High-fidelity heralded quantum squeezing gate based on entanglement

Kui Liu; Jiaming Li; Rongguo Yang; Shuqin Zhai

doi:10.1364/OE.398096

1. Introduction

As one of the core units of Gaussian quantum computing [1], the squeezing gate makes it possible to realize a controllable Z gate [2], controllable phase gate [3], and cluster state [4]. The squeezing gate [1,5] can achieve universal squeezing on a quantum state to reduce vacuum noise and better realize protocols such as quantum error correction [6] and decoherence mitigation of a quantum state [7].

Squeezing can be realized through nonlinear processes [8] such as the Kerr effect [9] in an optical fiber system, four-wave mixing [10–12] in an atomic system, and optical parametric processes [13], which mainly produce squeezing of a vacuum state and coherent state. Using an optical parametric amplifier, Zhang et al. [14,15] have achieved successive squeezing on a squeezed state. These systems are not easy to control or implement a large-scale integrate because of the complexity of the nonlinear process.

In 2009, Yoshikawa et al. [16] implemented universal squeezing based on auxiliary vacuum-squeezed fields and linear optical elements. In 2014, Su X. et al. [17] realized a squeezing gate and Fourier transformations using auxiliary entangled light. However, the low fidelity of a squeezing gate limits its applications in quantum protocols such as fault-tolerant quantum computing [4]. We realize a high-fidelity heralded squeezing gate [18] using a measurement-based noiseless linear amplifier (MB-NLA) with 6 dB auxiliary squeezing. For a specific target squeezing level for coherent states, we achieved measured fidelities higher than what would be possible using non-heralded schemes that use up to 15 dB [19] of the best available ancillary squeezing. The technique can be applied to non-Gaussian states and provides a promising path towards high-fidelity gate operations and fault-tolerant continuous-variable quantum computation.

This paper introduces squeezing gate schemes based on the MB-NLA. We analyze the scheme (scheme I) of Ref. [18] in detail, showing the advantages of the NLA, and analyze the influence of the purity of the auxiliary squeezed state on the fidelity of the squeezing gate using the NLA. This squeezing gate cannot meet the requirements under the present experimental conditions. Then we build a new heralded squeezing gate based on entanglement. Through calculation, we find that the fidelity of scheme II is immune to the purity of the auxiliary squeezed state. Therefore, compared with scheme I, scheme II better uses the current experimental resources and achieves better squeezing (15 dB) [19]. This makes it possible to realize a perfect squeezing gate for fault-tolerant continuous-variable quantum computation. The new scheme can also be used to realize other single-mode Gaussian operations and non-classical state squeezing operations. This promises efficient quantum information processing and fault-tolerant quantum computation [1,4].

2. Scheme I: heralded squeezing gate based on squeezing

The schematic of the squeezing gate implementation is shown in Fig. 1, where (a) is the deterministic squeezing gate and (b) is the heralded squeezing gate based on squeezing (HSQG-BS) [18]. Panel (b) includes the NLA (with gain ${g_f}$) and deterministic electronic amplification (with gain $g$), while (a) includes only deterministic electronic amplification (with gain $g$).

Fig. 1. Scheme I for a squeezing gate based on squeezing. (a) Deterministic squeezing gate. (b) Heralded squeezing gate with NLA. ${\hat a_{sq}}$: auxiliary squeezed vacuum state; BHD: balanced homodyne detector; AM: amplitude modulator; BS: 50/50 beam splitter; ${g_f}$: NLA gain.

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The NLA was first proposed by Ralph and Lund [20]. This scheme includes quantum scissors [21–23], photon addition-subtraction [24,25], quantum catalysis [26], and the MB-NLA [27,28].The MB-NLA is emulated by imposing conditions on the measurement records via a classical filter function. To realize an MB-NLA [29] with gain ${g_f}$, a dual-homodyne measurement is first performed on an input state ${\rho _{in}}$, resulting in a measured value ${\alpha _m}$ that follows the probability distribution $p\left ( {{\alpha _m}} \right )$. Then, a probabilistic filter $f\left ( {{\alpha _m}} \right )$ is used on the measurement records. The output $\tilde p\left ( {{\alpha _m}} \right )$ is the probability distribution of the amplified quantum state, which is:

(1)$$\tilde p\left( {{\alpha _m}} \right) = \frac{1}{P_s}p\left( {{\alpha _m}} \right)f\left( {{\alpha _m}} \right),$$

where

(2)$$f\left( {{\alpha _m}} \right) = \left\{ {\begin{array}{*{20}{c}} {exp\left[ {{{\left| {{\alpha _m}} \right|}^2} - {{\left| {{\alpha _c}} \right|}^2}\left( {1 - \frac{1}{{{{g_f}^2}}}} \right)} \right],\left| {{\alpha _m}} \right| < {\alpha _c}}\\ {\;\;\;\;\;\;\;\;\;\;1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\;\left| {{\alpha _m}} \right| \ge {\alpha _c}} \end{array}} \right.$$

${\alpha _c}$ is a cutoff parameter, and ${g_f}$ is the NLA gain. The success probability ${P_s}$ of the MB-NLA is given by $P_s = \int \!\!\!\int p\left ( {{\alpha _m}} \right )f\left ( {{\alpha _m}} \right ){d^2}{\alpha _m}$.

Consider a coherent state $\left \langle {{\alpha _{in}}} \right | = \left \langle {({x_{in}} + i{y_{in}})/\sqrt 2 } \right |$ input to an ideal squeezing gate $S\left ( r \right )$. The ideal output state is

(3)$$\begin{array}{l} \left\langle {\hat X_{out}^1} \right\rangle = {x_{in}}{e^{ - r}},\\ \left\langle {\hat Y_{out}^1} \right\rangle = {y_{in}}{e^r},\\ V\left( {\hat X_{out}^1} \right) = {e^{ - 2r}}\ \quad V\left( {\hat Y_{out}^1} \right) = {e^{2r}}. \end{array}$$

Here, $\hat X$ and $\hat Y$ are the amplitude and phase quadrature operators, $\left \langle { \ldots \;} \right \rangle$ denotes the expectation value, and $V(\cdots ) = \left \langle {{\Delta ^2}(\cdots )} \right \rangle$ is the variance of the quadrature operators.

Consider the squeezing gate in Fig. 1. The input state ${\rho _{in}}$ and auxiliary squeezed vacuum state ${\rho _{sq}}$ are coupled at beam splitter BS with transmission $T$; that is,

(4)$$\begin{array}{l} {{\hat a}_t} = \sqrt T {{\hat a}_{in}} + \sqrt {1 - T} {{\hat a}_{sq}},\\ {{\hat a}_r} = \sqrt {1 - T} {{\hat a}_{in}} - \sqrt T {{\hat a}_{sq}}. \end{array}$$

Here, ${\hat a_{sq}}$ is the annihilation operator of the auxiliary squeezed vacuum state, with $V\left ( {{{\hat X}_{sq}}} \right ) = {e^{ - 2r_1}}$ & $V\left ( {{{\hat Y}_{sq}}} \right ) = {e^{2r_2}}$.

Then the phase quadrature ${\hat Y_r}\;$of the reflected part ${\rho _r}$ is measured using a balanced homodyne detector (BHD) with quantum efficiency $\eta$:

(5)$${\hat Y_m} = \sqrt \eta {\hat Y_r} + \sqrt {1 - \eta } {\hat Y_\upsilon } + {\hat Y_{DN} }$$

where ${\hat Y_\upsilon }$ is the induced vacuum fluctuation caused by inefficient detection, and $V\left ( {{{\hat Y}_v}} \right ) = 1$.${\hat Y_{DN} }$ is additional dark noise of homodyne detection, and $V\left ( {{{\hat Y}_{DN}}} \right ) = V_d$.

We can express the transmitted mode ${\hat Y_t}$ as a function of the measurement quadrature ${\hat Y_m}$:

(6)$${\hat Y_t} = c{\hat Y_m} + y + {\hat Y_n}$$

Here, $c = \frac {{Cov\left ( {{{\hat Y}_m},{{\hat Y}_t}} \right )}}{{V\left ( {{{\hat Y}_m}} \right )}}$, $y = \left \langle {{{\hat Y}_t}} \right \rangle - c*\left \langle {{{\hat Y}_m}} \right \rangle$, and $V\left ( {{{\hat Y}_n}} \right ) = V\left ( {{{\hat Y}_t}} \right ) - {c^2}*V\left ( {{{\hat Y}_m}} \right )$, where

\begin{array}{c} Cov\left( {{{\hat Y}_m},{{\hat Y}_t}} \right) = \left\langle {\Delta {{\hat Y}_t}\Delta {{\hat Y}_m}} \right\rangle = \sqrt {\eta T\left( {1 - T} \right)} \left( {1 - {e^{2r_2}}} \right),\\ V\left( {{{\hat Y}_m}} \right) = \left\langle {{\Delta ^2}{{\hat Y}_m}} \right\rangle = \eta \left( {1 - T + T{e^{2r_2}}} \right) + 1 - \eta + V_d,\\ V\left( {{{\hat Y}_t}} \right) = \left\langle {{\Delta ^2}{{\hat Y}_t}} \right\rangle = T + \left( {1 - T} \right){e^{2r_2}},\\ \left\langle {{{\hat Y}_t}} \right\rangle = \sqrt T {y_{in}},\\ \left\langle {{{\hat Y}_m}} \right\rangle = \sqrt \eta \sqrt {1 - T} {y_{in}}. \end{array}

Next, we implement the MB-NLA operation with the filter function of Eq. (2) on measured outcomes of ${\hat Y_m}$. Then we transform ${\hat Y_m}$ into $\hat Y_m^{NLA}$ , and $\left \langle \hat Y_m^{NLA} \right \rangle = {g_f}*\left \langle {\hat Y_m}\right \rangle$, $V\left ( {\hat Y_m^{NLA}} \right ) = {g_f}*V\left ( {{{\hat Y}_m}} \right )$.

Finally, the rescaled signal is fed into the phase modulator to displace the transmitted mode ${\hat a_t}$, and the final output is

(7)$$\begin{array}{c} {{\hat X}_{out}} = {{\hat X}_t} = \sqrt {T} {{\hat X}_{in}} - \sqrt {1 - T} {{\hat X}_{sq}},\\ {{\hat Y}_{out}} = {{\hat Y}_t} + g\hat Y_m^{NLA} = \left( {c + g} \right)\hat Y_m^{NLA} + y + {{\hat Y}_n}. \end{array}$$

The gate is under the unity-gain condition to ensure it is universal; that is, $\left \langle {{{\hat X}_{out}}} \right \rangle = {x_{in}}{e^{ - r}}$, $\left \langle {{{\hat Y}_{out}}} \right \rangle = {y_{in}}{e^r}$. Under this condition, we get

(8)$$\begin{array}{c} T = {e^{ - 2r}},\\ g = \frac{{{\left\langle {{{\hat Y}_{out}}} \right\rangle} - y}}{{f\left\langle {{{\hat Y}_m}} \right\rangle }} - c,\\ V\left( {{{\hat X}_{out}}} \right) = T + \left(1 - T\right){e^{ - 2r_1}},\\ V\left( {{{\hat Y}_{out}}} \right) = {\left( {c + g} \right)^2}{g_f}*V\left( {{{\hat Y}_m}} \right) + {{\hat Y}_n}. \end{array}$$

The fidelity, which is used to quantify the performance of the squeezing gate, is given by [30,31]

(9)$$\begin{array}{c} F = \frac{2}{{\sqrt {\left( {V\left( {{{\hat X}_{out}}} \right) + V\left( {\hat X_{out}^1} \right)} \right)*\left( {V\left( {{{\hat Y}_{out}}} \right) + V\left( {\hat Y_{out}^1} \right)} \right)} }}\;\;\; = \;\;\sqrt {\frac{A}{B}},\\ A = 2{g_f}\eta \left( {{\textrm{{e}}^{2r}}\left( {1 + {V_d}} \right) - \eta + {\textrm{{e}}^{2{r_2}}}\eta } \right),\\ B = {\textrm{{e}}^{ - 2\left( {r + {r_1}} \right)}}\left( { - 1 + {\textrm{{e}}^{2r}} + 2{\textrm{{e}}^{2{r_1}}}} \right)\{ {g_f}\left( {1 + {V_d} - \eta } \right)\eta + {\textrm{{e}}^{2{r_2}}}{g_f}\eta \left( { - 1 - {V_d} + \eta } \right)\\ + {\textrm{{e}}^{2\left( {r + {r_2}} \right)}}{g_f}\eta \left( {1 + {V_d} + \eta } \right) + {\textrm{{e}}^{4r}}\left( {1 + {V_d}} \right)\left( {1 + {V_d} + {g_f}\eta } \right) - {\textrm{{e}}^{2r}}[{\left( {1 + {V_d}} \right)^2} + {g_f}{\eta ^2}].\} \end{array}$$

As shown in Fig. 2, the fidelity of the squeezing gate gradually decreases as the target squeezing increases. To obtain a better fidelity under high target squeezing with the traditional deterministic squeezing gate, the squeezing of the auxiliary squeezed state must be increased, as shown by the solid line in Fig. 2. However, because of the extra noise and loss, the squeezing gate cannot have high enough fidelity even with the best available squeezed state (15dB) [19] to achieve fault-tolerant CV quantum computing, which needs an auxiliary squeezed state with 20 dB (black solid line). The currently available auxiliary squeezing is 10 dB (red solid line) because of the prevailing optical loss. Through conditional postelection technology, the influence of additional noise can be overcome and high fidelity can be obtained with moderate auxiliary squeezing, as shown by the dashed line in Fig. 2. We have previously [18] realized fidelities higher than what would be possible using non-heralded schemes that used up to 15 dB of the best available ancillary squeezing.

Fig. 2. Relationship between fidelity of the squeezing gate and target squeezing. Deterministic squeezing gate with 10 dB auxiliary squeezing (solid red line), 15 dB auxiliary squeezing (solid blue line), and 20 dB auxiliary squeezing (solid black line); heralded squeezing gate with impure 10 dB auxiliary squeezing (dashed red line), pure 10 dB auxiliary squeezing (dotted red line), impure 15 dB auxiliary squeezing (dashed blue line) and pure 15 dB auxiliary squeezing (dotted blue line). The dark noise ${V_d} = 0.01$, corresponding to 20 dB dark noise clearance, and the detection efficiency $\eta = 0.98$.

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The fidelity of the heralded squeezing gate with pure 15 dB squeezing (blue dotted line) can surpass that of the deterministic squeezing gate with 20 dB auxiliary squeezing, which supports fault-tolerant CV quantum computing. The heralded squeezing state with pure 10 dB auxiliary squeezing can meet the requirements of fault-tolerant CV quantum computing to realize squeezing lower than 6 dB.

The inset of Fig. 2 also shows that under small target squeezing, the fidelity of the heralded squeezing gate based on pure 10 dB squeezing is better than that of the heralded squeezing gate based on pure 15 dB squeezing. This is a rather interesting phenomenon not revealed in our previous paper [18]. The reason may be that when the squeezing gate is small, its fidelity is more sensitive to the anti-squeezing noise, which cannot be suppressed completely by NLA, than to the auxiliary squeezing degree.

As shown in Fig. 2, under the current experimental conditions [19], the fidelity of the heralded squeezing gate with a impure 15 dB squeezed state (22 dB antisqueezing) (blue dashed line) and a impure 10 dB squeezed state (11dB antisqueezing) (red dashed line) have a big drop, especially for big target squeezing. Therefore, we analyzed the effect of the purity of the auxiliary squeezed state on the fidelity of the squeezing gate, as shown in Fig. 3. The purity of a squeezed state is defined as $P = \frac {1}{{\sqrt {\textrm {{det}}\left [ D \right ]} }} = \frac {1}{{{e^{r_2 - r_1}}}}$[32]; thermal noise increases with decreasing purity. The purity of the auxiliary squeezing state in Fig. 3 significantly affects the fidelity of the heralded squeezing gate. The fidelity of the squeezing gate decreases as the purity of the auxiliary squeezing state decreases, and the greater the target squeezing is, the greater the influence on the purity is. For example, the fidelity changes from 0.61 to 0.79 for a 20 dB target squeezing gate (brown line in Fig. 3(a)).

Fig. 3. Effect of the purity of the auxiliary squeezed state on the fidelity of the heralded squeezing gate. (a) 15 dB auxiliary squeezing for target squeezing of 5 dB (red line), 10dB (blue line),15 dB (black line), and 20 dB (brown line). (b) Fidelity for 2 dB target squeezing. The fidelity of deterministic squeezing gate with 20 dB auxiliary squeezing (solid black line), the fidelity of the heralded squeezed state with an auxiliary squeezed thermal state with 15 dB squeezing ($P=0.45$) (dashed red line) and with an pure auxiliary squeezed state with 6 dB squeezing (dashed blue line). The dark noise ${V_d}=0.01$ and the detection efficiency $\eta = 0.98$.

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To further analyze the influence of the purity of the auxiliary squeezing state on the fidelity of the squeezing gate, we analyzed the gate fidelity when the auxiliary squeezing state was a squeezed thermal state ($p=0.45$) with 15 dB squeezing [19] and a squeezed vacuum state ($p=1$) with 6 dB squeezing. We see in Fig. 3(b) that when the NLA gain is greater than a certain value, the fidelity of the pure squeezed state with 6 dB squeezing exceeds that of the squeezed thermal state with 15 dB squeezing. Furthermore, its maximum fidelity is close to that of the deterministic squeezing gate with 20 dB, higher than that of the squeezed thermal state.

Therefore, the heralded squeezing gate is more significantly affected by the purity of quantum resources than the degree of squeezing. It is difficult to achieve a pure, high squeezing in experiment [19], and purity gradually decreases with increasing squeezing. Therefore, the purity of the squeezing state restricts the implementation of scheme I.

3. Scheme II: heralded squeezing gate based on entanglement (SQG-BE)

Figure 4 shows scheme II for a heralded squeezing gate based on entanglement [17]. An input state ${\rho _{in}}$ is coupled with one beam ${\hat a_{E_1}}$ of a bipartite entanglement (EPR) at a 50/50 beam splitter (BS1); that is,

(10)$$\begin{array}{l} {{\hat a}_t} = \sqrt {1/2} ({{\hat a}_{in}} + i{{\hat a}_{E_1}}),\\ {{\hat a}_r} = \sqrt {1/2} ({{\hat a}_{in}} - i{{\hat a}_{E_1}}). \end{array}$$

The bipartite entanglement is equal to two auxiliary squeezed vacuum states ($sq_1$ and $sq_2$) with squeezing ${e^{ - 2r_1}}$ and anti-squeezing ${e^{2r_2}}$. The relations $\left \langle {{{\hat a}_{E_1}}} \right \rangle = \left \langle {{{\hat a}_{E_2}}} \right \rangle = 0$, $V\left ( {{{\hat X}_{E_1}}} \right ) = V\left ( {{{\hat X}_{E_2}}} \right ) = \left ( {{e^{ - 2r_1}} + {e^{2r_2}}} \right )/2$, and $V\left ( {{{\hat Y}_{E_1}}} \right ) = V\left ( {{{\hat Y}_{E_2}}} \right ) = \left ( {{e^{2r_1}} + {e^{ - 2r_2}}} \right )/2$ hold.

Fig. 4. Scheme II for heralded squeezing gate based on entanglement. EPR: bipartite entanglement; $sq_1$ and $sq_2$: auxiliary squeezed vacuum state; BHD: balanced homodyne detector; AM: amplitude modulator; PM: phase modulator; BS, BS1: 50/50 beam splitter; ${g_f}$: NLA gain.

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Then the reflected part of BS1 is measured using a dual-balanced homodyne detector (BHD) with quantum efficiency $\eta$:

(11)$$\begin{array}{c} {{\hat X}_{d_1}} = \sqrt \eta \hat X_r^\theta + \sqrt {1 - \eta } {{\hat X}_{\upsilon_1}} + {\hat X_{DN_1}},\\ {{\hat X}_{d_2}} = \sqrt \eta \hat X_t^\theta + \sqrt {1 - \eta } {{\hat X}_{\upsilon_2}} + {\hat X_{DN_2}},\\ \hat X_r^\theta = \sqrt {1/2} \left( {\cos \theta ({{\hat X}_{in}} - {{\hat Y}_{E_1}}} \right) + \sin \theta ({{\hat Y}_{in}} + {{\hat X}_{E_1}})),\\ \hat X_t^\theta = \sqrt {1/2} \left( {\cos \theta ({{\hat X}_{in}} + {{\hat Y}_{E_1}}} \right) - \sin \theta ({{\hat Y}_{in}} - {{\hat X}_{E_1}})). \end{array}$$

where modes $\upsilon _1$ and $\upsilon _2$ are the induced vacuum noise caused by inefficient detection; $V\left ( {{{\hat X}_{v_1}}} \right ) = V\left ( {{{\hat Y}_{v_2}}} \right ) = 1$; ${\hat X_{{DN_{1\left ( 2 \right )}}}}$ is additional dark noise of homodyne detection, and $V\left ( {{{\hat X}_{DN_1}}} \right ) = V\left ( {{{\hat X}_{DN_2}}} \right ) = {V_d}$; and $\theta$ is the detection direction of the BHD, which is the relative phase between the local oscillator beam and the signal.

Subtracting and adding the photocurrents of the two detectors, we get

(12)$$\begin{array}{l} {{\hat X}_m} = \left( {{{\hat X}_{d_1}} + {{\hat X}_{d_2}}} \right)/\sqrt 2 = \sqrt \eta (\cos \theta {{\hat X}_{in}} + \sin \theta {{\hat X}_{E_1}}) + \sqrt {\left( {1 - \eta } \right)} {{\hat X}_{v_1}} + {\hat X_{DN_1}},\\ {{\hat Y}_m} = \left( {{{\hat X}_{d_1}} - {{\hat X}_{d_2}}} \right)/\sqrt 2 = \sqrt \eta (\sin \theta {{\hat Y}_{in}} - \cos \theta {{\hat Y}_{E_1}}) + \sqrt {\left( {1 - \eta } \right)} {{\hat Y}_{v_2}} + {\hat X_{DN_2}}. \end{array}$$

We can express the other beam ${\hat a_{E_2}}$ of the bipartite entanglement (EPR) as a function of the measurement quadratures $({\hat X_m},{\hat Y_m})$ via

(13)$$\begin{array}{c} {{\hat X}_{E_2}} = c{{\hat X}_m} + x + {{\hat X}_n},\;\;\\ {{\hat Y}_{E_2}} = d{{\hat Y}_m} + y + {{\hat Y}_n}. \end{array}$$

where

\begin{array}{c} c = \frac{{\sqrt \eta \sin \theta *V\left( {{{\hat X}_{E_1}},{{\hat X}_{E_2}}} \right)}}{{V\left( {{{\hat X}_m}} \right)}},\\ x = - c\left\langle {{{\hat X}_m}} \right\rangle,\\ V\left( {{{\hat X}_n}} \right) = V\left( {{{\hat X}_{E_2}}} \right) - {c^2}*V\left( {{{\hat X}_m}} \right),\\ d = \frac{{ - \sqrt \eta \cos \theta *V\left( {{{\hat Y}_{E_1}},{{\hat Y}_{E_2}}} \right)}}{{V\left( {{{\hat Y}_m}} \right)}},\\ y = - c\left\langle {{{\hat Y}_m}} \right\rangle,\\ V\left( {{{\hat Y}_n}} \right) = V\left( {{{\hat Y}_{E_2}}} \right) - {c^2}*V\left( {{{\hat X}_m}} \right),\\ Cov\left( {{{\hat X}_{E_1}},{{\hat X}_{E_2}}} \right) = \left( {{e^{ - 2{r_1}}} - {e^{2{r_2}}}} \right)/2,\\ Cov\left( {{{\hat Y}_{E_1}},{{\hat Y}_{E_2}}} \right) = \left( {{e^{2r_1}} - {e^{ - 2r_2}}} \right)/2,\\ V\left( {{{\hat X}_m}} \right) = \eta \left( {{{\cos }^2}\theta + {{\sin }^2}\theta *V\left( {{{\hat X}_{E_1}}} \right)} \right) + 1 - \eta + {V_d},\\ V\left( {{{\hat Y}_m}} \right) = \eta \left( {{{\cos }^2}\theta *V\left( {{{\hat Y}_{E_1}}} \right) + {{\sin }^2}\theta } \right) + 1 - \eta + {V_d},\\ \left\langle{{\hat X}_m}\right\rangle = \sqrt \eta \cos \theta {x_{in}},\\ \left\langle{{\hat Y}_m}\right\rangle = \sqrt \eta \sin \theta {y_{in}}. \end{array}

After that, we implement the NLA operation on measured outcomes of ${\hat X_m}$ and ${\hat Y_m}$. Then we transform ${\hat X_m}$ into $\hat X_m^{NLA}$ and ${\hat Y_m}$ into $\hat Y_m^{NLA}$: $\hat X_m^{NLA} = {g_f}*\left \langle {{{\hat X}_m}} \right \rangle$, $V\left ( {\hat X_m^{NLA}} \right ) = {g_f}*V\left ( {{{\hat X}_m}} \right )$; $\left \langle {\hat Y_m^{NLA}} \right \rangle = {g_f}*\left \langle {{{\hat Y}_m}} \right \rangle$, $V\left ( {\hat Y_m^{NLA}} \right ) = {g_f}*V\left ( {{{\hat Y}_m}} \right )$.

Finally, the rescaled signal is fed into the amplitude modulator and phase modulator to displace the mode ${\hat a_{E_2}}$, and the final output is

(14)$$\begin{array}{l} {{\hat X}_{out}} = \left( {c + {g_x}} \right)\hat X_m^{NLA} + x + {{\hat X}_n},\\ {{\hat Y}_{out}} = \left( {d + {g_y}} \right)\hat Y_m^{NLA} + y + {{\hat Y}_n}. \end{array}$$

Under the unity-gain condition $\left \langle {{{\hat X}_{out}}} \right \rangle = {x_{in}}{e^{ - r}},\left \langle {{{\hat Y}_{out}}} \right \rangle = {y_{in}}{e^r}$, we get

(15)$$\begin{array}{c} {g_x} = \frac{{\left\langle{{\hat X}_{out}}\right\rangle - x}}{{{g_f}\left\langle{{\hat X}_m}\right\rangle}} - c,\\ {g_y} = \frac{{\left\langle{{\hat Y}_{out}}\right\rangle - y}}{{{g_f}\left\langle{{\hat Y}_m}\right\rangle}} - d. \end{array}$$

(16)$$\begin{array}{c} V\left( {{{\hat X}_{out}}} \right) = {\left( {c + {g_x}} \right)^2}{g_f}*V\left( {{{\hat X}_m}} \right) + {{\hat X}_n},\\ V\left( {{{\hat Y}_{out}}} \right) = {\left( {d + {g_y}} \right)^2}{g_f}*V\left( {{{\hat Y}_m}} \right) + {{\hat Y}_n}. \end{array}$$

because we operate at the unity gain point where the mean quadrature amplitudes of the output and target coincide. Similar to scheme I [18], for Gaussian inputs within the operational regime, this scheme including optimal detection direction is independent of the input state.

We can use the above formulas to obtain the fidelity of the heralded squeezing gate in Eq. (9). In particular, a target squeezing factor of zero ($r=0$) corresponds to a teleportation scheme of coherent states [33]. If we consider the pure EPR ($r_1=r_2$) and ideal experimental condition, then the fidelity of teleportation is:

(17)$$F = \frac{2}{{\sqrt {\left( {1 + \frac{{2{\textrm{{e}}^{ - 2r_1}}}}{{g_f}}} \right)\textrm{{*}}\left( {1 + \frac{{2{\textrm{{e}}^{ - 2r_1}}}}{{g_f}}} \right)} }}$$

This formula shows that the effect of the NLA is equivalent to enhancing the entanglement of the input EPR, so the heralded scheme can realize high-fidelity teleportation with limited entanglement.

When the NLA gain tends to infinity ( ${g_f} \to \infty$ ), the fidelity limit is

(18)$$\begin{array}{c} F = \frac{C}{D},\\ C = \{ - 1 - {\textrm{{V}}_d} + \eta + {\textrm{{e}}^{2r}}\left( { - 1 + T} \right)\eta + 2{\textrm{{e}}^{2{\textrm{{r}}_1}}}\left( { - 1 + T} \right)\eta + {\textrm{{e}}^{2r + 2{\textrm{{r}}_1} + 2{\textrm{{r}}_2}}}\left( { - 1 + T} \right)\eta \\ - T\eta - 2{\textrm{{e}}^{2r + 2{\textrm{{r}}_2}}}[1 + {\textrm{{V}}_d} + \left( { - 1 + T} \right)\eta ] - {\textrm{{e}}^{2{\textrm{{r}}_1} + 2{\textrm{{r}}_2}}}[1 + {\textrm{{V}}_d} + \left( { - 1 + T} \right)\eta ]\} \{ T\eta + \\ \left( {{\textrm{{e}}^{2r}} + 2{\textrm{{e}}^{2{\textrm{{r}}_1}}}} \right)\left( {1 + {\textrm{{V}}_d} - T\eta } \right) + {\textrm{{e}}^{2{\textrm{{r}}_2}}}[2{\textrm{{e}}^{2r}}T\eta + {\textrm{{e}}^{2{\textrm{{r}}_1}}}T\eta + {\textrm{{e}}^{2r + 2{\textrm{{r}}_1}}}\left( {1 + {\textrm{{V}}_d} - T\eta } \right)], \\ D = {\textrm{{e}}^{2r}}\{ \left( { - 1 + T} \right)\eta + {\textrm{{e}}^{2{\textrm{{r}}_1} + 2{\textrm{{r}}_2}}}\left( { - 1 + T} \right)\eta - 2{\textrm{{e}}^{2{\textrm{{r}}_2}}}[1 + {\textrm{{V}}_d} + \\ \left( { - 1 + T} \right)\eta ]\} \{ T\eta + {\textrm{{e}}^{2{\textrm{{r}}_1}}}[2 + 2{\textrm{{V}}_d} + \left( { - 2 + {\textrm{{e}}^{2{\textrm{{r}}_2}}}} \right)T\eta ]\} \end{array}$$

The BHD detection direction affects the fidelity of the squeezing gate. First, we analyzed the optimal detection direction of the squeezing gate without the NLA $({g_f} = 1)$, and found that the detection direction in Ref. [17] was not the optimal measurement scheme. As shown in Fig. 4, the solid lines are given using the detection direction ($T = \sin \theta = \frac {1}{{{\textrm {{e}}^{ - 2\textrm {{r}}}} + 1}}$ in Ref. [17], and the dotted lines are along the optimal detection direction. The value of $T$ in Ref. [17] is higher than the optimal direction value (Fig. 5(b)). The difference between the EPR and the optimal detection direction increases when the EPR entanglement and target squeezing increase (the black line in the figure), and the fidelity with the direction in Ref. [17] is significantly lower than the optimal condition.

Fig. 5. Effects of the BHD detection direction on the fidelity of the squeezing gate. (a) Fidelity vs. target squeezing. (b) BHD detection direction $\left ( {\textrm {{T}} = \sin \theta } \right )$ vs. target squeezing. BHDs are in the direction $\left ( {\frac {1}{{{\textrm {{e}}^{ - 2\textrm {{r}}}} + 1}}} \right )$ with 4 dB (red line) and 9 dB auxiliary squeezing (black line), and in the optimal direction with 4 dB (dashed red line) and 9 dB auxiliary squeezing (dashed black line).

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Figure 6 shows the relationship between the fidelity of the scheme II squeezing gate and the target squeezing with different auxiliary squeezed states. Under the current experiment condition, we consider the auxiliary state is the impure 10 dB and 15 dB squeezed state [19]. As in scheme I, the fidelity of the squeezing gate gradually decreases with increasing target squeezing. The traditional deterministic method needs to increase the auxiliary squeezing to obtain better fidelity with high target squeezing. The fidelity of scheme II without the NLA (blue solid line) is lower than that of scheme I without the NLA (black solid line) with the same auxiliary squeezing. However, scheme II can realize more general single-mode quantum state operations such as Fourier transformation and rotation. The NLA helps the heralder of scheme II to overcome the influence of extra noise to improve the fidelity, especially to realize high squeezing operation. When the target squeezing is less than the auxiliary squeezing, almost perfect squeezing can be achieved as shown by the dotted line in the figure. The fidelity of the heralded squeezing gate based on 10 dB squeezing is better than that of the heralded squeezing gate based on 15 dB squeezing for small target squeezing, that is due to higher auxiliary squeezing is more sensitive to the dark noise and loss.

Fig. 6. Relationship between fidelity of the squeezing gate and target squeezing in scheme II. Deterministic squeezing gate in scheme I 10 dB auxiliary squeezing (thick red line) and 15 dB auxiliary squeezing (thick blue line); deterministic squeezing gate in scheme I with 15 dB auxiliary squeezing (thick black line); heralded squeezing gate with 10 dB auxiliary squeezing (dashed red line) and 15 dB auxiliary squeezing (dashed blue line). The dark noise ${V_d}=0.01$ and the detection efficiency $\eta =0.98$.

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Similarly, we consider the effect of the purity of the auxiliary squeezed state on the fidelity of scheme II. Figure 7 shows that the purity of the auxiliary squeezed state has almost no effect on the fidelity of scheme II, especially on the fidelity of the squeezing gate, whose target squeezing is less than that of the auxiliary squeezing. The effect of purity increases with increasing target squeezing, and the purity changes slightly and the purity immunity is far better than that of scheme I (Fig. 3). This is because scheme II is mainly based on the quantum correlation part and does not involve extra thermal noise. Figure 7(b) shows that on the basis of the best available squeezed state with 15 dB squeezing ($P=0.45$) with different dark noise and detection efficiency. The fidelity of scheme II (dashed line) is better than that of scheme I (solid line). Moreover, we can find that the scheme II is sensitive to the dark noise and detection efficiency, and when under lower dark noise and higher detection efficiency (black dashed line), the scheme II can have a significant superiority. Therefore, scheme II is more conducive to practical operation and to obtaining a perfect-fidelity squeezing gate.

Fig. 7. Effect of the purity of the auxiliary squeezed state on the fidelity of the heralded squeezing gate. (a) 15 dB auxiliary squeezing for 5 dB target squeezing (red line), 10 dB (blue line), 15 dB (black line), and 20 dB (brown line). (b) Fidelity of the heralded squeezing gate with 15 dB impure auxiliary squeezing ($P=0.45$) in scheme I (solid line) and scheme II (dashed line). The dark noise ${V_d}=0.01$ and the detection efficiency $\eta =0.98$ (red line). The dark noise ${V_d}=0.001$ and the detection efficiency $\eta =0.99$ (black line).

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In addition, Fig. 8 illustrates the trade-off between fidelity, target squeezing and success probability ${P_s}$. In all cases, a substantial enhancement in fidelity is achieved with the heralded squeezing gate at the expense of success probability. The cutoff ${\alpha _c}$ [18] is chosen to include more than 98% of the total statistics to ensure the Gaussianity of the output is preserved. As shown in figure, because the traditional deterministic method (${P_s}=1$) in scheme I have a big advantage than scheme II, the fidelity of the scheme II is lower than of the scheme I, when with high success probability. With increasing the gain of NLA and decreasing success probability, the fidelity of scheme II can grow steadily, but the fidelity of scheme I can reach saturation point rapidly, which is limited by the impurity of auxiliary squeezing. When the success probability is above 1%, the scheme II (dotted red line) can surpass the scheme I (thick red line). For 10 dB (blue line), even 15 dB(black) target squeezing, the scheme II need a bigger postelection gain at expense of a smaller success probability to go beyond the scheme I. Therefore, considering the trade-off between fidelity, target squeezing and success probability ${P_s}$, we can experimentally conduct the scheme II for the below 10 dB target squeezing operation without dropping below 0.1% success probability.

Fig. 8. Fidelity against success probability with 15 dB impure auxiliary squeezing. The scheme I for target squeezing of 5 dB (thick red line), 10 dB (thick blue line), 15 dB (thick black line), and the scheme II for target squeezing of 5 dB (dashed red line), 10 dB (dashed blue line), 15 dB (dashed black line). The amplitude of input state $\left \langle {{{\hat a}_{in}}} \right \rangle = 0.5$. The dark noise ${V_d}=0.001$ and the detection efficiency $\eta =0.99$.

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4. Conclusion

The squeezing gate is the most important single-mode Gaussian operation whose fidelity directly affects the implementation of a quantum information scheme. Theoretical calculation shows that a high-fidelity squeezing operation can be completed using an NLA with limited quantum squeezing. Obtaining a higher fidelity and exceeding the bound of fault-tolerant quantum computation in scheme I require high squeezing and a pure auxiliary squeezed state. However, it is difficult to prepare a pure squeezed state with squeezing of more than 6 dB. To overcome this problem, we constructed a heralded squeezing gate (scheme II) based on entanglement. First, we optimized the deterministic squeezing gate in Ref. [17] and chose an optimal measurement direction to obtain higher fidelity. Most importantly, we found that the fidelity of scheme II is immune to the purity of the auxiliary squeezed states. Therefore, compared with scheme I, scheme II better uses the best available squeezing (15 dB) to achieve a unity-fidelity squeezing gate. In addition, more single-mode operations can be completed with scheme II. This paper has only considered squeezing of coherent states, whereas it is more meaningful to apply the scheme to non-classical states such as Fock and cat states.

Funding

National Natural Science Foundation of China (11674205, 11874248); National Key Research and Development Program of China (2016YFA0301404); Shanxi 1331 Project; Program for Outstanding Innovative Team of Higher Learning and Institution of Shanxi.

Acknowledgments

We thanks Dr. J. Zhao for helpful discussion and Mark Kurban for polishing up the manuscript.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012). [CrossRef]

2. S. Puri and A. Blais, “High-fidelity resonator-induced phase gate with single-mode squeezing,” Phys. Rev. Lett. 116(18), 180501 (2016). [CrossRef]

3. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79(6), 062318 (2009). [CrossRef]

4. N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112(12), 120504 (2014). [CrossRef]

5. R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-I. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. 106(24), 240504 (2011). [CrossRef]

6. S. L. Braunstein, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80(18), 4084–4087 (1998). [CrossRef]

7. H. Le Jeannic, A. Cavaillès, K. Huang, R. Filip, and J. Laurat, “Slowing quantum decoherence by squeezing in phase space,” Phys. Rev. Lett. 120(7), 073603 (2018). [CrossRef]

8. U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scr. 91(5), 053001 (2016). [CrossRef]

9. R. Dong, J. Heersink, J. F. Corney, P. D. Drummond, U. L. Andersen, and G. Leuchs, “Experimental evidence for raman-induced limits to efficient squeezing in optical fibers,” Opt. Lett. 33(2), 116–118 (2008). [CrossRef]

10. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef]

11. X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123(7), 070506 (2019). [CrossRef]

12. S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020). [CrossRef]

13. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986). [CrossRef]

14. J. Zhang, C. Ye, F. Gao, and M. Xiao, “Phase-sensitive manipulations of a squeezed vacuum field in an optical parametric amplifier inside an optical cavity,” Phys. Rev. Lett. 101(23), 233602 (2008). [CrossRef]

15. Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A 85(4), 040305 (2012). [CrossRef]

16. J.-I. Yoshikawa, T. Hayashi, T. Akiyama, N. Takei, A. Huck, U. L. Andersen, and A. Furusawa, “Demonstration of deterministic and high fidelity squeezing of quantum information,” Phys. Rev. A 76(6), 060301 (2007). [CrossRef]

17. S. Hao, X. Deng, X. Su, X. Jia, C. Xie, and K. Peng, “Gates for one-way quantum computation based on einstein-podolsky-rosen entanglement,” Phys. Rev. A 89(3), 032311 (2014). [CrossRef]

18. J. Zhao, K. Liu, H. Jeng, M. L. Gu, J. Thompson, P. K. Lam, and S. M. Assad, “A high-fidelity heralded quantum squeezing gate,” Nat. Photonics 14(5), 306–309 (2020). [CrossRef]

19. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 db squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016). [CrossRef]

20. T. C. Ralph and A. P. Lund, “Nondeterministic noiseless linear amplification of quantum systems,” Quantum Commun. Meas. & Comput. 1110, 155–160 (2008). [CrossRef]

21. S. Kocsis, G. Y. Xiang, T. C. Ralph, and G. J. Pryde, “Heralded noiseless amplification of a photon polarization qubit,” Nat. Phys. 9(1), 23–28 (2013). [CrossRef]

22. G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4(5), 316–319 (2010). [CrossRef]

23. F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104(12), 123603 (2010). [CrossRef]

24. J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80(5), 053822 (2009). [CrossRef]

25. A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5(1), 52–56 (2011). [CrossRef]

26. A. E. Ulanov, I. A. Fedorov, A. A. Pushkina, Y. V. Kurochkin, T. C. Ralph, and A. I. Lvovsky, “Undoing the effect of loss on quantum entanglement,” Nat. Photonics 9(11), 764–768 (2015). [CrossRef]

27. H. M. Chrzanowski, N. Walk, S. M. Assad, J. Janousek, S. Hosseini, T. C. Ralph, T. Symul, and P. K. Lam, “Measurement-based noiseless linear amplification for quantum communication,” Nat. Photonics 8(4), 333–338 (2014). [CrossRef]

28. J. Zhao, J. Dias, Y. H. Jing, T. Symul, and K. L. Ping, “Quantum enhancement of signal-to-noise ratio with a heralded linear amplifier,” Optica 4(11), 1421 (2017). [CrossRef]

29. J. Zhao, J. Y. Haw, T. Symul, P. K. Lam, and S. M. Assad, “Characterization of a measurement-based noiseless linear amplifier and its applications,” Phys. Rev. A 96(1), 012319 (2017). [CrossRef]

30. J. Richard, “Fidelity for mixed quantum states,” J. Mod. Opt. 41(12), 2315–2323 (1994). [CrossRef]

31. S. Olivares, M. G. A. Paris, and U. L. Andersen, “Cloning of gaussian states by linear optics,” Phys. Rev. A 73(6), 062330 (2006). [CrossRef]

32. G. Adesso and F. Illuminati, “Entanglement in continuous-variable systems: recent advances and current perspectives,” J. Phys. A: Math. Theor. 40(28), 7821–7880 (2007). [CrossRef]

33. A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998). [CrossRef]

High-fidelity heralded quantum squeezing gate based on entanglement

Abstract

1. Introduction

2. Scheme I: heralded squeezing gate based on squeezing

3. Scheme II: heralded squeezing gate based on entanglement (SQG-BE)

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (8)

Equations (20)

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