## 1 Introduction

The key part of quantum information processing is controlling a quantum state. Quantum teleportation is a scheme to transfer a quantum state utilizing not only the classical channel but also quantum entanglement and an essential protocol for realizing quantum computing [1]. In previous experimental studies, light has been extensively used to demonstrate quantum teleportation [2, 3] and entanglement-based quantum computing [4, 5].

There are two natural ways to encode information in the quantum domain. One is to use two discrete levels of a single photon (e.g. two orthogonal polarizations or two optical paths of a single photon) that describe discrete variables (DVs). The other is to use a quadrature phase amplitude of an electromagnetic field (e.g. a coherent light or squeezed light) that describes continuous variables (CVs). In accordance with the two methods of encoding quantum information, there are mainly two types of quantum entanglement. One is DV entanglement, typically a pair of entangled photons [6], or NOON states [7]. The other is CV entanglement such as Einstein-Podolsky-Rosen (EPR) [8] beams, which correspond to two-mode squeezed light in the case with infinite squeezing. In this article, however, we represent EPR beams for both cases with infinite and finite squeezing.

The DV systems benefit from high-fidelity operations like a digital information technology, but are limited by the imperfect generation and detection of single photons. In contrast, CV schemes can achieve a deterministic and unconditional operation, while the fidelities are not so high for the majority of quantum protocols. A newly developed hybridization approach can overcome these problems by combining the benefits of DVs and CVs in a single system [9]. One of the significant experiments is hybrid quantum teleportation of photonic quantum bits using CV quantum teleportation schemes [10]. In this experiment, quantum information is encoded on a single-photon; EPR beams are used as a quantum entanglement resource; and balanced homodyne measurement is applied to make deterministic Bell-state measurements.

Another important approach in quantum information processing is measurement-based one-way quantum computation with multi-partite quantum entanglement such as cluster states [11–14]. CV systems have an advantage over DVs, as CV cluster states can be unconditionally prepared with squeezed light sources and beam splitters [15]. In addition, balanced homodyne detection is available as a deterministic measurement. CV cluster states with four optical modes are generated and applied to unconditional one-way quantum computation in the frequency domain [16]. An ultra-large-scale CV cluster state containing > 10,000 modes is also generated by using a time-domain multiplexing method [17]. A current theoretical work shows that fault-tolerant quantum computing is possible with CV cluster states obtained by highly squeezed light source [18] and Gottesman-Kitaev-Preskill encoding [19] with cat states [20].

These important proof-of-principle experiments have been performed in conventional free-space optics in which photons or light beams propagate through complex optical circuits consisting of a lot of mirrors, beam splitters, and lenses. Such a scheme has limited stability due to the relatively large-sized optical setup. Practical applications of quantum information technology require an integrated waveguide circuit in which photons or light beams are guided, manipulated, and interfered. Integrated waveguide circuits have been successfully developed for DV applications by O’Brien’s group in Bristol [21, 22], and they have demonstrated high-fidelity and reliable operations [23]. More recently, integrated waveguide circuits are applied for CV applications [24]. Generation and verification of EPR-entangled beams, including balanced homodyne measurement, is demonstrated by using an integrated waveguide circuit. A benefit of this optical integration is stabilization and miniaturization of quantum circuits and attainment of a high degree of spatial mode matching, which is essential for both classical and quantum interference. Fiber optics is also important for guiding light beams and stabilizing an optical system, and then applied for various CV applications [25–29].

In this article, we mainly review the current development in integrated photonic implementation of EPR beams and balanced homodyne measurement, which are important for CV as well as hybrid quantum information processing. In Section 2, we briefly review an integrated waveguide circuit and the generation of photon-entangled states as an example of DV entanglement to better understand CV experiments in the following sections. Section 3 shows balanced homodyne detection of a CV quantum state, such as squeezed states of light. Section 4 shows generation and verification of EPR beams. Finally, we conclude this review article in Section 5.

## 2 DV entanglement in a photonic chip

In this section, we review the currently developed quantum optics experiments that generate and characterize DV entanglement using a photonic chip where waveguide Mach-Zender interferometers are integrated [21, 22]. As a simple example, we focus on the states described as *N* = 1 and 2, respectively [7]. The latter state is known as the Hong-Ou-Mandel (HOM) state [30]. The entangled HOM state is an important resource for phase-estimation technology based on quantum metrology [31] and precise processing technology based on quantum optical lithography [32]. Two-photon interference is also an essential interaction for the KnillLaflamme-Milburn (KLM) scheme, which yields a fully scalable quantum computer [33].

### 2.1 Construction of integrated waveguide circuits

Figure 1A [22] presents a schematic of a waveguide circuit integrated in a silica-on-silicon chip. The circuit is composed of a pair of directional couplers and a resistive heater (R) lithographically defined on top of one waveguide arm. These elements construct a waveguide interferometer. A precise design and fabrication of the directional coupler enables mixing two light beams with high mode-matching efficiency and an exact beam-splitting ratio [21]. Inset B shows the cross section view of the chip. The waveguide circuit is fabricated on silicon wafer (I). The waveguide core (III) has 3.5 μm squared dimensions and is surrounded by lower refractive index cladding (II, IV). The spacing between each waveguide core is 250 μm. Optical wave is confined in the core region and propagates through the waveguide. Inset C shows the simulation result of the intensity distribution of the guided optical wave and a feature of almost single mode. By tuning the electrical voltage applied for resistive heater (R) on contact with pads p_{1} and p_{2}, it is possible to raise the temperature of one waveguide arm and then control the phase difference *φ* between two arms. So, the resistive heater is used as a voltage-controlled phase shifter. Owing to this feature, the waveguide interferometer can work as a continuously variable beam splitter. More details will be described in Section 3.3.

### 2.2 Generation of photon-entangled states

Figure2A and B [22] show the experimental result of one-photon and two-photon interference using the waveguide interferometer as shown in Figure 1A. A single-photon or indistinguishable two single-photon state is generated outside the chip by a spontaneous-parametric-down conversion process with a type-I non-linear optical crystal and a pulsed laser light source. In Figure 2A, a single photon is introduced from input port a. The one-photon count rates are measured by two single-photon counting modules at output ports g and h, by varying a phase of the waveguide interferometer. The observed sinusoidal pattern arises from one-photon interference. The output states correspond to one-photon entangled states *π*/2. In Figure 2B, the two-photon coincidental count rate is measured at output ports g and h by introducing two single-photon states from a and b. The significant sinusoidal fringe arises from two-photon interference as two photons are indistinguishable and arrive simultaneously at the waveguide interferometer. The output state is the HOM state, *π*/2.

Figure 3 [22] shows the visibility of the HOM dip at certain phases obtained by scanning a relative delay time between incident two single-photons outside the photonic chip as shown in the insets. At the phase of -*π*/2, the maximum HOM dip is obtained.

The experiments reviewed already are demonstrated with only one waveguide interferometer. However, in a recent work, a reprogrammable optical circuit in a single photonic chip where many waveguide interferometers and phase shifters are integrated is applied to the realization of universal linear optics [34, 35]. The universal linear optical component is a required element for constructing a fully scalable quantum computer based on KLM scheme together with single photon detectors and single photon sources [33]. The above experiments use a single-photon resource generated outside a chip. However, current experimental works demonstrate on-chip direct generation of DV entanglement by using a directional coupler, waveguide arrays consisting of a non-linear optical crystal [36, 37], and integrated waveguide circuits with a silicon-on-insulator photonic platform, which utilize spontaneous four-wave mixing process [38].

## 3 Demonstration of balanced homodyne measurement for squeezed light in a photonic chip

### 3.1 Balanced homodyne measurement using a waveguide beam splitter

We review the balanced homodyne measurement of squeezed light [24] by using a waveguide beam splitter that is the same as Figure 1A. Balanced homodyne measurement is also applied to the verification of quantum entanglement in EPR beams in Section 4. Historically, Yuen and Chan theoretically developed balanced homo-dyne measurement as a method for measuring quadrature squeezing in 1983 [39]. Since then, it has become the standard method to measure squeezed light and has been applied to many quantum optics experiments. In order to perform balanced homodyne measurement, it is required to combine squeezed light and local oscillator (LO), which is in a coherent state with high intensity using an exact 50/50 beam splitter. Two output beams from the beam splitter are detected by two photodiodes of a balanced homodyne detector. A subtraction signal of the photocur-rents is detected and amplified by low-noise electrical circuits. In this balancing process, an excess classical noise of the local oscillator is cancelled.

In free-space optics, however, a dielectric multilayer beam splitter is used. The accuracy of the beam-splitting ratio depends on the process of optical coating. There is usually a margin of error with a few percent. So, in the actual experiment, fine tuning of beam-splitting ratio is performed by tilting a beam splitter angle. However, by using a waveguide beam splitter, the final tuning of beam-splitting ratio can be easily achieved by adjusting the electrical currents of heaters. This is one of the biggest advantages of a waveguide beam splitter.

For efficient balanced homodyne measurement, it is necessary to combine squeezed light and local oscillator beams with a high level of spatial mode matching. In free-space optics, high mode-matching efficiency is obtained by a precise optical alignment by using optical lenses for mode shaping or adjusting optical path length. By using a waveguide beam splitter, a high level of mode matching can be easily obtained, as the spatial mode is automatically formed by the waveguide mode, if once the optical beams are injected into waveguides. This is another advantage of a waveguide beam splitter.

In waveguide optics, stray light arising from insufficient coupling to waveguides, waveguide bending losses, and cladding guiding causes a crosstalk between different waveguide modes. Such an unwanted crosstalk is a major limitation factor in quantum optical networks, reducing the fidelities of quantum operations. In quantum optics experiments with CVs, the quantum information is often encoded on side bands of optical fields, which are far from the noise spectrum caused by stray light. Moreover, balanced homodyne measurement can sensitively detect the signal spectrum at side-band modes. So, quantum optics experiments with CVs are suitable for the optical integration. In Sections 3 and 4, all balanced homodyne measurements are performed at the 1.5MHz side bands of the laser frequency to avoid the harmful effects of stray light.

### 3.2 Experimental setup

The experimental setup for balanced homodyne measurement is shown in Figure 4. Figure 5 is a picture of a silica-on-silicon chip where a number of waveguide interferometers are integrated in parallel. A continuous-wave Ti:Sapphire laser at 860 nm is used as a light source. Squeezed light SL_{1} at 860nm is generated by subthreshold optical parametric oscillator (OPO_{1}) outside the chip. The OPO_{1} consists of an optical cavity with a bow-tie configuration and a periodically poled potassium titanyl phosphate (PPKTP) crystal as a non-linear optical medium [40, 41]. The optical cavity consists of two spherical mirrors with curvature radius of 50mm and two flat mirrors. One of the flat mirrors has a partial transmittance of 0.113 at 860nm and is used as an output coupler. The other three mirrors have high reflectivity at 860 nm. All mirrors have high transmissivity at 430 nm. The round-trip length of the cavity is 500 mm. The PPKTP crystal has 10mm length and 1*1 mm^{2} of cross section, and is placed between two spherical mirrors. The crystal temperature is kept at 40 C for achieving a phase matching condition. A pump beam at 430nm for OPO_{1} is generated by an optical frequency doubler, which has the same optical configuration with the OPO_{1} but includes a periodically poled MgO-doped lithium niobate crystal [42]. Squeezed light SL_{1} is generated by pumping the OPO_{1} with the 430nm beam below an oscillation threshold power. Local oscillator LO_{1} is a part of the light source and has the same wavelength with squeezed light SL_{1}. A weak coherent beam of 860 nm is also introduced into the OPO_{1} for the purpose of phase locking. For balanced homodyne measurement, squeezed light SL_{1} and local oscillator LO_{1} are introduced into the chip and combined at beam splitter BS_{2}, which is tuned at a 50/50 beam-splitting ratio. Output beams from beam splitter BS_{2} are measured by balanced homodyne detector HD_{1} outside the chip.

In order to efficiently inject squeezed light SL_{1} and local oscillator LO_{1} to the waveguides and eject the output beams, a pair of fiber arrays is used for both entrance and exit facets of the chip. On fiber array assembly, single-mode, polarization-maintaining fibers are arranged on a V-grooved Si substrate at an interval of 250 μm. The fiber core diameter is chosen as 5 μm, which is comparable to waveguide core size. The fiber clad diameter is 125 μm. The chip is attached on an aluminum plate that is fixed on a rigid stage. In order to realize a high fiber-to-waveguide coupling efficiency, two fiber arrays are separately mounted on two six-axis positioning stages. One is set at the entrance side and the other is at the exit side of the chip for aligning the position and orientation of fiber arrays with a high degree of accuracy, as shown in Figure 6A. When the optical alignment is performed, views of the interface between the fiber array and chip from a top and side are monitored by using stereomicro-scopes, as shown in Figure 6B and C. Firstly, positioning stages are moved in order to align the fiber arrays as close and parallel as possible. Finally, the gaps between fiber arrays and chip are filled with an index matching fluid to eliminate reflection losses. Then, the fiber-to-waveguide coupling efficiency is improved.

All beams, squeezed light SL_{1}, and local oscillator LO_{1} are coupled to the fibers whose end faces are designed as ferrule connector/angled physical contact (FC/APC). To make better fiber coupling efficiency, aspheric lenses with focal length of 8 mm and numerical aperture of 0.5 are used. Then, the beam size is focused comparable to the fiber core diameter. An overall coupling efficiency *η _{c}* = 0.72 is measured by introducing a weak coherent beam into the fiber coupling lens and collecting the output intensity at the fiber exit. The overall coupling efficiency is given as

*η*is the fiber coupling efficiency and

_{f}*η*is the fiber-to-waveguide coupling efficiency.

_{w}*η*is obtained as 0.87 with a direct measurement, and then

_{f}*η*is estimated as 0.91 on average. The waveguide transmission loss is negligibly small.

_{w}### 3.3 Balance tuning

In order to perform balanced homodyne measurement, it is required to use an exact 50/50 beam splitter and achieve high balance of optical powers at each photodiode of a homodyne detector. In this work, it is especially required to achieve high balance at the 1.5MHz side band of the laser frequency, as all of the balanced homodyne measurement is performed at this frequency in order to avoid harmful effects from stray light. Figure 7 shows an experimental result of variable beam splitter BS_{2}. An electric current for the phase shifter of beam splitter BS_{2} is supplied through metal contact pads p_{1} and p_{2}, as shown in Figure 1A, with a current source. In this experiment, only local oscillator LO_{1} is introduced to beam splitter BS_{2}. The intensities from two-split local oscillator beams are monitored at output fibers with photodiodes separately. The vertical axis shows normalized intensity of output beams *I*_{1} from the lower exit and *I*_{2} from the upper exit of beam splitter BS_{2}. They correspond to the beam-splitting ratio of the waveguide interferometer at a certain current. Solid lines indicate fitting curves using the following equations:

Both directional couplers are assumed to have the same value of intensity transmissivity *T _{r}*. Optical phase

*ϕ*is assumed as a polynomial function of electric current

*I*[18]

*I*is varied from 0 to 90 (mA) in this experiment. The parameters are estimated to achieve the best fit for experimental results and found at

*T*= 0.7644,

_{r}*α*= 0.6637,

*β*= 1.087*10

^{-3},

*γ*= -1.646*10

^{-5}, and

*δ*= 1.318*10

^{-7}.

It is obvious that the beam-splitting ratio of the waveguide interferometer is controlled by electric current *I*. The waveguide interferometer can therefore work as a variable beam splitter at a certain range, 0.36–1 for output *I*_{1} and 0–0.64 for output *I*_{2}. Actually, it is difficult technology to fabricate a directional coupler with just beam-splitting ratio as designed. In balanced homodyne measurement, however, an exact 50/50 beam splitter is required for noise cancellation. So, the variable beam splitter consisting of a waveguide interferometer is suitable for practical use.

Figure 8 shows an experimental result of precise balance tuning by using beam splitter BS_{2}. In this experiment, only local oscillator LO_{1} is introduced to beam splitter BS_{2}, divided in two beams, and then observed by homodyne detector HD_{1}. There are two types of electrical signals analyzed by a spectrum analyzer in Figure 8. Trace (i) shows a signal from one of the photodiodes of homodyne detector HD_{1} with measuring one of the divided local oscillator LO_{1} beams. A large signal at 1.5MHz is the amplitude noise created by an electro-optical modulator outside the chip. Another noise at around low frequency is a laser-originated noise. Trace (ii) is a subtraction signal from both photodiodes with measuring both of divided local oscillator LO_{1} beams. While measuring trace (ii), the electrical current for the phase shifter of beam splitter BS_{2} is firstly kept at around 26 mA, which corresponds to an approximate 50/50 splitting ratio, as shown in Figure 7. Then, the current is finely tuned in order to minimize signals at 1.5 MHz. The final result shows the noise reduction of about -40 dB. The beam splitter BS_{2} can provide an almost exact 50/50 splitting ratio and be used for balanced homodyne measurement.

### 3.4 Result of balanced homodyne measurement of squeezed light

Figure 9 [24] presents a typical result of balanced homo-dyne measurement of squeezed light SL_{1}. A pump beam power for OPO_{1} is 100 mW. The resolution bandwidth is 30 kHz, and the video bandwidth is 300 Hz. The measurement is performed at the 1.5MHz side band of the laser frequency in order to avoid a noise from stray light. Trace (iv) is a shot noise level normalized to 0dB with only the local oscillator LO_{1} injecting to balanced homodyne detector HD_{1}. Therefore, it corresponds to a vacuum noise level amplified by the local oscillator amplitude. Trace (i) is a noise level observed while the local oscillator phase is scanned by using a piezo-electric controller. The observed signal varied between a squeezed and anti-squeezed noise level. Such a phase-sensitive noise is a characteristic property of squeezed light. Traces (ii) and (iii) are noise levels, while the local oscillator phase is locked at the squeezed and anti-squeezed quadrature, respectively. The observed noise levels are -4.02±0.13dB for squeezing and ±11.85±0.15dB for anti-squeezing, respectively.

The measurement was repeated at different pump powers. The squeezing and anti-squeezing levels as the pump power is varied are summarized in Figure 10 [24] and indicated by blue circles and red squares, respectively. The normalized pump parameter *x* is defined as *P* mW is a pump power and *P*_{th} = 179 mW is the oscillation threshold power, which is obtained by measurement of the classical parametric amplification for the OPO_{1}. The solid curves in Figure 10 show the calculation results based on the theoretical formula described in Section 3.5, and agree well with the experimental results.

The saturation of the squeezing level at around -4 dB, corresponding to about 0.2 of an average photon number, at a higher pump power is predominantly caused by an insufficient overall coupling efficiency *η _{c}* (=0.72) described in Section 3.2. If perfect coupling efficiency is assumed, -8.4dB of the squeezing level is expected. Another important factor for efficient homodyne detection is visibility

*η*, which is a degree of mode matching between squeezed light SL

_{V}_{1}and local oscillator LO

_{1}in beam splitter BS

_{2}. In these calculations, visibility

*η*is set at 0.995. The agreement between the experimental values and calculations verify a high degree of mode-matching efficiency. Almost perfect visibility (0.995) is also achieved by direct measurement of an interference signal between two coherent beams using beam splitter BS

_{V}_{2}.

As a result, the waveguide beam splitter makes it possible to achieve an exact 50/50 splitting ratio by tuning the current of the phase shifter and a high degree of mode matching (0.995), and then to realize efficient balanced homodyne measurement. By using the side-band measurement, it is also possible to avoid the harmful effects of stray light and to detect high levels of squeezing. These results demonstrate the high performances of an integrated device for balanced homodyne measurement, which is necessary for CV operations.

### 3.5 Theoretical formula for squeezing and anti-squeezing levels

In this section, a theoretical formula is provided for the on-chip balanced homodyne measurement of squeezed light. The solid curves in Figure 10 represent the theoretical values of noise levels at various values of the normalized pump parameter *x*. The noise spectrum *R*_{±} for the anti-squeezed (+) and squeezed (-) quadrature can be written as [27, 28]

*f*is a measurement frequency normalized by an optical cavity line width,

*ρ*is escape efficiency, and

*η*is homodyne detection efficiency. In this experiment, a cavity line width is estimated as 11.8 MHz and a measurement frequency at side-band mode is 1.5 MHz. Then, normalized measurement frequency

*f*is obtained as 0.127. Homodyne detection efficiency

*η*is described as

*η*=

*η*

_{PD}*

*η*

_{P}*

*η*

_{C}*

*η*

_{V}

^{2}.

*η*

_{PD}(=0.998) is the quantum efficiency of photodiodes used in the homodyne detector.

*η*(=0.99) is the propagation efficiency of an optical path in a free space between the OPO

_{P}_{1}output coupler and the fiber coupling lens.

*η*(=0.72) is the overall coupling efficiency as described in Section 3.2.

_{C}*η*(=0.995) is the visibility between the OPO

_{V}_{1}output mode and local oscillator LO

_{1}at beam splitter BS

_{2}, as described in Section 3.4. These values yield homodyne detection efficiency

*η*of 0.704.

Escape efficiency *ρ* is expressed as *T* = *T*/(*T*+*L*), where *T* (=0.113) is transmissivity and *L* is intracavity loss of OPO_{1}. In PPKTP crystals, it is reported that an intracavity loss *L* increases with pump beam power at 430 nm [41]. This phenomenon is called as blue-light-induced infrared absorption (BLIIRA). If BLIIRA exists, an intracavity loss is modeled as a function of pump power *P* and expressed as *L* = *L*_{0}+*aP*, where *L*_{0} is a passive loss without a pump beam and *a* is a coefficient for pump-induced losses. The pump-induced loss is precisely measured in a PPKTP crystal. *L*_{0} and *a* are evaluated as 0.00254 and *a*0.00922 (W^{-1}), respectively. These results yield escape efficiency *ρ* at around 0.97.

In order to evaluate precise values for the squeezed and anti-squeezed spectra, it is necessary to consider another experimental imperfection. In actual homodyne measurement, phase fluctuation *δθ* between squeezed light and local oscillator, which is caused by the imperfection of phase locking, degrades the observed squeezing and anti-squeezing levels. The noise levels given by Eq. (4) are modified as

*δθ*[43]. In the optical configuration here, phase fluctuation

*δθ*is estimated as 1.5°. It is also necessary to consider the effect of an electrical circuit noise of homodyne detection. If clearance

*C*of the shot noise level above the electrical circuit noise is taken into account, the final noise spectra are obtained as

The clearance *C* is evaluated as 13.5dB with the local oscillator power of 3.5 mW. By using the experimental parameters described above, the theoretical values of noise levels are calculated as shown by solid lines in Figure 10. The results agree well with the experimental values.

## 4 EPR beams in a photonic chip

### 4.1 Heisenberg representation of EPR beams and inseparability criterion

This section reviews the generation of EPR beams and the verification of quantum entanglement in a photonic chip. EPR beams are generated by combining two squeezed light beams with a 50/50 beam splitter. In this process, the relative optical phase *ϕ* between two squeezed light beams is an important experimental parameter. Here, we schematically show typical cases with different values of *ϕ* in phase space representation. When *ϕ* is 90°, the final outputs seem to be phase-insensitive noises as shown in Figure 11A. However, there is quantum entanglement between quadrature phase amplitude of outputs EPR_{1} and EPR_{2}. On the other hand, when *ϕ* is 0°, the outputs are two-independent squeezed light SL_{1}' and SL_{2}', as shown in Figure 11B, and there is no quantum correlation between them.

The theoretical expression for the above explanations is presented here by using the Heisenberg representation with the optical configuration of Figure 12. Firstly, two single-mode squeezed light beams, SL_{1} and SL_{2}, are obtained by using two-independent squeezers that act on vacuum states Vac_{1} and Vac_{2}, and are described as squeezing operators *S*(*r*) and *S*(-*r*) with squeezing parameter *r*, respectively. The phase shifter gives the optical phase shift *ϕ* to the squeezed light SL_{2}. The quadrature phase amplitude operators _{i} (*i* = 1, 2) are introduced. They correspond to cosine and sine components of the optical field. Then, the complex amplitude operators of squeezed light beams SL_{1} and SL_{2} can be written as

It is assumed that two squeezers are characterized by the same squeezing parameter *r*, and both squeezed light beams SL_{1} and SL_{2} are pure states. The squeezed light beams SL_{1} and SL_{2} are combined using 50/50 beam splitter BS. Then, the complex amplitude operators of outputs Out_{1} and Out_{2} are derived as

*x̂*and

_{i}*p̂*are quadrature phase amplitude operators of output Out

_{i}_{i}(

*i*= 1, 2). The variances of

*x*and

*p*quadratures from output Out

_{i}are calculated as

To verify quantum correlation in final outputs, an inseparability criterion is necessary. For that purpose, it is required to introduce correlation variance in quadrature phase amplitudes between two outputs Out_{1} and Out_{2} as

It has been proven by Duan et al. [44] and Simon [45] that the two outputs Out1 and Out2 are inseparable and show quantum entanglement when

Equation (14) gives the sufficient condition for quantum entanglement and is often used as the inseparability criterion. In the optical configuration of Figure 12, the correlation variance is derived as

Notice that the value of Eq. (15) is 1 when the squeezing parameter *r* is zero. It means that the outputs are actually two independent vacuum states, so there is no quantum correlation between them.

When *ϕ* is 90°, the variances of *x _{i}* and

*p*quadrature of output Out

_{i}*(i = 1, 2) are given as*

_{i}The correlation variance is calculated as

The final outputs are entangled EPR_{1} and EPR_{2} beams, as the correlation variance satisfies the inseparability criterion *r* is larger than zero. They show a phase-insensitive noise as *x* and *p* quadratures have the same noise variances as shown by Eq. (16) and in Figure 11A schematically. Notice that the value of *r*, achieving perfect correlation. On the other hand, when *ϕ* is 0°, the variances of *x _{i}* and

*p*quadrature of output Out

_{i}_{i}(

*i*=1, 2) are given as

The correlation variance is calculated as

The final outputs are two independent squeezed light SL_{1}' and SL_{2}' and show phase-sensitive noise as shown by Eqs. (18) and (19) and in Figure 11B schematically. It can be easily seen that

### 4.2 Experimental setup

Figure13 [24] shows the experimental setup for generating and characterizing EPR beams utilizing four variable beam splitters integrated in a photonic chip. Beam splitters BS_{1}, BS_{2}, and BS_{4} are tuned at a 50/50 beam-splitting ratio, and BS_{3} is tuned at 99/1. To guide the output from one beam splitter to another, two fibers are connected mechanically using mating sleeves designed for FC/APC. The additional squeezed light SL_{2} is generated by extra optical parametric oscillator OPO_{2} outside the chip, which has the same optical configuration with OPO_{1}. Firstly, two squeezed light beams SL_{1} and SL_{2} are combined at beam splitter BS_{2}. At the same time, weak coherent beams are introduced into both OPOs for the purpose of phase locking. One percent of the interference signals from those beams is picked up by beam splitter BS_{3}. Then, this signal is used for phase locking the relative phase *ϕ* between squeezed light beams SL_{1} and SL_{2} at 90° using a servo-amplifier and piezo-actuator. By this means, the entangled beams EPR_{1} and EPR_{2} are generated within the photonic chip. The output beam EPR_{1} (EPR_{2}) is combined with local oscillator LO_{1} (LO_{2}) at beam splitter BS_{1} (BS_{4}). Then, the noise levels for both quadratures _{1} (EPR_{2}) beam are detected by balanced homodyne detector HD_{1} (HD_{2}). Finally, the variance of difference signals 〈[Δ(*x̂*_{1}-*x̂*_{2})]^{2}〉 and sum signals 〈[Δ(*p̂*_{1}-*p̂*_{2})]^{2}〉 are measured using a hybrid junction to check the inseparability criterion.

### 4.3 Generation and verification of EPR beams

Figure 14A and B [24] show the noise levels of output beams as the phases of local oscillator LO_{1} and LO_{2} are varied, respectively. The experiment is performed at the 1.5 MHz side band of the laser frequency. The setting of spectrum analyzer is same as the experiments shown in Figures 9 and 10. Trace (i) represents the noise of one of the output beams as relative phase *ϕ* between incident squeezed light beams SL_{1} and SL_{2} is varied while simultaneously scanning the LO phase on a faster timescale. So, the output seems to continuously vary depending on *ϕ* [46]. At the moment that *ϕ* is 0°, which is indicated by dashed arrows, the outputs are two independent squeezed light beams. The noise levels periodically change from anti-squeezed to squeezed quadrature depending on the LO phase. On the other hand, at the moment that *ϕ* is 90°, which is indicated by solid arrows, the outputs are the entangled EPR_{1} and EPR_{2} beams. Traces (ii) and (iii) represent the noise levels of the *x* and *p* quadratures when *ϕ* is locked at 90°. The output EPR_{1} beam shows the noise levels of _{1}. Moreover, the output EPR_{2} shows the noise levels of _{2} above the shot noise level without incident squeezed light beams represented by trace (iv). This phase-insensitive behavior is one of the essential properties of EPR beams.

Finally, 〈[Δ(*x̂*_{1}-*x̂*_{2})]^{2}〉 and 〈[Δ(*p̂*_{1}-*p̂*_{2})]^{2}〉 are measured to check the inseparability criterion. Trace (i) in Figure 14C shows the variance of the difference signal between *x̂*_{1} from HD_{1} and *x̂*_{2}and from HD_{2}. Trace (ii) represents the variance of the sum signal between 1 *p̂*_{1} and *p̂*_{2} obtained in the same way. The measurement shows 〈[Δ(*x̂*_{1}-*x̂*_{2})]^{2}〉 = 1.44±0.12 dB and 〈[Δ(*p̂*_{1}-*p̂*_{2})]^{2}〉 = 1.49±0.12 dB dB below the noise level without quantum correlation, as shown by trace (iii). These results yield a correlation variance of

## 5 Conclusion and outlook

Entanglement is an essentially important resource for the majority of quantum information science and technologies. In this article, we reviewed the currently developed scheme for generating quantum entanglement using integrated waveguide circuits. As an example of DV entanglement, the experiment for generating a photon-entangled state was reviewed. As an example of CV entanglement, we reviewed the current study of generation of EPR beams. It also includes balanced homodyne measurement as a method to detect a quantum state of light and verify the inseparability in CV entangled states. The use of high-quality interference due to the almost perfect mode matching in waveguide circuits and side bands of the optical field to avoid the noise caused by stray light enables simultaneous operation on non-classical light and intense coherent beams in a single photonic chip. The demonstration of all the key components for Gaussian operations within a photonic chip, including the generation and characterization of EPR beams and the verification of entanglement, points the way to full optical integration of CV and hybrid quantum information processing.

For a future outlook, integrated squeezing and non-Gaussian operations like photon counting are required to realize universal quantum information processing [47]. Generation of single-mode squeezed light is demonstrated by using a waveguide device consisting of a non-linear optical crystal such as LiNbO_{3} [48–50] or KTiOPO_{4} [51, 52]. Generation of CV entanglement is also demonstrated by mixing two single-mode squeezed light beams generated by two LiNbO_{3} waveguides with a bulk beam splitter [53]. Recent theoretical work proposes the possibility of full optical integration for generating CV entanglement using non-linear waveguide arrays [54]. Another approach to achieve integrated squeezing is to use an on-chip microresonator such as a silicon micromechanical resonator [55] or a silicon nitride microring resonator that utilizes a four-wave mixing process [56]. Integrated semiconductor detectors [57] or superconducting single-photon detectors [58, 59] are promising and will enable the use of side-band techniques using a recently developed optical high-pass filter [60], thereby eliminating the influence of stray light that would affect the dark counts. These results will yield the way to universal quantum information processing within a monolithic photonic chip.

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