant-ph/0010107) introduce alternate criteria for setting this boundary and as a result claim that the appropriate boundary should be F = 2/3. Although larger fidelities would lead to enhanced teleportation capabilities, we show that the new conditions of Grangier and Grosshans are largely unrelated to the questions of entanglement and Bell-inequality violations that they take to be their primary concern. With regard to the quantum-classical boundary, we demonstrate that fidelity F{classical} = 1/2 remains the appropriate point of demarcation. The claims of Grangier and Grosshans to the contrary are simply wrong, as we show by an analysis of the conditions for nonseparability (that complements our earlier treatment) and by explicit examples of Bell-inequality violations.
quant-ph↗
Optimal cloning of coherent states with a linear amplifier and beam splitters
A transformation achieving the optimal symmetric N-to-M cloning of coherent states is presented. Its implementation only requires a phase-insensitive linear amplifier and a network of beam splitters. An experimental demonstration of this continuous-variable cloner should therefore be in the scope of current technology. The link between optimal quantum cloning and optimal amplification of quantum states is also pointed out.
quant-ph↗
Quantum information distributors: Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit
We show that for any Hilbert-space dimension, the optimal universal quantum cloner can be constructed from essentially the same quantum circuit, i.e., we find a universal design for universal cloners. In the case of infinite dimensions (which includes continuous variable quantum systems) the universal cloner reduces to an essentially classical device. More generally, we construct a universal quantum circuit for distributing qudits in any dimension which acts covariantly under generalized displacements and momentum kicks. The behavior of this covariant distributor is controlled by its initial state. We show that suitable choices for this initial state yield both universal cloners and optimized cloners for limited alphabets of states whose states are related by generalized phase-space displacements.
quant-ph↗