Manipulating the quantum information of the radial modes of trapped ions: Linear phononics, entanglement generation, quantum state transmission and nonlocality tests
Abstract
We present a detailed study on the possibility of manipulating quantum information encoded in the “radial” modes of arrays of trapped ions (i.e., in the ions’ oscillations orthogonal to the trap’s main axis). In such systems, because of the tightness of transverse confinement, the radial modes pertaining to different ions can be addressed individually. In the first part of the paper we show that, if local control of the radial trapping frequencies is available, any linear optical and squeezing operation on the locally defined modes – on single as well as on many modes – can be reproduced by manipulating the frequencies. Then, we proceed to describe schemes apt to generate unprecedented degrees of bipartite and multipartite continuous variable entanglement under realistic noisy working conditions, and even restricting only to a global control of the trapping frequencies. Furthermore, we consider the transmission of the quantum information encoded in the radial modes along the array of ions, and show it to be possible to a remarkable degree of accuracy, for both finitedimensional and continuous variable quantum states. Finally, as an application, we show that the states which can be generated in this setting allow for the violation of multipartite nonlocality tests, by feasible displaced parity measurements. Such a demonstration would be a first test of quantum nonlocality for “massive” degrees of freedom (i.e., for degrees of freedom describing the motion of massive particles).
1 Prologue: the promise of alternative continuous variable degrees of freedom
The last decade saw a boom in the development of experimental capabilities available for quantum information processing. The ability to manipulate the information of discrete variables encoded in polarisation, spin and internal atomic degrees of freedom has by now reached very high standards. On the other hand, the control and manipulation of continuous variable (CV) quantum information is still almost exclusive to light fields in quantum optical settings. Even though purely optical systems rely on well established tools and are the natural choice for communication tasks over long distances, they also suffer from significant drawbacks. Notably, the entanglement generation in such systems is strongly limited by the efficiency of parametric processes in nonlinear crystals; moreover, ‘static’ optical degrees of freedom – i.e. light resonating in cavities – are seriously affected by losses and decoherence over their typical dynamical time scales.
Such limitations motivate the question of whether the full potential of infinite dimensional Hilbert spaces could be better harnessed by “massive” (i.e., related to the position of a massive particle) CV degrees of freedom. Of course, to compete with the so far very successful quantum optical toolbox, such degrees of freedom would have to allow for a range of coherent manipulations at least as exhaustive as what quantum optics currently permits. Besides, to be considered advantageous over quantum optical systems, such degrees of freedom should allow for notable improvements in the generation of quantum entanglement and squeezed states under realistic working conditions.
In the present paper, we argue that trapped ions meet both such requirements, and present an extensive study to substantiate this argument. In particular, we shall focus on the radial motion of trapped ions, that is on oscillations along a direction orthogonal with respect to the array of ions. These oscillations are described by continuous variable quantum degrees of freedom which we will refer to as radial modes [1, 2]. Radial modes have attracted considerable interest in the last few years, mostly in view of the fact that they allow for a tighter confinement (which also permits one to define the phonons locally).
The paper is organised as follows. After having introduced the description of the physical system (Sec. 2), in Sec. 3 we demonstrate that any linear optical and squeezing operation can be obtained for radial modes of trapped ions by controlling the individual radial trapping frequencies, indicating that trapped ions can at least match the processing capabilities possible for light modes. In Sec. 4 we show that, even restricting to cases where only global control of the trapping potentials is possible, such systems are actually apt to outperform optical modes in the generation of entanglement, both bipartite and multipartite. As applications we consider, in Sec. 5, the propagation of quantum information along the array of ions, at both qubit and continuous variable levels and, in Sec. 6, the violation of multipartite non locality tests, and show them to be within the reach of current technology. Radial modes will thereby turn out to be promising not only for information processing but also as probes of fundamental physics.
2 The trap
We shall consider the radial modes of ions of mass and charge in a linear Paul trap [3]. Let and be the position and momentum operators associated to the radial degree of freedom of the th ion, which is trapped in the radial direction with angular frequency (see Fig 1). In the following, the longitudinal trapping frequency will be the unit of frequency and will set the unit of length as well (equal to , where is the dielectric constant); also, we shall set , so that all the quantities will be dimensionless. The Coulomb interaction affects the local radial oscillation frequencies: for convenience, let us then define the ‘effective’ local radial frequencies
being the equilibrium positions of the ions [3].
Rescaling the canonical operators according to
,
, and grouping them
in a vector of operators
, allows one to express
the global Hamiltonian of the system
in the harmonic approximation
(1) 
where is a diagonal matrix: , while the potential matrix has diagonal entries and offdiagonal entries for .
Let us recall the canonical commutation relations , where the matrix has entries for , and that Gaussian states are defined as states with Gaussian characteristic functions: a Gaussian state is thus uniquely determined by its “covariance matrix” (CM) , with entries given by
(2) 
and by the vector of first moments , with components [4, 5].
The ground state of Hamiltonian is a Gaussian state with a block diagonal CM , where
and vanishing first moments.
Finally, let us remember that the evolution for the time of an initial Gaussian state with CM under a quadratic Hamiltonian (where is any symmetric matrix) is a Gaussian state with CM given by
(3) 
for .
2.1 Dissipation
In our study, we will take into account the decoherence of the radial modes in an environment of phonons with temperature and ‘loss rate’ (for simplicity assumed to be the same for every mode). Under such conditions, the evolution of the ions’ state at frequencies is described by the following master equation in interaction picture [6]
(4) 
where the number of phonons in the radial mode is given by , according to Bose law ( being Boltzmann constant) and . In accordance with the experimental terminology, we will also refer to the quantity as the ‘heating rate’, essentially representing the rate at which thermal phonons are injected into mode .
Eq. 4 preserves the Gaussian character of the initial state and can results, for the CM of Gaussian states, into the following equation:
(where , ), with solution:
3 Linear phononics and beyond
In this section, we shall assume that the trapping frequencies ,
and thus , can be controlled locally and changed
suddenly (i.e., much faster than ).
Such a local control may be achieved by building small, local radial electrodes, by
adding local optical standing waves,
(where is the antisymmetric symplectic form, defined
at page 2), to preserve the canonical commutation
relations. Notice that such a class of operators includes squeezing
transformations.
Let us first remark that any symplectic operation on a system of many canonical degrees of freedom (“modes”) can be decomposed into a combination of generic singlemode symplectic transformations and twomode rotations (“beam splitters”, in the quantum optical terminology) [14, 15]. This fact follows from the Euler decomposition of symplectic operations [14] and from the possibility of decomposing energy preserving operations into a network of beamsplitters between pairs of modes [15]. It is therefore sufficient for us to establish the possibility of performing these two subclasses of operations (singlemode symplectic transformations and twomode beam splitters) on our system of ions by manipulating the local frequencies. In turn, again because of Euler decomposition, singlemode operations can always be reduced to combinations of squeezings and phaseshifts.
3.1 Single mode operations
In what follows we
assume that the original effective frequencies of the ions are different but
commensurate, as given by, say, , and that is
large enough so that interaction between any two ions is
suppressed when the relevant coherent manipulation sets off.
(5) 
The first and last matrix of this decomposition correspond to ‘squeezing’ operations in the quantum optical terminology, whereas the second factor is known as a ‘phase shift’ (i.e., a rotation in the singlemode phase space). Combinations of squeezings and phaseshifts make up any possible singlemode symplectic operation: we thus need to show that such operations can be implemented individually on any ion of the system in a controllable manner (as pointed out above, this is sufficient because of the Euler decomposition of single mode operations [14]).
Phaseshift
To realise a phaseshift operation on the th ion, without any squeezing, it is sufficient to change the frequencies of all the other ions in the same way, such that, in the notation defined above, and for ( being the factor by which the frequency of ion is multiplied by manipulating the local trapping potential). As apparent from Eq. (5), after a time from the change one has for : all the ions which undergo a change in trapping frequency are back to their initial state (let us recall that by assumption, so that after the central rotation of Eq. (5) reduces to the identity and the two opposite squeezing transformations just nullify each other). On the other hand, the transformation will be equal to the central rotation (with no squeezing, as is kept equal to ) of angle : the oscillation of the th ion will have acquired such a phase, analogous to the “optical phase” of light modes. If the frequencies are switched back to the original values after a time , the net effect of the evolution is then analogous to an ‘optical’ phaseshift of angle on the ion .
Squeezing
In order to squeeze the state of ion , leaving all the other ions
unaffected, one can conversely
change only the pertinent frequency, so that
and for . Then, after a time period
, all the other ions will have returned to the
initial state, while ion will be squeezed and phaseshifted
according to Eq. (5).
The phaseshift can always
be arbitrarily corrected by applying the strategy to obtain “pure” phase shifts
described in Sec. 3.1.1, which conclusively shows that
any linear operation on a single mode can be implemented by controlling
the local trapping frequencies.
The degree of squeezing achievable with such a strategy depends crucially on the phaseshift operation since, as shown by Eq. (5), the two squeezing operations act along orthogonal directions and are the inverse of each other. In the case for , the two squeezings act effectively along the same phase space direction: in this instance the phaseshift can be balanced by a counterrotation of in phase space and the final squeezing operation is a diagonal matrix given by . In principle, the value of is only limited by the stability of the system and the breakdown of the harmonic approximation [2], occurring when the squeezing is comparable to the ratio between the size of the wave packet and the distance between the atoms. In actual experiments, the considered setup would thus allow for . Notice that such values are by far out of reach in optical systems, where squeezings corresponding to were recently reported [16]. Also notice that, by placing the ions inside cavities, the squeezing of the massive degrees of freedom could be transferred to light, so that radial modes could act as an effective source of squeezing (and potentially even entanglement) for optical systems as well [17].
Beamsplitters
To extend our proof to any linear operation, on any number of ions, let us now turn to ‘beamsplitting’ operations between the radial modes of any two ions in the array. Such operations are achieved by bringing the two modes in question (hereafter labeled by and ) to the same frequency , so that the Coulomb interaction between them is no longer suppressed. Switching to the interaction picture, one has the following coupling Hamiltonian between the two modes:
where the ladder operators are
defined as and is an entry of the matrix
defined in Eq. (1).
If the frequency is
sufficiently large
the rotating wave approximation reliably applies and the rotating terms can be
neglected to yield
This Hamiltonian realises exactly the desired beam splitterlike evolution, resulting into a symplectic transformation which mixes with and with (rotating such pairs equally, by the angle ). For instance, a ’’ beam splitter is achieved after a time . Since the interaction requires a change of the local frequencies it includes automatically in it a local operation, which may however be corrected before or after the ‘beamsplitting’ procedure, as detailed above.
Summing up, the combined arguments presented in Secs. 3.1.1, 3.1.2 and 3.1.3 show that any “linear optical” operation, including squeezing operations and entangling operations acting on multiple different modes, can be implemented for radial modes of trapped ions by properly tuning the frequencies of the radial microtraps of the individual ions.
Linear optical operations, complemented by displacements (see Sec. 3.2), correspond to all the unitary transformations that preserve the Gaussian character of the initial state. Therefore, all the developments based on Gaussian states in quantum optics, in particular concerning entanglement manipulation [4] and information protocols [18], can be carried over to radial modes of ion traps if local control is achieved. Arguably, the framework we have outlined for the implementation of linear optical operations on the local oscillations of distinct ions, to which one may refer as “linear phononics”, could offer further practical advantages over its quantum optical counterparts, essentially related to the static nature of the quantum information processed (for instance, ‘mode matching’, which hampers substantially the implementation of linear optics, is not a problem here).
3.2 Further manipulations and measurements
Beside the linear operations treated so far,
implemented through varying trapping potentials and Coulomb interactions,
the motional degrees of freedom of trapped ions allow for other controlled manipulations,
some of which take advantage of the coupling between motion and the internal degrees of freedom
that can be engineered by applying standing or travelingwave pulses to the ions.
In order to give a complete account of the possibilities offered by radial modes, let us here review
such strategies, and briefly comment about their applicability to
radial modes. As a general remark let us mention that, because of
the tighter confinement they allow for, radial modes easily meet
the LambDicke condition (depending on the width of the ground
state’s wavepacket), which means that the coupling with the internal
degrees of freedom can be tailored to a high degree of accuracy
(generally better than for longitudinal modes).
Displacement operations, which shift the operators by real numbers, can be realized in several ways: by classical driving fields, by standing waves, or by shifting the radial equilibrium positions of the ions [19]. In particular, transverse driving fields could be applied to displace the radial modes, with no particular hurdles. In the following, the unitary displacing the canonical operators of mode by, respectively, and will be denoted by .
Notably, even nonGaussian states can be engineered in this setup with relative ease (with respect to quantum optics, where nonGaussian manipulations require higher order nonlinearities, usually extremely weak), either by entering the nonlinear regime of the Coulomb interactions or by coupling the internal degrees of freedom of the ions. The experimental realisation of a crossKerr coupling for the longitudinal oscillations of trapped ions has been recently reported [2].
The coupling to internal degrees of freedom also allows for Gaussian and non Gaussian measurements on individual ions. The measurement of quadrature operators, corresponding to homodyning, was proposed in [20, 21, 22]. Quantum nondemolition measurement of local number states and parity could be measured as well, by applying the scheme suggested and realised for cavity QED in [23], based on the dispersive coupling of the number of oscillations to two internal levels of the ion. Quite remarkably, such a scheme would allow one to measure the phonon number’s parity on a single copy of the state and run of the apparatus. As we will see in Sec. 6, this possibility is consequential for the violation of Belllike inequalities. Notice also that such nonGaussian measurements pose a considerable technological challenge for light fields, where resolving photon numbers with high detector efficiency is still daunting despite recent progress [24, 25]. On the other hand, the schemes recalled above are bound to be comparatively slow, requiring to wait for half a period of coherent interaction between motions and internal levels and then subsequent readout of the internal levels by, e.g., fluorescence. When internal degrees of freedom are involved, radial modes require longer times, roughly on the order of tens of microseconds, but achieve remarkable precision.
Finally, concerning the preparation of initial states for coherent manipulations, let us remind that cooling of ion oscillations to their ground state can be achieved very efficiently by sidebandRaman pulses [26, 27] (or more recent variations over such a strategy [28, 29]). Note, since in the described scenario the initial local potential is much larger than the interaction, simultaneous cooling of all the chain could be done by cooling each ion to its local ground state, which is to a good approximation the global ground state. The local single excitations on such ground states may be prepared by addressing the first ion with a proper sequence of blue and red sideband pulses, as detailed in [19].
In the remainder of the paper we shall demonstrate the potential of linear operations on radial modes in specific applications, with a particular focus on settings fully accessible to current experiments.
4 Entanglement generation
The set of operations described in Sec. 3, including local squeezing and twomode beam splitters between distant modes, allows for the generation of entanglement between the radial modes of two ions in the chain. In fact, as is well known, two singlemode states squeezed along orthogonal quadratures, entering a beam splitter, give rise to an entangled state for the outgoing modes (this is a standard procedure to create CV entanglement in quantum optics). In principle, for the system of trapped ions under examination, the local control of the transverse trapping frequencies allows for the creation of tailored entangled states, where the entanglement can be build up between any two ions in the chain. Moreover, and possibly even more intriguingly, the possibility of controlling a system of parties (the ions) all constantly interacting with each other, paves the way for the creation of multipartite entanglement between the distinct ions.
In this section we show that unprecedented degrees of bipartite entanglement, as well as interesting and robust multipartite entangled states, can be obtained in settings accessible to current experiments, starting from a ground state of the system and requiring only global control of the trapping potential (where, i.e., the trapping frequencies are the same for all ions at any time, and are only changed simultaneously). Further, we will briefly discuss the possibilities a local control of the frequencies that would open up for the controlled dynamics of multipartite entanglement.
The specific situation we shall address starts off from the ground state of Hamiltonian [see Eq. (1)] – with all frequencies being equal, i.e. for – as the initial state. Next, the trapping frequencies are changed to the common value , so that the state will not be stationary anymore under the modified Hamiltonian. For large , the initial state contains very little entanglement but, if the frequencies are suddenly changed, entanglement builds up during the subsequent evolution (see [30] for an analogous scheme in chains of nanomechanical oscillators). Entanglement between different subsystems will be quantified by the logarithmic negativity , where stands for the trace norm of the ‘partially transposed’ density matrix of the subsystem [31, 5]. Different partitions will be denoted by superscripts referring to the ions: for instance, the logarithmic negativity between ions  and  will be denoted by . The intuition behind this entanglement generation method is that operating faster than the speed of sound in the system, i.e., essentially, faster than the inverse of the energy gaps, is analogous to operating locally. Thus, by changing the local potentials (in the same way for each ion) fast enough one generates local squeezing, which is then ‘converted’ into entanglement by the timeevolution through the harmonic coupling between the ions, analogous to a set of beam splitters.
The case portrayed in Fig. 2 represents four ions starting from the ground state for , which then evolves under the frequency for (for a transverse trapping frequency ). As can be seen, the global state gets entangled under any possible bipartition of the four modes, a situation which is referred to in the literature as “complete inseparability” [32]. A more extensive inspection shows that this is a general property of the Hamiltonian at hand, even for larger number of ions. Of course, as shown in Fig. 3, complete inseparability also occurs for three ions: in Sec. 6 we shall see how the multipartite entanglement for three ions analysed in Fig. 3 could serve to violate Belllike inequalities. Our plots also show that such an entanglement exhibits considerable resilience under realistic heating rates. In this respect, notice that the heating rate for the dashed (green) curved plotted in the graphs is , and that heating rates as low as have been recently observed in ion traps (even though for a single ion only [33, 34]): robust multipartite entanglement (between and of logarithmic negativity) between the ions can thus be created. Such degrees of entanglement are inconceivable for multipartite systems in photonic systems where, furthermore, the manipulation of several modes tends to become awkward due to the increasing number of modematching conditions to be fulfilled.
In the plots, we also report the entanglement of pairs of modes in the traps for four, three
and two ions.
For bipartite entanglement as well, as shown in Fig. b,
the degree of robust entanglement achievable is by far beyond the maximal values
obtained in quantum optical systems,
Also, in the traps with more than two ions, the dynamics of the entanglement between pairs of modes displays a ‘monogamous’ behaviour [36]: for four ions, when the entanglement between ions and fades, the entanglement between and starts raising and then is replaced by entanglement between and , which in turn fades out before the revival of the quantum correlations between and . The entanglement between and – strongly favoured by the fact that such ions are both nearest neighbours and at perfect resonance with each other (not true in general, as the Coulomb corrections on the trapping frequencies depend on the position in the trap) – is always positive but, significantly, its peaks and dips follows those of the entanglement between and , as one would heuristically expect by virtue of monogamy. As reminded above, the reported values of and represent the ‘bare’ trapping frequencies, not taking into account the corrections due to Coulomb repulsion: therefore, the ions are not all at resonance during the evolution. Now, if one assumes that local control were available, one could correct for such a mismatch by applying different bare frequencies to ions in different positions along the trap’s longitudinal axis. We applied such corrections in the four ions case considered above and found that, quite interestingly, one can spread correlations more evenly in this way and get closer to the ultimate bounds imposed by monogamy: after (all the parameters being the same aside from the corrections) one ends up with a state where each radial mode is individually entangled with all the other radial modes (a situation which never arises without corrections, see Fig. 2).
The continuous variable entanglement between two radial modes of the ions could be swapped to light if cavities were added to the setup, with the potential to achieve unprecedented degrees of optical entanglement, as the parametric processes are definitely outperformed by the strategy outlined above based on the trapping frequency’s control: this possibility will be the subject of a detailed investigation in a forthcoming paper [17].
5 Propagation of quantum information
The locally defined radial modes of the ions, interacting via the Hamiltonian (1), are an example of harmonic chain, where the coupling between different oscillators can be, to a very good extent, controlled. The propagation of quantum information in such a setting has been proposed and analysed theoretically in [37], but not yet realised in practice since mechanical oscillators have yet to enter the quantum regime (due to the technical difficulties still encountered in controlling both the couplings and the decoherence and dissipation of chains of nanooscillators [38]). Trapped ions could thus provide a very promising alternative to implement harmonic chains, realize quantum data buses [39] and test the results predicted by theoretical studies. More generally, such a demonstration would be a further application of ion traps as quantum simulators, as already proposed in several past studies [40, 41]. In this Section, we shall present an overview of the possibilities offered by the radial modes as harmonic chains for the propagation of quantum states, with quantitative investigations, in scenarios similar to those detailed in the preceding sections.
5.1 Transmission of twodimensional quantum states



The transmission of quantum information encoded in finite dimensional quantum states (most often qubit states) through a chain of interacting quantum systems has drawn much attention in recent years [42], as such systems are envisaged as possible quantum buses, linking different parts of future quantum processors. However, despite considerable efforts, finding systems where such a transmission can be realised with sufficiently small level of noise has proven to be very challenging. Therefore, good practical candidates for such tasks are still of interest. Proposals in this context have so far focused on the transmission of finite dimensional states in chains of interacting (effective or proper) ‘spins’. Here, instead, we consider qubit states encoded in the bosonic Hilbert space of the radial modes of trapped ions and, guided by the results of the Section 3, we show that such modes are able to send qubit states through chains of trapped ions.
As a preliminary remark, let us notice that, if the ratio between the radial and the longitudinal
trapping frequencies is very large, , then
the effect of the Coulomb interaction on the Hamiltonian governing the radial modes
is negligible
Neglecting, for the time being, thermal noise (see below), one has that the state can be sent perfectly through the chain. Therefore, the overall fidelity of the transmission depends solely on the fidelity with which can be sent from the first to the last ion. The amplitude of such a transfer can be determined analytically by tracking the evolution of the initial operator (creating the excitation in the first ion) in Heisenberg picture. Because the system is harmonic the evolution of results in a linear combination of field operators, which can be determined with standard techniques. We could thus consider different situations. The histograms we plot show the probability amplitude for finding a single excitation at different sites in the chain. The resulting transmission fidelity (averaged over the Haar measure of a single qubit space) is given by , monotonically increasing with . We will consider a chain of ions and an initial ground state for trapping frequencies .
Fig. a. shows the case where the trapping frequencies are left unchanged after the initial preparation. Because the interacting terms, while very small, are still present, the excitation propagates because the local mode of the first ion is not exactly a normal mode of the system. After an evolution time , of the probability amplitude is transferred to the final state. However, the propagation is rather dispersive, as each mode is on resonance with all the others.
In order to boost the probability of transmitting the excitation, sender and receiver can switch the local trapping frequencies of their respective ions from the initial value down to a common value : in this way they can realise the beam splitting operation at distance described in Sec. 3.1.3 which, in principle, would allow for perfect swapping. This specific example thus also illustrates our previous general discussion and shows with which precision and over which operating times are linear operations actually possible over a chain of ions. As one can see from Fig. b, where has been assumed, the beam splitting operation is virtually perfect: all the probability amplitude is gradually transferred to the final ion, while the other ions are never involved in the process. This quality in the transfer comes at the expense of transfer time: over ions the beam splitting operation takes roughly an order of magnitude more () than the imperfect transfer considered in Fig. a. This is simply due to the fact that, by keeping the middle ions on resonance, one takes advantage of the interaction between nearby ions, which are clearly stronger (as the interaction decays like the cube of the distance) and propagate the excitation through the chain. For given trapping frequencies, the time needed to achieve the beam splitter between radial modes at the two end of a chain of ions scales very accurately as . Notice also, as a side remark, that since the Coulomb corrections to the local trapping frequencies are symmetric with respect to the longitudinal centre of the trap, the transfer between extremal ions is somewhat favoured in practice, as equal trapping potentials will result in equal effective trapping frequencies for such ions.
Finally, in Fig. c, we consider a ‘compromise’ between the two cases analysed above: all the frequencies are changed to the frequency obtaining, after a time , a transferred probability amplitude of . It is important to remark that the slower beam splitting operation, where the quantum information does not disperse through the chain at all, is not only more precise than the other two options examined, but also much more stable with respect to imperfections in the allowed interaction times.
Let us now turn to inspect the effect of decoherence:
clearly, thermal phonons are deleterious for the transfer of information encoded in
single excitation sectors. The analytical estimates for the transfer times we just presented allow us to
determine the restrictions on the heating rates that would allow such systems to transfer information
in practical situations. Assuming a longitudinal trapping frequency of and a
(less ambitious) chain of ions, one would realise a beam splitting operation (perfect transfer)
between extremal ions in . In such a case, heating rates as low as
would be needed for a coherent transfer to take place.
For such heating rates, the action of the thermal phonons on the fidelity between initial and final state
can be assumed to be linear resulting, after average over the Haar measure,
into a mean fidelity .
5.2 The role of the internal degrees of freedom
In the preceding section, while discussing the transfer of finite dimensional quantum information, we started off from a situation where a qubit state is encoded in the first two number states of the first ion. As already mentioned above, one of the most expedient ways to create such a finite dimensional state for the motional degrees of freedom is creating the desired state for the ‘internal’ degrees of freedom of the ion (embodied by its internal electronic levels) and then coupling them to the local radial modes to achieve the swapping. In order to provide the reader with a more comprehensive treatment and give him/her a flavour of the way the internal degrees of freedom enter in such dynamics, we briefly give account of a specific example where the internal degrees of freedom are involved throughout the whole transmission of quantum information.
We shall consider a trap with four ions, with for . To keep the example down to earth, we will not assume any capability of local control nor any change during the evolution for the radial frequencies here. The only manipulation required will be the switching on and off of the coupling between the local phonons and the two levels of each ion where the initial qubit state is encoded. Denoting such levels and , and setting , we shall assume the following interaction Hamiltonian for each ion :
which describes appropriately the coupling realised in experiments (see, e.g., [19]). For simplicity, the interaction strengths will be all set to (notice that, because the oscillators’ frequency are set to , the interaction reduces, up to a very good approximation, to a rotating wave one). As initial state, at , we assume a Bell state between the internal degrees of freedom of ions and , and the ground state for the remainder f the system:
where is the ground state of the radial modes of the ions. As a signature for the transmission of quantum information, we will consider the evolution of the entanglement between the internal and motional degrees of freedom of ions (initially entangled) and ions (which gets entangled through the process), which is reported in Fig. 6 in tems of logarithmic negativity (the logarithmic negativity between internal degrees of freedom of ions and is denoted ). The dynamics is then split into three stage: from time to the internal and motional degrees of freedom of ions and are coupled and the entanglement is dynamically swapped from the former to the latter. Next, from to , the coupling is switched off and the entanglement propagates through the chain via the motional degrees of freedom, and the radial modes of ions and gets entangled. Finally, from to , the entanglement is swapped to the internal degrees of freedom of ions and . One can see that, though the quantum information partially disperses under such conditions (essentially due to the stringent restrictions we put, in this instance, on the control of the local frequencies), the process is capable of transferring the entanglement from the internal degrees of freedom of ions and to those of ions and through the radial motional degrees of freedom.
5.3 Propagation of continuous variable states and entanglement
Clearly, quantum information (and entanglement) can be propagated through the chains under examination also at a continuous variable level, when populating the whole infinite dimensional Hilbert space. The study of information and entanglement propagation over harmonic chains has been proposed and discussed in detail in [37, 39]. Here, we shall investigate cases of such propagations along the chains of ions we are considering, addressing specific instances which can be realised in the laboratory. Our aim is pointing out at what degree and under what conditions can the theoretical schemes based on harmonic chains be implemented on radial modes of trapped ions.
As argued in [37], the capacity of transmitting quantum states between distant parties is closely related to the capacity of transmitting, or “swapping”, entanglement between them. In fact, the latter also critically requires the transfer to happen coherently through the whole process. In view of this fact we will limit ourselves to consider the propagation of CV entanglement (rather than the fidelity between sent and received quantum states). Also, in order to focus on a feasible scenario, we shall consider a common initial condition: the chain of ions starts off from the (completely separable) ground state for and, then, entanglement between the first and the second ion is created, as described in Sec. 4, by switching their frequencies to the same frequency . From now on the first ion is left offresonance at frequency (and thus effectively isolated from the chain), and the aim is transmitting its initial entanglement with the second ion through the chain over to the final th ion. Two options will be examined:

the entanglement is swapped by a chain of beam splitters between neighbouring ions (achieved by setting such neighbouring ions at the original frequency while putting the others offresonance);

the entanglement is swapped directly by a beam splitter between the ions at the far ends of the chain.
Notice that in case i) the beamsplitters operate at the original frequency so that no local squeezing takes place, as evident from Eq. (5). Local squeezing would affect the entanglement between the modes but also, crucially, alter the original state to be transmitted and is thus undesirable in the present context.
The two strategies i) and ii) are compared in Fig. (b) for a chain of four ions. Such a small system already permits one to highlight all the essential features of the two strategies. As apparent, a complete dynamical swapping of the entanglement can be achieved in both cases, with remarkable accuracy. A chain of beam splitters allows for a faster transfer, as one should expect since the coupling between distant ions is inversely proportional to their cubed distance: in general, the time needed to send information across ions by a ‘relay’ of beam splitters roughly scales like , whereas the time needed to achieve a beamsplitting operation between first and last ion scales like . A ‘relay’, taking less time, is thus less sensitive to decoherence and dissipation. On the other hand, a single beamsplitting operation between distant ions only requires a single adjustment of the trapping frequencies (after the initial entanglement is created), whereas a chain of beamsplitters requires manipulations, which would be possibly difficult to master in practice and could involve more errors and imperfections. Therefore, we have investigated the effect of imperfections in frequencies and evolution times on the transfer for both cases: the series of beamsplitters turns out to be more robust. In the instance depicted in Fig. b, the transfer by a series of beamsplitters is left virtually unaffected by uncertainties of on frequencies and on operating times ( is the longitudinal trapping frequency) whereas, on average, the entanglement transferred by a single beam splitter is reduced to to ebits of logarithmic negativity (out of ebits initially present between ions and ) by the same imperfections. Summing up, strategy i) proved to be more reliable and faster (and thus less subject to decoherence and dissipation) and is hence to be preferred for the transmission of CV quantum information over chains of ions. Considering once again the example of Fig. b, for a longitudinal frequency , an effective coherent transfer would take about and thus require heating rates around to be carried out effectively.
6 Nonlocality tests
The presence of strong Gaussian multipartite entanglement highlighted in Sec. 4 can be experimentally demonstrated and put to use in testing quantum nonlocality. Central to this endeavour is the capability of performing nonGaussian measurements on the motional state of the ions, pointed out in Sec. 3.2. In particular, we already recalled that phononnumer parity measures are possible on single copies of the system, and so are displacement operations. In this section, we will explore the possibility of violating multipartite Belllike inequalities (the socalled “Klyshko” inequalities [43]) by measuring “displaced parity” observables, as proposed for generic CV systems by Banaszek and Wodkiewicz [44]. Although bound to be subject to the locality loophole (considering that the distances between the ions are typically of the order of ), such an experiment would be a remarkable test of quantum nonlocality with massive particles, which is still lacking.
To fix ideas and address a situation within the reach of current experiments, we shall study the test on the threemode Gaussian state whose entanglement is described in Fig. 3, setting the evolution time to and the heating rate to (resulting from a coupling to the bath and a temperature ), whose CM will be denoted by . The family of (nonGaussian) local, bounded, dichotomic observables for the displaced parity test is given by , where and are respectively the displacement and number of phonons operators of ion . The three observers, pertaining to the three ions, randomly apply two different displacements [ and ] on their ions and then measure parity locally. Such a measurement is clearly nonGaussian, and allows one to violate Bell inequalities with Gaussian states. The expectation value of the operator is simply proportional to the Wigner function of the composite system evaluated in the point : [45]. For a threemode Gaussian state with covariance matrix one has
The BellKlyshko inequality finally reads:
(6)  
Quantum mechanics allows for .
Fig. 8 show a region in the space of displacements where the violation of the inequality is substantial and remarkably stable. It is apparent that, for such choices of displacements, the tolerable error on the displacement operation needed to maintain the violation is around , which is within reach of current experimental capabilities. Also, as shown in the plot on the right, heating rates of are still compatible with a violation of the inequalities.
This preliminary study reveals very promising perspectives concerning the violation of Bell inequalities with massive degrees of freedom. Even if subject to a locality loophole, this endeavour would still stand out as a major, not yet probed, testing ground for fundamental quantum mechanics [46], and epitomises the considerable promise radial modes hold for quantum information and fundamental investigations alike.
7 Summary and outlook
We demonstrated how the local control of the trapping frequencies would allow one to reproduce any linear optical manipulation on radial modes of trapped ions. Drawing from previous studies on similar settings, we pointed out that phonon detection and homodyne detection as well as the implementation of nonGaussian operations is possible in this setting. Next, we emphasized that, even restricting to global control, such manipulations enjoy a high efficiency in entanglement generation and a considerable resilience in the face of currently achievable dissipation rates. We then made manifest that radial modes could be used for the transmission of quantum information, stored in finite dimensional subspaces of the bosonic Hilbert space as well as in the full continuous variable domain. Finally we showed that, through (achievable) measurements of the displaced parity operator, Belllike inequalities can be violated under realistic decoherence and dissipation rates. As a rule of thumb, applying to systems with ions or less and longitudinal trapping frequencies , our study indicates that heating rates around are sufficiently low for coherent manipulations and robust entanglement generation in the continuous variable regime, whereas nonlocality tests and transmission of finitedimensional quantum dimensional information (stored in single phonons) are more delicate, requiring heating rates around to be carried out efficiently.
In the light of the above, the experimental pursuit of the presented programme holds considerable promise, concerning both technological developments, such as the storage and manipulation of quantum information or the efficient generation of entanglement, and tests of fundamental physical aspects, as in the nonlocality test for massive degrees of freedom here discussed. As a first step to push this analysis further, a study is currently under way to examine the possibility to swap the remarkable entanglement produced in the traps to light modes, to ultimately beat parametric processes in the generation of quantum optical CV entanglement [17].
Footnotes
 Notice that, in the harmonic approximation (i.e., at second order in the ions’ displacements) the coupling between radial modes and longitudinal ones vanishes. Fourth order couplings are around times smaller in the considered experimental conditions and will thus be safely neglected.
 Currently, standing waves realize, at most, trapping potentials of about in experiments where the internal degrees of freedom are controlled. However, the manipulation of radial modes can tolerate much higher scattering rates than that of internal degrees of freedom, and would thus allow for much higher trapping frequencies.
 In the literature, squeezing transformations have a tendency to be excluded from the class of “linear” transformations, because their implementations require ‘nonlinear’ (third order) interactions. However, the effective evolution of the modes of interest (‘signal’ and ‘idle’, usually) is actually linear in the sense specified above (linear evolution of the vector in Heisenberg picture), and we will thus generally refer to ‘linear’ transformations as including squeezing.
 The linear scaling of the coupling is not a stringent requirement. Because the coupling between ion and falls off like , one can safely assume, say, a ‘triangle’ profile (alternately increasing and decreasing) for the couplings in long chains of ions. Note also that, for a linear scaling, the population of the levels of ion due to the interaction with roughly scales as : a frequency step is already enough to make the effects of all the interactions essentially negligible.
 In fact, arbitrary adjustment of the phase allows for “pure” squeezing operations along the phase space directions and , and thus implies the possibility to construct any linear operation, because of the symplectic Euler decomposition previously discussed.
 This always holds at the trapping frequencies we shall consider in specific examples, going from to few tens of , for which stability around are achievable.
 The price to pay for such an accuracy is longer operation times, which could be ultimately reduced if stronger lasers became viable.
 To the best of our knowledge, the highest measured value for the logarithmic negativity in optical systems (inferred from state reconstruction) is [35]. A simple evaluation also shows that, exploiting the degree of squeezing reported in [16] and assuming perfect beamsplitting operations, one could achieve at most .
 See page 2 and notice that, quite remarkably, the ratios does not depend on the trapping frequencies nor on the masses of the ions but only on the total number of ions (see [3] for details).
 The effect of losses and thermal phonons for small (so that over the interesting timescales) can be easily estimated by letting the master equation Eq. (4) for act at first order on an otherwise perfectly transferred generic state , and by setting . The fidelity between initial and final states (corresponding to the overlap, for pure initial state) can be then determined and averaged over the Haar measure of a single qubit Hilbert space, to obtain the mean value .
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