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The interference of two indistinguishable photons from a photon pair on a balanced beam splitter, where each photon enters in one of the input ports, yields a maximally entangled N00N-state shared between the outputs of the beam splitter through Hong-Ou-Mandel interference. At the same time, interference between the two modes of a two-mode squeezed state - the generalized output of a parametric photon-pair source - yields two independent, hence separable, single-mode squeezed states. Notably, both experiments use the exact same experimental setup. We resolve this seeming contradiction by measuring photon-number resolved quantum interference of the two modes of a two-mode squeezed state from an engineered parametric down-conversion source and show that the notion of inseparability is a result of post-selection on the presence of photons. Indeed, the complete joint photon statistics including the vacuum component are separable, as expected from textbook calculations.

There is a long-standing apparent conflict between the existence of diamagnetism and causality as expressed through the Kramers-Kronig relations. In essence, using causality arguments, along with a handful of other seemingly well-justified assumptions, one can show that diamagnetism is impossible. Nature naturally does not mind this, and experiments show diamagnetic responses from magnetic media. We present a resolution to this issue, which also explains the absence of observed dia-electric responses in optical media., Comment: 5 pages + supplemental, 1 figure. Comments welcome

Does the cross-product of the position and the electromagnetic momentum density, g, include the optical spin momentum? We answer this long-standing question in the affirmative by evaluating, explicitly, the torque exerted on a particle by a beam of light carrying angular momentum.

Optical helicity quantifies the handedness of light, and plays a central role in the description of interactions between light and chiral matter. In free space, it is related to the duality symmetry of the electromagnetic field, a continuous symmetry encapsulating the invariance of Maxwell's equations under the interchange of electric and magnetic fields. However, in materials the situation is not so straightforward, as the free space transformation must be extended to encompass mixing of both the $\mathbf{E}$/$\mathbf{H}$ and $\mathbf{D}$/$\mathbf{B}$ field pairs. The simultaneous direct interchange of $\mathbf{E}$/$\mathbf{H}$ and of $\mathbf{D}$/$\mathbf{B}$ is incompatible with the presence of linear constitutive relations. In this work, we extend the duality transform in a way that resolves this incompatibility, and use this to define the optical helicity in a general medium, which may be dispersive, lossy, chiral or nonreciprocal. We find that the helicity density must contain an explicit contribution associated with the polarisation and magnetisation of the matter, and we show that the form of this matter contribution is independent of the details of the medium. We also show that the in-medium helicity can be naturally expressed in terms of the elementary quantised excitations of the system., Comment: 4 pages + supplemental. 2 figures. Comments welcome

We investigate the effects of the electromagnetic vacuum field on a harmonically bound electron. We show that in the electric-dipole approximation the model atom couples only to an effective one-dimensional electric field. In a simplified form, in which the problem is reduced to a single spatial dimension, we determine, analytically, the form of the ground state and discuss the significance of this., Comment: 11 pages, 2 figures. This article is a contribution to the special issue of Physics, "Matter-Radiation Interactions -- In Memory of Professor Francesco Saverio Persico"

We determine the form of the unitary transformation that diagonalizes the Jaynes-Cummings Hamiltonian. This leads to operators the action of which has a simple interpretation in terms of the dressed states, the energy eigenstates. This suggests a set of coherent states and spin coherent states based on the dressed states., Comment: 5 pages, no figures

We reanalyse the quantum damped harmonic oscillator, introducing three less than common features. These are (i) the use of a continuum model of the reservoir rather than an ensemble of discrete oscillators, (ii) an exact diagonalisation of the Hamiltonian by adapting a technique pioneered by Fano, and (iii) the use of the thermofield technique for describing a finite temperature reservoir. We recover in this way a number of well-known and some, perhaps, less familiar results. An example of the latter is an ab initio proof that the oscillator relaxes to the mean-force Gibbs state. We find that special care is necessary when comparing the damped oscillator with its undamped counterpart as the former has two distinct natural frequencies, one associated with short time evolution and the other with longer times., Comment: 38 pages, 6 figures. This article, part review and part new results, is a contribution to the special issue of Physica Scripta "Challenges in quantum dynamics'', in honour of Professor Igor Jex on the occasion of his 60th birthday. arXiv admin note: substantial text overlap with arXiv:1508.02442

We show that Skyrmion field lines, constructed from the local Stokes parameters, trace out lines of constant optical polarisation., Comment: Updated version published in Europhysics Letters. Only minor changes

The Glauber-Sudarshan P-representation is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. For simplicity only one type of boson or fermion will be considered., Comment: To appear in Physica Scripta (2023)

It has often been remarked that single-photon interference experiments, however complicated, seem to behave very much in the same way as those performed in the classical regime, using the field generated by a laser. This observation has the status of being `well-known to those who know it', but perhaps mysterious to others. We discuss the reasons underlying the similarity and also some of the limitations of this simple idea.

Stated simply the paradox is as follows: it is clear that the orbital angular momentum of a light beam in its direction of propagation is an intrinsic quantity, and therefore has the same value everywhere in the beam. How then can a Gaussian beam, with precisely zero orbital angular momentum, drive a (single-photon) quadrupole transition which requires the transfer of angular momentum 2$\hbar$ to an absorbing atom?, Comment: Submitted to J.Phys.A. Special Issue "Claritons and the Asymptotics of Ideas: the Physics of Michael Berry"

We present a method to measure the optical torque applied to particles of arbitrary shape held in an optical trap, inferred from the change of angular momentum of light induced by the particle. All torque components can be determined from a single interference pattern recorded by a camera in the back focal plane of a high-NA condenser collecting all forward scattered light. We derive explicit expressions mapping the measured complex field in this plane to the torque components. The required phase is retrieved by an iterative algorithm, using the known position of the optical traps as constraints. The torque pertaining to individual particles is accessible, as well as separate spin or orbital parts of the total torque., Comment: V2, with updated figures

Any treatment of magnetic interactions between atoms, molecules and optical media must start at the form of the interaction energy. This forms the base on which predictions about any number of magnetic atom-light properties stands -- spontaneous decay rates and forces included. As is well-known, the Heaviside-Larmor duality symmetry of Maxwell's equations, where electric and magnetic quantities are exchanges, is broken by the usual form of the magnetic interaction energy. We argue that this symmetry can be restored by including general local-field effects, and that local fields should be treated as a necessity for correctly translating between the microscopic world of the dipole and the macroscopic world of the measured fields. This may additionally aid in resolving a long standing debate over the form of the force on a dipole in a medium. Finally, we compute the magnetic dipole decay rate in a magneto-dielectric with local-field effects taken into account, and show that macroscopic quantum electrodynamics can be made to be dual symmetric at an operator level, instead of only for expectation values., Comment: 9 pages, 4 figures, plus appendices. Comments welcome

Prediction is the making of statements, usually probabilistic, about future events based on current information. Retrodiction is the making of statements about past events based on current information. We present the foundations of quantum retrodiction and highlight its intimate connection with the Bayesian interpretation of probability. The close link with Bayesian methods enables us to explore controversies and misunderstandings about retrodiction that have appeared in the literature. To be clear, quantum retrodiction is universally applicable and draws its validity directly from conventional predictive quantum theory coupled with Bayes' theorem., Comment: New version, accepted by journal: Symmetry

A simple and flexible scheme for high-dimensional linear quantum operations on optical transverse spatial modes is demonstrated. The quantum Fourier transformation (QFT) and quantum state tomography (QST) via symmetric informationally complete positive operator-valued measures (SIC POVMs) are implemented with dimensionality of 15. The matrix fidelity of QFT is 0.85, while the statistical fidelity of SIC POVMs and fidelity of QST are ~0.97 and up to 0.853, respectively. We believe that our device has the potential for further exploration of high-dimensional spatial entanglement provided by spontaneous parametric down conversion in nonlinear crystals.

The emission properties of atoms lie at the foundations of both quantum theory and light-matter interactions. In the context of macroscopic media, exact knowledge thereof is important both in current quantum technologies as well as in fundamental studies. While for isotropic media, this is a very well-studied problem, there are still big gaps in the theory of anisotropic media. In particular, to the best of our knowledge, an explicit expression for the spontaneous emission rate in general anisotropic media has not been presented. In this work, we first derive the quantised electromagnetic field operators to calculate the emission rate in uniaxial media. For the more general case of biaxial media we propose an approximate expression based on interpolation between the limiting cases of uniaxial media. We support our model with numerical simulations which are in strong agreement for typical media configurations, and furthermore show how local field effects can be taken into account in the model., Comment: 11 pages, 3 figures

We consider the energy and helicity densities of circularly polarised light within a lossless chiral medium, characterised by the chirality parameter $\beta$. A form for the helicity density is introduced, valid to first order in $\beta$, that produces a helicity of $\pm\hbar$ per photon for right and left circular polarisation, respectively. This is in contrast to the result obtained if we use the form of the helicity density employed for linear media. We examine the helicity continuity equation, and show that this modified form of the helicity density is required for consistency with the dual symmetry condition of a chiral medium with a constant value of ${\epsilon}/{\mu}$. Extending the results to arbitrary order in $\beta$ establishes an exact relationship between the energy and helicity densities in a chiral medium.

Multiple-copy state discrimination is a fundamental task in quantum information processing. If there are two, pure, non-orthogonal states then both local and collective schemes are known to reach the Helstrom bound, the maximum probability of successful discrimination allowed by quantum theory. For mixed states, it is known that only collective schemes can perform optimally, so it might be expected that these schemes are more resilient to preparation noise. We calculate the probability of success for two schemes, one local and one collective, in the regime of imperfect preparation fidelity. We find two surprising results. Firstly, both schemes converge upon the same many-copy limit, which is less than unity. Secondly, the local scheme performs better in all cases. This highlights the point that one should take into account noise when designing state discrimination schemes.

We exploit the vorticity, familiar from fluid mechanics and the theory of superfluids, as a tool to track the birth and subsequent development of optical vortices at a spiral phase plate., Comment: 16 pages, 7 figures, see ancillary files section

Vector vortex beams possess a topological property that derives both from the spatially varying amplitude of the field and also from its varying polarization. This property arises as a consequence of the inherent Skyrmionic nature of such beams and is quantified by the associated Skyrmion number, which embodies a topological property of the beam. We illustrate this idea for some of the simplest vector beams and discuss the physical significance of the Skyrmion number in this context., Comment: 6 pages, 6 figures

Phase distortions, or aberrations, can negatively influence the performance of an optical imaging system. Through the use of position-momentum entangled photons, we nonlocally correct for aberrations in one photon's optical path by intentionally introducing the complementary aberrations in the optical path of the other photon. In particular, we demonstrate the simultaneous nonlocal cancellation of aberrations that are of both even and odd order in the photons' transverse degrees of freedom. We also demonstrate a potential application of this technique by nonlocally cancelling the effect of defocus in a quantum imaging experiment and thereby recover the original spatial resolution., Comment: 6 pages, 4 figures

The subtraction or addition of a prescribed number of photons to a field mode does not, in general, simply shift the probability distribution by the number of subtracted or added photons. Subtraction of a photon from an initial coherent state, for example, leaves the photon statistics unchanged and the same process applied to an initial thermal state increases the mean photon number. We present a detailed analysis of the effects of the initial photon statistics on those of the state from which the photons have been subtracted or to which they have been added. Our approach is based on two closely-related moment generating functions, one well-established and one that we introduce.

The Janszky representation constructs quantum states of a field mode as a superposition of coherent states on a line in the complex plane. We show that this provides a natural Schr\"{o}dinger picture description of the interference between a pair of modes at a beam splitter.

The models we use, habitually, to describe quantum nonlinear optical processes have been remarkably successful yet, with few exceptions, they each contain a mathematical flaw. We present this flaw, show how it can be fixed and, in the process, suggest why we can continue to use our favoured Hamiltonians.

The usual multipolar Hamiltonian for atom-light interaction features a non-relativistic moving atom interacting with electromagnetic fields which inherently follow Lorentzian symmetry. This combination can lead to situations where atoms appear to experience a friction force, when in fact they only change their internal mass-energy due to the emission or absorption of a photon. Unfortunately the simple Galilean description of the atom's motion is not sufficient to distinguish between a change in momentum due to acceleration and a change in momentum due to a change in internal mass-energy. In this work we show how a low-order relativistic correction can be included in the multipolar atom-light Hamiltonian. We also give examples how this affects the most basic mechanical interactions between atoms and photons.

Quantum coherence, the ability to control the phases in superposition states is a resource, and it is of crucial importance, therefore, to understand how it is consumed in use. It has been suggested that catalytic coherence is possible, that is repeated use of the coherence without degradation or reduction in performance. The claim has particular relevance for quantum thermodynamics because, were it true, it would allow free energy that is locked in coherence to be extracted $\textit{indefinitely}$. We address this issue directly with a careful analysis of the proposal by $\AA{}$berg. We find that coherence $\textit{cannot}$ be used catalytically, or even repeatedly without limit., Comment: 23 pages with 2 figures

We investigate the optimal measurement strategy for state discrimination of the trine ensemble of qubit states prepared with arbitrary prior probabilities. Our approach generates the minimum achievable probability of error and also the maximum confidence strategy. Although various cases with symmetry have been considered and solution techniques put forward in the literature, to our knowledge this is only the second such closed form, analytical, arbitrary prior, example available for the minimum-error figure of merit, after the simplest and well-known two-state example.

Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary unitary and non-unitary linear operations for N-dimensional phase-coherent spatial modes with meticulously designed phase gratings. As implemented on spatial light modulators (SLMs), the unitary transformation matrix has been realized with dimensionalities ranging from 7 to 24 and the corresponding fidelities are from 95.1% to 82.1%. For the non-unitary operators, a matrix is presented for the tomography of a 4-level quantum system with a fidelity of 94.9%. Thus, the linear operator has been successfully implemented with much higher dimensionality than that in previous reports. It should be mentioned that our method is not limited to SLMs and can be easily applied on other devices. Thus we believe that our proposal provides another option to perform linear operation with a simple, fixed, error-tolerant and scalable scheme.

We know that in empty space there is no preferred state of rest. This is true both in special relativity but also in Newtonian mechanics with its associated Galilean relativity. It comes as something of a surprise, therefore, to discover the existence a friction force associated with spontaneous emission. he resolution of this paradox relies on a central idea from special relativity even though our derivation of it is non-relativistic. We examine the possibility that the physics underlying this effect might be explored in an ion trap, via the observation of a superposition of different mass states., Comment: 8 pages, 2 figures. Published in Journal of Modern Optics on 14 September 2017. Version 2 with a corrected typo on page 6

Sir Peter Knight is a pioneer in quantum optics which has now grown to an important branch of modern physics to study the foundations and applications of quantum physics. He is leading an effort to develop new technologies from quantum mechanics. In this collection of essays, we recall the time we were working with him as a postdoc or a PhD student and look at how the time with him has influenced our research.

Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important application of the new method is demonstrated, providing an axiomatic derivation of important results of the BB84 protocol in quantum cryptography., Comment: 27 pages, 3 figures

The Roentgen term is an often neglected contribution to the interaction between an atom and an electromagnetic field in the electric dipole approximation. In this work we discuss how this interaction term leads to a difference between the kinetic and canonical momentum of an atom which, in turn, leads to surprising radiation forces acting on the atom. We use a number of examples to explore the main features of this interaction, namely forces acting against the expected dipole force or accelerations perpendicular to the beam propagation axis.

State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bi-partite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal sequential measurement must satisfy, analogous to the well-known Helstrom conditions for minimum error discrimination in the unrestricted case. We give several examples and compare the optimal probability of correctly identifying the state via global versus sequential measurement strategies., Comment: 17 pages, 3 figures

We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities., Comment: 8 pages, 1 figure

The four-qubit states $\lvert\chi^{ij}\rangle$, exhibiting genuinely multi-partite entanglement have been shown to have many interesting properties and have been suggested for novel applications in quantum information processing. In this work we propose a simple quantum circuit and its corresponding optical embodiment with which to prepare photon pairs in the $\lvert\chi^{ij}\rangle$ states. Our approach uses hyper-entangled photon pairs, produced by the type-I spontaneous parametric down-conversion (SPDC) process in two contiguous nonlinear crystals, together with a set of simple linear-optical transformations. Our photon pairs are maximally hyper-entangled in both their polarisation and orbital angular momentum (OAM). After one of these daughter photons passes through our optical setup, we obtain photon pairs in the hyper-entangled state $\lvert\chi^{00}\rangle$, and the $\lvert\chi^{ij}\rangle$ states can be achieved by further simple transformations., Comment: Submitted to Journal of Optics

Non-locality without entanglement is a rather counter-intuitive phenomenon in which information may be encoded entirely in product (unentangled) states of composite quantum systems in such a way that local measurement of the subsystems is not enough for optimal decoding. For simple examples of pure product states, the gap in performance is known to be rather small when arbitrary local strategies are allowed. Here we restrict to local strategies readily achievable with current technology; those requiring neither a quantum memory nor joint operations. We show that, even for measurements on pure product states there can be a large gap between such strategies and theoretically optimal performance. Thus even in the absence of entanglement physically realizable local strategies can be far from optimal for extracting quantum information., Comment: 5 pages, 1 figure

Two particle interference phenomena, such as the Hong-Ou-Mandel effect, are a direct manifestation of the nature of the symmetry properties of indistinguishable particles as described by quantum mechanics. The Hong-Ou-Mandel effect has recently been applied as a tool for pure state tomography of a single photon. In this article, we generalise the method to extract additional information for a pure state and extend this to the full tomography of mixed states as well. The formalism is kept general enough to apply to both boson and fermion based interferometry. Our theoretical discussion is accompanied by two proposals of interferometric setups that allow the measurement of a tomographically complete set of observables for single photon quantum states., Comment: 10 pages, 3 figures. Submitted to Journal of Optics

We show how a simple calculation leads to the surprising result that an excited two-level atom moving through vacuum sees a tiny friction force of first order in v/c. At first sight this seems to be in obvious contradiction to other calculations showing that the interaction with the vacuum does not change the velocity of an atom. It is yet more surprising that this change in the atom's momentum turns out to be a necessary result of energy and momentum conservation in special relativity.

As an alternative to conventional multi-pixel cameras, single-pixel cameras enable images to be recorded using a single detector that measures the correlations between the scene and a set of patterns. However, to fully sample a scene in this way requires at least the same number of correlation measurements as there are pixels in the reconstructed image. Therefore single-pixel imaging systems typically exhibit low frame-rates. To mitigate this, a range of compressive sensing techniques have been developed which rely on a priori knowledge of the scene to reconstruct images from an under-sampled set of measurements. In this work we take a different approach and adopt a strategy inspired by the foveated vision systems found in the animal kingdom - a framework that exploits the spatio-temporal redundancy present in many dynamic scenes. In our single-pixel imaging system a high-resolution foveal region follows motion within the scene, but unlike a simple zoom, every frame delivers new spatial information from across the entire field-of-view. Using this approach we demonstrate a four-fold reduction in the time taken to record the detail of rapidly evolving features, whilst simultaneously accumulating detail of more slowly evolving regions over several consecutive frames. This tiered super-sampling technique enables the reconstruction of video streams in which both the resolution and the effective exposure-time spatially vary and adapt dynamically in response to the evolution of the scene. The methods described here can complement existing compressive sensing approaches and may be applied to enhance a variety of computational imagers that rely on sequential correlation measurements., Comment: 13 pages, 5 figures

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum - atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics. This paper presents a phase space theory for fermion systems based on distribution functionals, which replace the density operator and involve Grassmann fields representing anti-commuting fermion field annihilation, creation operators. It is an extension of a previous phase space theory paper for fermions (Paper I) based on separate modes, in which the density operator is replaced by a distribution function depending on Grassmann phase space variables which represent the mode annihilation and creation operators. This further development of the theory is important for the situation when large numbers of fermions are involved, resulting in too many modes to treat separately. Here Grassmann fields, distribution functionals, functional Fokker-Planck equations and Ito stochastic field equations are involved. Typical applications to a trapped Fermi gas of interacting spin 1/2 fermionic atoms and to multi-component Fermi gases with non-zero range interactions are presented, showing that the Ito stochastic field equations are local in these cases. Solutions for spin 1/2 fermions either free or in lattices are found., Comment: 63 pages. 0 figures. Version 2. References updated

We introduce chiral rotational spectroscopy: a new technique that enables the determination of the orientated optical activity pseudotensor components $B_{XX}$, $B_{YY}$ and $B_{ZZ}$ of chiral molecules, in a manner that reveals the enantiomeric constitution of a sample and provides an incisive signal even for a racemate. Chiral rotational spectroscopy could find particular use in the analysis of molecules that are chiral solely by virtue of their isotopic constitution and molecules with multiple chiral centres. The principles that underpin chiral rotational spectroscopy could be exploited moreover in the search for molecular chirality in space, which, if found, might add weight to hypotheses that biological homochirality and indeed life itself are of cosmic origin. A basic design for a chiral rotational spectrometer together with a model of its functionality is given. Our proposed technique offers the more familiar polarisability components $\alpha_{XX}$, $\alpha_{YY}$ and $\alpha_{ZZ}$ as by-products, which could see it find use even for achiral molecules.

We recognise that Stegosaurus exhibited exterior chirality and could, therefore, have assumed either of two distinct, mirror-image forms. Our preliminary investigations suggest that both existed. Stegosaurus's exterior chirality raises new questions such as the validity of well-known exhibits whilst offering new insights into long-standing questions such as the function of the plates. We inform our discussions throughout with examples of modern-day animals that exhibit exterior chirality.

We analyse the properties of a strongly-damped quantum harmonic oscillator by means of an exact diagonalisation of the full Hamiltonian, including both the oscillator and the reservoir degrees of freedom to which it is coupled. Many of the properties of the oscillator, including its steady-state properties and entanglement with the reservoir can be understood and quantified in terms of a simple probability density, which we may associate with the ground-state frequency spectrum of the oscillator.

We respond to recent works by Bradshaw and Andrews on the discriminatory optical force for chiral molecules, in particular to the erroneous claims made by them concerning our earlier work.

We determine the shared information that can be extracted from time-bin entangled photons using frame encoding. We consider photons generated by a general down-conversion source and also model losses, dark counts and the effects of multiple photons within each frame. Furthermore, we describe a procedure for including other imperfections such as after-pulsing, detector dead-times and jitter. The results are illustrated by deriving analytic expressions for the maximum information that can be extracted from high-dimensional time-bin entangled photons generated by a spontaneous parametric down conversion. A key finding is that under realistic conditions and using standard SPAD detectors one can still choose frame size so as to extract over 10 bits per photon. These results are thus useful for experiments on high-dimensional quantum-key distribution system., Comment: 18 pages, 6 figures

We show how it is possible to operate end-to-end relays on a QKD network by treating each relay as a trusted eavesdropper operating an intercept/resend strategy. It is shown that, by introducing the concept of bit transport, the key rate compared to that of single-link channels is unaffected. The technique of bit transport extends the capability of QKD networks. We also discuss techniques for reducing the level of trust required in the relays. In particular we demonstrate that it is possible to create a simple quantum key exchange scheme using secret sharing such that by the addition of a single extra relay on a multi-relay channel requires the eavesdropper to compromise all the relays on the channel. By coupling this with multi-path capability and asynchronous quantum key establishment we show that, in effect, an eavesdropper has to compromise all relays on an entire network and collect data on the entire network from its inception.