Optical Generation of Ultra-Large-Scale Continuous-Variable Cluster States

Optical Generation of Ultra-Large-Scale Continuous-Variable Cluster States

Shota Yokoyama, Ryuji Ukai, Seiji C. Armstrong,
Chanond Sornphiphatphong, Toshiyuki Kaji, Shigenari Suzuki,
Jun-ichi Yoshikawa, Hidehiro Yonezawa,
Nicolas C. Menicucci, Akira Furusawa

Department of Applied Physics, School of Engineering, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Centre for Quantum Computation and Communication Technology,
Department of Quantum Science, The Australian National University,
Canberra, ACT 0200, Australia
School of Physics, The University of Sydney, NSW, 2006, Australia

To whom correspondence should be addressed; E-mail: akiraf@ap.t.u-tokyo.ac.jp

Entanglement is a uniquely quantum phenomenon and a physical resource that can be used for quantum information processing 1. Cluster states are multipartite states with an entanglement structure that is robust to local measurements 2 and can be used for measurement-based quantum computation 3. To date, the largest optically generated cluster state consists of 8 entangled qubits 4; 5, while the largest multipartite entangled state of any sort to date involves 14 trapped ions 6. These implementations consist of quantum entities separated in space, and in general each experimental apparatus is used only once. This inherent inefficiency prohibits the generation of larger entangled states due to the experimental setup requiring additional components as the state grows.

Here, we report the experimental generation and full characterisation of an ultra-large-scale entangled state containing more than 10,000 entangled modes that is equivalent to a continuous-variable cluster state 7; 8 up to local phase shifts. This is an improvement by 3 orders of magnitude over the largest entangled states created to date. The entangled modes are individually addressable, finite-duration wavepackets of light in two beams that are multiplexed in the time domain and created deterministically. Our ultra-large-scale entangled states are an important contribution to investigations of quantum information processing.

Originally formulated as a demonstration as to why quantum mechanics must be incomplete in the famous 1935 EPR paradox 9, entanglement is now recognized as a signature feature of quantum physics 10, and it plays a central role in various quantum information processing (QIP) protocols1; 11. For example, the bipartite entangled state known as an Einstein-Podolsky-Rosen (EPR) state 9 is a resource for quantum teleportation, whereby a quantum state is transferred from one location to another without physical transfer of the quantum information12; 13; 14.

Measurement-based quantum computation (MBQC)3; 7; 8; 15; 16; 17; 18, which is based on the quantum teleportation of information and logic gates, requires the special class of multipartite entangled resource states known as cluster state 2. The number of entangled quantum entities and their entanglement structure (represented by a graph) determines the resource space available for computation. Ultra-large-scale QIP (which could be based on MBQC) will require ultra-large-scale entangled states.3; 7; 8.

To date, entangled states containing up to quantum entities have been generated and exploited in QIP using various physical systems, such as ionized or neutral atoms6; 19; 20, superconducting devices21, and spins in diamond22. In the vast majority of optical experiments, quantum modes are distinguished from each other by their spatial location. This leads to an inherent lack of scalability as each additional entangled party requires an increase in laboratory equipment, and dramatically increases the complexity of the optical network 23; 24. Further, due to the probabilistic nature of photon pair generation, demonstrations involving the post selection of photonic qubits4; 15; 16 suffer from dramatically reduced event success rates with each additional qubit. For example, it was reported that the creation of eight-qubit entanglement produced only several hundred events during hours 4.

One method to overcome this problem of scalability is to deterministically encode the modes within one beam. Entanglement between quadrature-phase amplitudes in continuous-wave laser beams has been deterministically created and exploited in QIP 5; 13; 14; 17; 18; 23; 25, even though the quantum correlations are finite. Previous attempts to deterministically create cluster states within one beam have exploited the spatial or spectral orthogonality of quantum modes25; 26; 27. However, the complexity of the setups was not suitably reduced, and as a result, the graphs in previous demonstrations were limited in size. A novel method proposed in 28 lets quantum modes propagate within the same beam – distinguished and ultimately made orthogonal by their separation in time. This time-domain multiplexing approach allows each additional quantum mode to be manipulated by the same optical components at different times.

Here, we experimentally generate an ultra-large entangled state that is locally equivalent (up to phase shifts) to a continuous-variable cluster state consisting of more than entangled wave-packets of light. The entangled states are multiplexed in the time domain and are deterministically created. We can individually access each wave-packet in the ultra-large-scale entangled states, which is an important consideration for QIP. The generated states, which we call extended EPR states, are equivalent to topologically one-dimensional continuous-variable cluster states up to local phase shifts28 and are therefore a universal resource for single-mode MBQC with continuous-variables8 (see supplementary section ‘S2’). Fully universal multimode MBQC is achievable simply by combining two extended EPR states (with differing time delays) on two additional beam-splitters 28. Importantly for experiment, verifying the entanglement is much more efficient in an extended EPR state than in a cluster state.

Extended EPR states are generated by entangling together sequentially propagating EPR states contained within two beams. This can be viewed as four distinct steps, as illustrated in Fig. 1a (see supplementary section ‘S1’ for complete experimental details). First, two continuous-wave squeezed light beams are generated from two optical parametric oscillators (OPOs) (Step i in Fig. 1a). We divide the squeezed light beams into time-bins of time period , where is sufficiently narrower than the bandwidths of the identical OPOs. Wave-packets of light in each of the time-bins represent mutually independent (orthogonal) squeezed states. Second, a series of EPR states separated by time interval are deterministically created by combining the two squeezed light beams on the first balanced beam-splitter (Step ii in Fig. 1a). The quantum correlations that manifest from the beam-splitter interaction are represented by links between the nodes (coloured spheres). The nodes here represent the orthogonal wave-packets. Third, the bottom-rail node of the EPR state is delayed for the duration after passing through a fiber delay line (Step iii in Fig. 1a). After the delay the top-rail node of each EPR state is synchronised in time with the bottom-rail node of the previous EPR state. By combining the staggered EPR states on the second balanced beam-splitter, each EPR state interacts with the previous and successive EPR states (Step iv in Fig. 1a). This leads to all of the wave-packets in each of the two rails being connected to neighbouring wave-packets by entanglement links, thereby producing the extended EPR state. We created arbitrarily large extended EPR states by continuing this procedure, limited only by technical issues.

Ideal quadrature-entangled states are simultaneous eigenstates of particular linear combinations of the quadrature operators, called nullifiers. For example an ideal EPR state , which is the ideal state approximated by the Gaussian state at step ii in Fig. 1a, is specified by the following nullifier relations:

(1a)
(1b)

Here, superscripts and denote two wave-packets. In our setup, they refer to the top rail and bottom-rail wave-packets, respectively. and are the quadrature operators of a wave-packet , which do not commute for the same wave-packet: , where is the Kronecker delta, and is normalised to . While and of a single wave-packet cannot be determined simultaneously due to the Heisenberg uncertainty principle, equation (1) shows that the correlations between the two wave-packets are perfectly determined. More precisely, the quadrature amplitudes of the two wave-packets are perfectly correlated (), and the amplitudes are perfectly anticorrelated ().

In its ideal form, the extended EPR state generated in our experiment (Fig. 1b) is specified by

(2a)
(2b)

Here, and denote the two independent beams (top and bottom rail, respectively), while represents the temporal index. See supplementary section ‘S2’ for a full treatment of this state. We consider to be a natural extension of EPR states to an ultra-large basis. This follows from the nullifiers being composed of either or quadratures, but not both, familiar from the EPR formalism.

We see that the addition of two quadratures (respectively, ) at any given time index  have negative (positive) correlations with the difference of the two quadratures (respectively, ) at the subsequent time index . The representation in equation (2) leads to the nullifiers having zero variances: and , where a bracket denotes the expectation value of an operator .

In reality, the generated states and therefore the nullifiers have excess noise due to the unphysical nature of infinite squeezing. Despite this, the full inseparability of the state can be shown when the resource squeezing level is high enough. The sufficient conditions for fully inseparable entanglement29 are given by the variances as

(3a)
(3b)

for all (see supplementary section ‘S2’).

The quadrature amplitudes , , and of the first wave-packets are plotted in Fig. 2(a–d). We see they are randomly distributed around zero. Linear combinations of the quadrature amplitudes at neighbouring times exhibit quantum correlations as in equation (2) and are shown in Fig. 2(e,f). For clarity, the sign of one of the quadrature terms is flipped in order to show both quadratures exhibiting correlations. The amplitudes almost perfectly overlap, showing strong anti-correlations and correlations in both quadrature combinations.

In order to quantify the quantum correlations, we repeat the single-shot generation of the entire state more than times, allowing us to measure the variances at each temporal position. We then evaluate the multi-partite inseparability criteria given in equation (3). The measurement results are shown in Fig. 3, as well as in supplementary section ‘S3’. The variances for the states are shown by traces (i). The bound of inseparability given in equation (3) corresponds to  dB, shown by the dashed lines (iii). We see that is clearly below the bound of inseparability in the entangled region. Vacuum-state inputs are used as references, and traces around  dB (ii) show the variances of the nullifiers for the vacuum states.

The mean variances of the first 1,000 points in the and quadratures are  dB and  dB, respectively. Note that absolutely no corrections for any losses are performed.

The variances of the state steadily increase with time for technical reasons related to our control scheme. During the data acquisition process we switch off all active feedback control of the optical setup. This is in order to avoid any unwanted noise arising from the feedback that will degrade our measurements. The increase in variance is therefore explained by the relative phase drifts of the entangled state caused by disturbances from the environment. Although the variances are degrading in time, the state is comfortably entangled for at least the first temporal positions shown here. Given the dual-rail structure of the state, the two beams and each contain the same number of wave-packets so that temporal positions corresponds to wave-packets. Figure S14 in the supplementary information shows the variances for the entire state, showing where the variances degrade into the separable region. We see that up to about wave-packets are entangled and fully inseparable.

In summary, we have experimentally demonstrated the generation of ultra-large-scale entangled states in a deterministic fashion. More than wave-packets of light are shown to be fully inseparable within a state that is locally equivalent to a continuous-variable cluster state with one-dimensional topology 28. Fault-tolerance will additionally require efficient encoding and error correction 30. Compared to the largest entangled states previously engineered, the entangled state created here is larger by three orders of magnitude. Due to their sheer size, regular structure, and deterministic method of creation, we fully expect that these ultra-large-scale states will enable other QIP applications in addition to MBQC.

Methods

The squeezing levels of our OPOs were  dB with a bandwidth of  MHz. The optical fiber length was  m, corresponding to the time duration of the wave-packets  ns. Homodyne detection is employed to measure the quadrature amplitudes of each wave-packet. The signals of the homodyne detectors are integrated with the non-overlapping temporally-localised mode functions of the wave-packets.

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Acknowledgments:

This work was partly supported by PDIS, GIA, G-COE, APSA, and FIRST commissioned by the MEXT of Japan, and ARC coE for CQC2T. S.Y. acknowledges financial support from ALPS. R.U. acknowledges support from JSPS. S.A. acknowledges financial support from the A-A Prime Minister’s Award. N.C.M. was supported by the ARC under grant No. DE120102204.

Fig. 1: Schematic of the experimental setup and ultra-large-scale XEPR state. a, Generation of ultra-large-scale entanglement. Squeezed temporal modes are mixed on a beam-splitter to create EPR states, which are then staggered by a fiber delay line. The staggered EPR-state wave-packets are then mixed to generate a continuous graph structure, where each wave-packet is entangled to neighbouring wave-packets. Nodes (coloured spheres) and links between them represent optical wave-packets and entanglement, respectively. Independent quantum states exist in each temporally localised wave-packet with time duration . OPO, optical parametric oscillator; 50:50 BS, balanced beam-splitter; HD, homodyne detector; LO, local oscillator. See supplementary section ‘S1’ for details, including link to animation of experiment. b, The extended EPR state. While only a small section is shown here for brevity, it is fully entangled for more than wave-packets ( temporal node positions in rails and ). See supplementary section ‘S2’ for full details and graph representation.
Fig. 2: Measured quantum correlations of the first wave-packets. ad, Measured quadrature amplitudes of , , and of temporally localised wave-packets at temporal node positions . They are randomly distributed about the mean. e,f, Additions and time-shifted subtractions of certain quadratures show quantum correlations displayed by the near perfect overlaps.
Fig. 3: Quantum correlations of the extended EPR state for the first wave-packets. a,b, (i) Measured nullifier variances for the first 5,000 temporal mode positions ( wave-packets in both rails) are shown for the quadrature in blue (a) and for the quadrature in red (b), as in equation (3). (ii) measured variances for vacuum inputs, used as a reference for noise power. (iii) dB lines indicate the bounds of inseparability; data points below this line demonstrate entanglement.

Supplementary Information for

Optical Generation of Ultra-Large-Scale Continuous-Variable Cluster States

Appendix S1 Experimental Setup

Fig. S4: Experimental setup.

Figure S4 shows the schematic of the experimental setup. In this setup, we employ a continuous-wave (CW) Ti:Sapphire laser (SolsTiS-SRX, M Squared Lasers) operating at  nm as the primary optical source. The pump laser is a laser-diode-pumped and frequency-doubled Nd:YVO laser (Verdi V-10, Coherent). 10 W of the pump beam from the Nd:YVO results in  W output power of the Ti:Sapphire laser.

The 860 nm fundamental beam passes through the optical isolator (ISO; FI850-5SV, Linos) and the electro-optic phase modulator (EOM; PM25, Linos). The phase modulation by the EOM adds 16.5 MHz sideband components which are utilised for locking all the optical cavities via the Pound-Drever-Hall locking technique.

Almost half of the fundamental beam is sent to a second harmonic generator (SHG) and converted to approximately  mW of a  nm beam. The SHG is a bow-tie cavity consisting of two spherical mirrors (radius of curvature  mm), two flat mirrors, and contains a  mm  mm  mm KNbO crystal. Its round trip length is  mm, and the input coupler transmissivity is . The rest of the fundamental beam is further split and distributed for controlling the sub-threshold optical parametric oscillators (OPOs), the interferometers, the homodyne detectors, and so on.

The second harmonic from the SHG is injected into the OPOs as their pump beam ( mW for each OPO) to generate the squeezed vacuum beams, which are the quantum resources for this experiment. The OPO is a bow-tie cavity consisting of two spherical mirrors (radius of curvature  mm), two flat mirrors and a  mm  mm  mm periodically poled KTiOPO (PPKTP) crystal. Its round trip length is  mm, the half width at half maximum is  MHz, the output coupler transmissivities are (OPO-A) and (OPO-B), and intra cavity losses are (OPO-A) and (OPO-B). The OPO is locked via the Pound-Drever-Hall locking method by introducing a locking beam which is appropriately frequency-shifted and counter-propagating in the cavity so as to avoid interference with the squeezed vacuum beam.

The squeezed vacuum states are combined by a Mach-Zehnder interferometer (MZI) with asymmetric arm lengths. After the first balanced beam-splitter, they are converted into two-mode EPR states. The EPR states become staggered due to the asymmetric arm lengths. The staggered EPR states are then combined in sequence at the second balanced beam-splitter, forming the extended EPR state.

For the optical delay line to asymmetrise the MZI, we employ an optical fiber. To minimize the insertion loss of the delay line, we employ fiber patchcodes with special anti-reflection (AR) coatings at  nm on both ends (PMJ-3A3A-850-5/125-1-2-1-AR2, OZ Optics). We fabricated arbitrary lengths of fiber cables by splicing the patchcodes and bare fibers (SM85-PS-U40A, Fujikura). The lenses for focusing and collimating the beam are single aspheric lenses with AR coatings (C240TME-B, Thorlabs). Furthermore, in the aim of improving the spatial mode matching between the TEM in free-space and LP in PM fibers, we developed a special fiber alignment device (FA1000S, FMD Corporation). As a result, we obtained in the throughput of the entire fiber delay line, which was kept as high as during 6 hours. Since small temperature changes around the fiber cause drastic changes of the optical pass length resulting in the instability of phase locking, the fiber is placed inside a box consisting of heat insulating material and vibration-proofing materials.

The Extended EPR states are measured by homodyne detectors. The two homodyne measurements are performed with balanced beam-splitters and continuous-wave local oscillator beams. To optimise the spatial mode-matching, the local oscillator beams are first filtered by a separate cavity which is a bow-tie cavity consisting of two spherical mirrors (radius of curvature  mm) and two flat mirrors. Its round trip is  mm length, finesse is , the half width at half maximum is  MHz, and the input and output coupler transmissivities are both . Visibilities are above for every pair of signal beams through free space including the LO beams. The average visibility through the fiber is . The propagation efficiencies from the OPOs to the homodyne detectors are . The homodyne detectors’ quantum efficiencies are higher than (special order, Hamamatsu), while the bandwidth of the detectors are above 20 MHz. The LO power is set to 10 mW for every homodyne measurement.

The signals from the homodyne detectors in the time-domain are stored by an oscilloscope (DPO 7054, Tektronix). The sampling rate of the oscilloscope is set to  MHz in order to sample enough data points in each wave packet. Each frame of  ms contains points, corresponding to about wave-packets. For each quadrature measurement of each wave-packet we measure frames in order to gather enough statistics to calculate variances. The quadrature amplitudes are elicited from them by using the temporal mode function .

(S4)

where are discretized quadratures, are continuous quadratures from the homodyne detectors, and is a Gaussian filter which is normalised as . While the bandwidth can be chosen arbitrarily, the interval of the wave-packet depends on the fiber length. The parameters used here are  ns and  MHz ( ns).

It is necessary to lock the relative phases of beams at every point where the beams interfere. For this purpose, we utilise bright laser beams of about  W, which are directed by homodyne detectors. However, their laser noise interferes with our measurement results. To avoid this problem, we switch between data acquisition and feedback control periodically at a switching frequency of  Hz. Furthermore, to realise strong and reliable phase locking, we implement a custom-made digital feedback control system via field programmable gate arrays (NI PXI-7853R, National Instruments) in PXI chassis (NI PXI-1033, National Instruments).

s1.1 Animation

Fig. S5: Legend for animation of experimental setup.

An animation can be found at (http://www.alice.t.u-tokyo.ac.jp/Graph-animation.avi) that shows temporal modes propagating through the experimental setup. Figure S5 is provided as a legend for the animation.

Appendix S2 Theory

s2.1 Derivation of Extended EPR States

An equivalent linear optics network to our experimental setup is represented in Fig. S6. In this section we derive the expressions of the extended EPR state by following this circuit with both Schrödinger and Heisenberg evolutions. In the Schrödinger picture, we assume the ideal case where the resource squeezing levels are infinite. On the other hand we can calculate experimentally realistic expressions in the Heisenberg picture.

Fig. S6: The equivalent quantum circuit to the temporal-mode method setup.

s2.1.1 Schrödinger Picture in the Ideal Case

Here we utilise infinite squeezing for simplicity. As per the following calculations with Schrödinger evolution, the output state is a simultaneous eigenstate of nullifiers. First, there are position and momentum eigenstates with zero eigenvalue in each temporal location . Each row in Fig. S6 shows the spatial mode index which the temporal-mode method would correspond to. The ket vector in step (i) is represented as

(S5)

Note that we omit the interval of an integral in this text when the integral interval is from to . Second, and eigenstates in each temporal location are combined by the first beam-splitters. Here, the beam-splitter operator for mode and is defined as . Therefore becomes

(S6)

Then an optical delay is implemented to spatial mode . It is equivalent to a delay in the temporal mode index: ,

(S7)

Finally, the beams are combined on the second beam-splitters, giving

(S8)

In the same manner the expressions in the basis can also be calculated as

(S9)

Since the output extended EPR state is equal to , the nullifiers obviously become zero as

(S10a)
(S10b)

s2.1.2 Heisenberg Evolution with Finite Squeezing

In the Heisenberg evolution, the variances of nullifiers in the case of finite resource squeezing levels can be calculated. The complex amplitudes and of the initial state in step (i) are represented as \@fleqntrue\@mathmargin=

(S11)
\@fleqnfalse\@mathmargin

where and are the squeezed quadratures of the -th squeezed state in the spatial location and , respectively. So we have

(S12a)
(S12b)

After combining the terms through the action of a beam-splitter they become \@fleqntrue\@mathmargin=

(S13)

Subsequently, the of optical delay for mode is represented as

(S14)

Finally, by combining them on the last beam-splitter the final complex amplitudes of the output state are given:

(S15)
\@fleqnfalse\@mathmargin

Since and , the ideal nullifiers of the extended EPR state are expressed as

(S16a)
(S16b)

Therefore, we can calculate the nullifier variances which determine the theoretical value of the inseparable condition as shown in the main text [Eq. (3)],

(S17a)
(S17b)

This shows that the sufficient condition for inseparability is satisfied when  dB resource squeezing in each OPO is available.

s2.2 Inseparability Criteria for Extended EPR States

Here, we discuss sufficient conditions of entanglement for the extended EPR states, based on the van Loock-Furusawa inseparability criteria 29. We consider all of the cases where an approximate extended EPR state is separable into two subsystems and . A necessary condition of separability is obtained as an inequality for each case. If all of the separable cases are denied by not satisfying the inequalities, the state is proved to be in an entangled state with full inseparability.

First, we consider the combinations of four nodes distributed into the two subsystems. When all of the four are not distributed into either subsystem, the possible cases are as below. Here we abbreviate the nullifiers as and .

Therefore, when the inequalities and are satisfied, any of the seven inequalities (S18a)–(S18g) is not satisfied, which means that the four nodes are not separable into two subsystem and .

Then, we apply the same discussion for all temporal indices . When the inequalities and are satisfied for all , any partitioning of the whole system is denied, which means that all nodes are entangled. We may take a more severe but simpler sufficient condition for entanglement as

(S19)

for all , which is shown in Eq. (3) of the main text.

s2.3 Graph Correspondence

In this section, we discuss the intuitive representation of the extended EPR state in terms of the graphical calculus for Gaussian pure states 31. Every -mode zero-mean Gaussian pure state can be uniquely represented by an undirected complex-weighted graph , whose imaginary part (i.e., that of the adjacency matrix for the graph) is positive definite. (In what follows, we make no distinction between a graph and its adjacency matrix.) The graph  shows up directly in the position-space wavefunction for the corresponding state  (with ):

(S20)

Any Gaussian pure state  satisfies a set of exact nullifier relations based on its associated complex matrix  (ref. 31):

(S21)

where and are column vectors of momentum and position operators, respectively. The special case of the -mode ground state () is easy to verify by noting that the vector of nullifiers in that case is just the vector of annihilation operators.

The extended EPR state is exactly the state originally proposed by Menicucci in ref. 28.In that proposal, it was shown that such a state is locally equivalent (up to phase shifts on half the modes) to a CV cluster state, which is a universal resource for measurement-based quantum computing with continuous variables 8; 30. Since measurement-based quantum computation requires the ability to do homodyne detection of any (rotated) quadrature, plus photon counting 30, the phase shifts required to transform the generated state (the extended EPR state) into a CV cluster state do not need to be physically performed on the state after generation. Instead, one can account for them entirely just by updating the measurement-based protocol to be implemented (i.e., redefine quadratures and on the appropriate modes) 28. Because of this equivalence, the original proposal 28 used a simplified graphical calculus that blurred the distinction between the extended EPR state and corresponding CV cluster state since the two were, for quantum computational purposes, effectively the same resource. The distinction between these states turns out to make a huge difference, however, when one tries to experimentally characterize the generated state. In this case, it is much easier to work with the mathematics of the extended EPR state.

For clarity and completeness, here we present the full complex-weighted graph  (ref. 31) corresponding to the extended EPR state originally proposed in ref. 28 and reported on in this work:

Fig. S7: The full graph 31 for the extended EPR state.

In this graph, and , with being the initial squeezing parameter of the states emitted by the OPO. (See ref. 31 for more details on such graphs.) The green (purple) edges have positive- (negative-)imaginary weight , and the green self-loops have positive-imaginary weight . This state can be transformed, by phase shifts on both modes of all odd (or all even) time indices, into the following CV cluster state  (ref. 28):

Fig. S8: The full graph 31 for the CV cluster state obtained by phase shifting both spatial modes in every other time index of the extended EPR state.

In this graph, and . The blue (yellow) edges have positive- (negative-)real weight , and the green self-loops have positive-imaginary weight . Darker colours indicate larger magnitude of the corresponding edge weight. In the large-squeezing limit, , and , which allows us to define an unphysical, ideal CV cluster-state graph  to which is a physical approximation:

Fig. S9: The ideal CV cluster-state graph in the limit .

Notice that

(S22a)
(S22b)

The crucial properties of that enable such a simple connection between , , and are (a) that is bipartite and (b) that is self-inverse (i.e., as a matrix). These mathematical properties allow all three of these graphs to be visually similar.

With this simplification in hand, we can derive new nullifier relations for in terms of . We start by observing that premultiplying both sides of Eq. (S21) by gives the additional exact nullifier relation

(S23)

Substituting [Eq. (S22a)] and noting that , we have the two exact nullifier relations

(S24a)
(S24b)

By premultiplying by , respectively, we obtain

(S25a)
(S25b)

In the large-squeezing limit (), and , and we have the following approximate nullifiers for the extended EPR state:

(S26a)
(S26b)

This is the state we have created. For completeness, however, we can compare these to the exact and approximate nullifiers for the associated CV cluster state  obtained by phase shifting particular nodes as described above (either actively or by simply redefining quadratures used for the measurements). Substituting [Eq. (S22b)] and noting that , the exact nullifiers are

(S27a)
(S27b)

which, in the large-squeezing limit, reduce to the following approximate nullifiers:

(S28a)
(S28b)

Once again, these simple and symmetric expressions in terms of are unusual and only possible because is bipartite and self-inverse 31.

s2.4 Equivalence to Sequential Teleportation-based Quantum Computation Circuit

In reference 28, Menicucci proposed that by erasing half of the state (one rail), the cluster states can be used as resources for measurement-based quantum computation (MBQC). However, erasing half of the state is a wasteful process and it is experimentally hard to perform the necessary feedforwards to future and past modes in the time axis. Here, we show that the extended EPR state is a resource for MBQC, and that we can fully utilise every degree of freedom without erasing half of the state. We devise a much more efficient method of using this resource state for quantum computation than the method originally proposed in ref. 28, in terms of its use of the available squeezing resources.Specifically, arbitrary Gaussian operation may be implemented by only measurements, which is more efficient than the 8 measurements necessary with the original method 32. Furthermore, we show that non-Gaussian operations may be performed on the extended EPR state by introducing non-Gaussian measurements, leading to one-mode universal MBQC.

s2.4.1 Gaussian Operation

Fig. S10: Teleportation-based circuit

First of all, let us consider the quantum teleportation-based circuit shown in Fig. S10. The resource EPR state is generated by combining position and momentum eigenstates via the first beam-splitter. After the input state is coupled via a second beam-splitter, two observables and are measured, giving the measurement results and . Then, the corresponding feedforward operations and are performed on the remaining mode, where and are the position and momentum displacement operators, and is . The resulting ket vector becomes

(S29)

Here, and are the rotation operator and squeezing operator, respectively, in phase space.

We may combine this teleportation-based circuit sequentially as shown in Fig. S11a. It can be experimentally realised by modifying our experimental setup as shown in Fig. S12, where the input-output coupling port is realised by a switching device to a Mach-Zehnder interferometer containing a configurable phase-shifter. In Fig. S11a, the input-coupling beam-splitters and displacement operators can be exchanged as and . As a result, an equivalent circuit is Fig. S11b. Here, in the region enclosed by the dotted line, is the part of the circuit that creates the extended EPR state, which is represented as a graph as shown in Fig. S11c. This shows that the extended EPR state can be used as a resource for MBQC. Therefore, by adding input-coupling, measurement and feedforward optics to our setup, it will become a MBQC circuit.

Fig. S11: The resource for MBQC with extended EPR states. a, Sequential teleportation circuit. b, Equivalent circuit. c, Abstract illustration of sequential teleportation with an extended EPR state.
Fig. S12: Experimental setup (left) and abstract illustration (right) for MBQC with extended EPR states. a, Input coupling. b, Sequential MBQC. c, Output. d, Switching (SW) device with Mach-Zehnder interferometer including electro optical modulator (EOM).

s2.4.2 Non-Gaussian Operation

Non-Gaussian operations may also be implemented by using the teleportation-based circuit shown in Fig. S13. It is derived in the following way. The ket vector after input coupling is represented as

(S30)

Therefore, by measuring on the input mode giving measurement results , and performing the displacement operation and , the resulting state becomes . Further, it is also expressed as

(S31)

where is the arbitrary function of and is the Fourier transform operator. In the same way, by measuring on mode , giving measurement results , and performing the displacement operation , the output state becomes . When is higher than a quadratic polynomial, it is a non-Gaussian operation. Note that since it can be accomplished by only using displacement feedforwards, input coupling beam-splitters can also be exchanged. Therefore, by using the extended EPR state, non-Gaussian operations can be performed sequentially, resulting in a resource for universal one-mode universal MBQC.

Fig. S13: Teleportation-based non-Gaussian operation circuit

Appendix S3 Data Analysis

s3.1 Influence of Experimental Losses

Experimental imperfections lead to degraded resource squeezing levels. In particular, the unbalanced losses between the optical fiber and free space channels cause the degradation of nullifier variances. Taking into account these losses, instead of equation (S17) we get more involved theoretical values given by:

(S32a)
(S32b)

where and are squeezing and anti-squeezing terms for beam , and and are the effective efficiencies of squeezing levels for beam through the free space and optical fiber channels, respectively. To be more precise, is given by: (quantum efficiency at homodyne detector) (influence of intracavity loss ) (visibility between probe and LO beams) (propagation efficiency), where is the transmittance of the output coupler and is the intracavity loss in the OPO. In the experiment, these efficiencies are , , and . Squeezing levels are calculated as

(S33a)
(S33b)

where is the Fourier transformation of mode function , is the ratio of electrical noise to shot noise at angular frequency , is the pump parameter which is related to the classical parametric amplification gain as , and is the angular frequency half width at half maximum of the OPO. By substituting in these experimental values, we get  dB and  dB. They agree well with experimental results.

s3.2 Influence of Phase Fluctuations

Fig. S14: Quantum correlations of the extended EPR state over time. a,b, First 15,000 temporal positions. Each trace shows measured (blue) and (red) nullifier variances of each temporal index (i)with and (ii)without resource squeezing. (iii)  dB lines show the bound of inseparability; variances below sufficiently indicate entanglement.

In the experiment, we alternate between active feedback control (phase locking) and measurement. We switch between the two because the feedback control will interfere with the measurement, and we do not want to corrupt the measurements with phase noise. However, this leads to the degradation of entanglement as there is no active phase control during the measurement. Figure S14 shows the experimental results of nullifier variances for the first temporal positions. The degradation in variances over time is explained by dephasing due to environmental noise. Although the experimental results shown in Fig. S14 include these experimental imperfections, we see here that the statistical variances on average obviously satisfy the sufficient condition for entanglement up to about temporal positions. This corresponds to about entangled wave-packets.

Supplementary References

  • 31 Menicucci, N. C.  Flammia, S. T. & van Loock, P. “Graphical calculus for Gaussian pure states,” Phys. Rev. A 83, 042335 (2011).
  • 32 Alexander, R. et al., Optimising the temporal mode scheme for single qumode Gaussian operations. in preparation.
Acknowledgments:

N.C.M. is grateful to R. Alexander, and P. van Loock for helpful discussions.

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