Realization of the tradeoff between internal and external entanglement

TL;DR

This paper experimentally demonstrates a new entanglement monogamy relation showing that internal and external entanglement in twin photons limit each other, confirming theoretical predictions and advancing understanding of quantum entanglement constraints.

Contribution

The study provides the first experimental observation of the tradeoff between internal and external entanglement, revealing a new monogamy relation in quantum entanglement.

Findings

Internal and external entanglement limit each other

Maximal internal entanglement leads to vanishing external entanglement

Experimental results match theoretical predictions

Abstract

We experimentally realize the internal and external entanglement tradeoff, which is a new kind of entanglement monogamy relation different from that usually discussed. Using a source of twin photons, we find that the external entanglement in polarization of twin photons, and the path-polarization internal entanglement of one photon, limit each other. In the extreme case, when the internal state is maximally entangled, the external entanglement must be vanishing, that illustrate entanglement monogamy. Our results of the experiment coincide with the theoretical predictions, and therefore provide a direct experimental observation of the internal and external entanglement monogamy relation.

Tables

Table 1. Table 1: Fidelities for the quantum state Eq. (8) of the main text. The θ 𝜃 \theta is fixed at 45 ∘ superscript 45 45^{\circ} but the ϕ italic-ϕ \phi is changing from 0 ∘ superscript 0 0^{\circ} to 90 ∘ superscript 90 90^{\circ} . The average fidelity of these states is 99.11 ± 0.04 % plus-or-minus 99.11 percent 0.04 99.11\pm 0.04\% .

$\varphi$	$0^{\circ }$	${15}^{\circ }$	${30}^{\circ }$	${45}^{\circ }$	${50}^{\circ }$
Fidelity	$99.05\pm 0.11\%$	$99.16\pm 0.07\%$	$99.51\pm 0.02\%$	$99.16\pm 0.05\%$	$98.94\pm 0.05\%$
$\varphi$	${60}^{\circ }$	${70}^{\circ }$	${80}^{\circ }$	${90}^{\circ }$	average
Fidelity	$99.10\pm 0.06\%$	$98.77\pm 0.03\%$	$98.20\pm 0.12\%$	$99.39\pm 0.05\%$	$99.11\pm 0.04\%$

Equations

$$C_{A|B}^2+C_{A|C}^2\le C_{A|BC}^2,$$

$${\tilde{E}}_{A_1|A_2}+E_{A_1{A}_2|B}\le {\tilde{E}}_{\max },$$

$$E_F({\varrho }_{A_1{A}_2})+E_F^{\prime }(|\psi {⟩}_{A_1{A}_2|B})\le 1,$$

$$E_F^{\prime }(|\psi {⟩}_{A_1{A}_2|B})$$

$$|\psi ⟩=\cos \varphi |110⟩+\sin \varphi \frac{|01⟩+|10⟩}{\sqrt{2}}|1⟩.$$

$$E_F({\varrho }_{A_1{A}_2})$$

$$E_F^{\prime }(|\psi {⟩}_{A_1{A}_2|B})$$

$$|\psi ⟩=\cos \varphi |1{⟩}_{A_1}|1{⟩}_{A_2}|0{⟩}_B+\sin \varphi |\phi {⟩}_{A_1{A}_2}|1{⟩}_B,$$

$$E_N({\varrho }_{A_1{A}_2})+g[E_N({\varrho }_{A_1{A}_2|B})]\le E_{N,\max },$$

$$C({\varrho }_{A_1{A}_2})+\tilde{g}[C({\varrho }_{A_1{A}_2|B})]\le C_{\max },$$

$$E_N({\varrho }_{A_1{A}_2})+E_N^{\prime }(|\psi {⟩}_{A_1{A}_2|B})\le \frac{1}{2},$$

$$E_N^{\prime }(|\psi {⟩}_{A_1{A}_2|B})=\frac{1}{2}-E_N({\varrho }_{A_1{A}_2}^{\prime }),$$

$$E_N({\varrho }_{A_1{A}_2}^{\prime })$$

$$|\psi ⟩=\cos \varphi |110⟩+\sin \varphi \frac{|01⟩+|10⟩}{\sqrt{2}}|1⟩.$$

$$E_N({\varrho }_{A_1{A}_2})$$

$$E_N^{\prime }(|\psi {⟩}_{A_1{A}_2|B})$$

$$E_N({\varrho }_{A_1{A}_2})+g[E_N(|\psi {⟩}_{A_1{A}_2|B})]\le \frac{1}{2},$$

$$g(x)=\frac{3}{4}-\frac{\sqrt{1-2x^2}}{2}-\frac{\sqrt{1-4x^2}}{4},$$

$$|\psi ⟩=\cos \varphi |110⟩+\sin \varphi \frac{|01⟩+|10⟩}{\sqrt{2}}|1⟩.$$

$$E_N({\varrho }_{A_1{A}_2})$$

$$g[E_N(|\psi {⟩}_{A_1{A}_2|B})]$$

$$C({\varrho }_{A_1{A}_2})+\tilde{g}[C({\varrho }_{A_1{A}_2|B})]\le 1,$$

$$\tilde{g}(x)=\frac{1-\sqrt{1-x^2}}{2}.$$

$$C(|\psi {⟩}_{A_1{A}_2|B})=\sqrt{2(1-\mathrm{Tr}{\varrho }_{A_1{A}_2}^2)}=2\sqrt{{\lambda }_1(1-{\lambda }_1)},$$

$$C({\varrho }_{A_1{A}_2|B})\le 2\sqrt{{\lambda }_1(1-{\lambda }_1)},$$

$$C({\varrho }_{A_1{A}_2})$$

$$C({\varrho }_{A_1{A}_2})+\tilde{g}[C({\varrho }_{A_1{A}_2|B})]$$

$$|\psi ⟩=\cos \varphi |110⟩+\sin \varphi \frac{|01⟩+|10⟩}{\sqrt{2}}|1⟩.$$

$$C({\varrho }_{A_1{A}_2})$$

$$\tilde{g}[C(|\psi {⟩}_{A_1{A}_2|B})]$$

$${\lambda }_1=\max \{{\cos }^2\varphi ,{\sin }^2\varphi \}.$$

$$E_F^{\prime }({\varrho }_{A_1{A}_2|B})=\underset{\{p_i,|{\psi }_i{⟩}_{A_1{A}_{2}B}\}}{\inf }\sum _i{p}_i{E}_F^{\prime }(|{\psi }_i{⟩}_{A_1{A}_2|B}).$$

Full text

Entanglement monogamy is one of the most fundamental properties for multipartite quantum states, which means that if two qubits $A$ and $B$ are maximally entangled, then $A$ or $B$ cannot be entangled with the third qubit $C$ mono1 ; mono2 . The quantitative entanglement monogamy inequality was first proved by Coffman, Kundu, and Wootters (CKW) for three-qubit states CKW ,

[TABLE]

where $C^{2}$ denotes the squared concurrence for quantifying bipartite entanglement Wootters . From Eq. (1), one can easily find that there is a consequent tradeoff between the amount of entanglement shared by qubits $A$ and $B$, and the entanglement shared by qubits $A$ and $C$. For three-qubit pure states, the difference between the right hand side and left hand side of Eq. (1) is defined as the so-called “three-tangle” CKW , which is a genuine three-qubit entanglement measure. After the CKW inequality, several entanglement monogamy inequalities NCKW ; gaussian1 ; gaussian2 ; gaussian3 ; gaussian4 ; SEntangle1 ; SEntangle2 ; negativity ; CREN ; Kim2 ; Allen ; SEOF ; SEOF2 ; SEOF3 ; bai2 ; Kim3 ; Renyi ; Kim4 ; song ; song2 ; exp ; exp2 ; Gour ; Yu ; Sen ; Sen2 ; Fei ; Fei2 ; Fei3 ; ou2 ; Regula ; Regula2 ; Osterloh ; Luo ; Luo2 ; Winter ; Eltschka ; Eltschka2 ; Eltschka3 and even monogamy equalities equa ; equa2 were introduced. Osborne and Verstraete proved the CKW monogamy inequality for $N$-qubit states NCKW . In Refs. gaussian1 ; gaussian2 , the CKW inequality was generalized to Gaussian states. Moreover, other entanglement measures, such as the squashed entanglement SEntangle1 ; SEntangle2 , the negativity negativity ; CREN ; Kim2 ; Allen , and the squared entanglement of formation SEOF ; SEOF2 ; SEOF3 , were also employed to derive the corresponding entanglement monogamy inequalities.

Recently, new kinds of monogamy relation have been derived by Camalet Camalet ; Camalet2 ; Camalet3 ; Camalet4 , i.e. internal entanglment (or local quantum resource) and external entanglement have a tradeoff. The usually discussed entanglement monogamy inequalities in Refs. NCKW ; gaussian1 ; gaussian2 ; SEntangle2 ; negativity ; CREN ; Kim2 ; SEOF ; SEOF2 ; SEOF3 indicate the trade-off relation between $E(\varrho_{AB})$ and $E(\varrho_{AC})$ (or its extension to $N$-partite case), where $E$ is one kind of entanglement measure, $\varrho_{AB}$ and $\varrho_{AC}$ are reduced density matrices from a three-qubit state. Unlike these previously derived inequalities, Camalet has proposed a new entanglement monogamy inequality Camalet . Consider a tripartite quantum state $\varrho_{A_{1}A_{2}B}$ illustrated in Fig. 1, where $A_{1}$ and $A_{2}$ come from the same physical system but have been encoded in different degrees of freedom, and $B$ is encoded in another physical system. This inequality shows the tradeoff relation between the internal entanglement $\tilde{E}_{A_{1}|A_{2}}$ and the external entanglement $E_{A_{1}A_{2}|B}$, where $\tilde{E}$ and $E$ are two different but related entanglement measures, and $E_{A_{1}A_{2}|B}$ ($\tilde{E}_{A_{1}|A_{2}}$) denotes the entanglement of $\varrho_{A_{1}A_{2}B}$ ($\varrho_{A_{1}A_{2}}$) under the bipartition $A_{1}A_{2}|B$ ($A_{1}|A_{2}$).

Here we experimentally demonstrate the entanglement monogamy relation between the internal and external entanglement, with a source of twin photons. As shown in Fig. 1, there are two qubits (the polarization qubit $A_{1}$ and the path qubit $A_{2}$) encoded in photon $A$; but only one qubit (the polarization qubit $B$) is encoded in photon $B$. Here we provide a direct experimental observation of the tradeoff between the internal entanglement in $A_{1}|A_{2}$ and the external entanglement in $A_{1}A_{2}|B$.

Theoretical framework.— Let us focus on a tripartite state $\varrho_{A_{1}A_{2}B}$ where $A_{1}$ and $A_{2}$ are encoded in the same physical system $A$ by using different degrees of freedom, see Fig. 1. The third party is encoded in system $B$. Camalet’s entanglement monogamy inequality is

[TABLE]

where $\tilde{E}_{A_{1}|A_{2}}$ denotes the internal entanglement measure between $A_{1}$ and $A_{2}$, $E_{A_{1}A_{2}|B}$ is the external entanglement measure between $A_{1}A_{2}$ and $B$, and $\tilde{E}_{\max}$ is the value of $\tilde{E}_{A_{1}|A_{2}}$ when $A_{1}$ and $A_{2}$ are maximally entangled Camalet . It is worth noting that $\tilde{E}$ and $E$ are strongly related, although they are two different entanglement measures. From the inequality (2), one can see that $E_{A_{1}A_{2}|B}$ is also bounded by $\tilde{E}_{\max}$. When the state $\varrho_{A_{1}A_{2}B}$ is pure and the reduced density operator $\varrho_{A_{1}A_{2}}$ is absolutely separable AS ; MEMS , the external entanglement $E_{A_{1}A_{2}|B}$ is equal to the maximum value $\tilde{E}_{\max}$. On the other hand, when the internal entanglement $\tilde{E}_{A_{1}|A_{2}}$ is maximal, the external entanglement $E_{A_{1}A_{2}|B}$ must be vanishing.

The internal entanglement measure $\tilde{E}_{A_{1}|A_{2}}$ in Eq. (2) can be arbitrary entanglement measures, such as the entanglement of formation $E_{F}$ Wootters , the negativity $E_{N}$ neg1 ; neg2 , and the relative entropy of entanglement $E_{R}$ ER . When we choose the entanglement of formation $E_{F}$ to quantify the internal entanglement between $A_{1}$ and $A_{2}$, the inequality (2) for a general three-qubit pure state $|\psi\rangle_{A_{1}A_{2}B}$ becomes to

[TABLE]

where the internal entanglement $\tilde{E}_{A_{1}|A_{2}}$ is E_{F}(\varrho_{A_{1}A_{2}})=H\big{(}1/2+\sqrt{1-C^{2}(\varrho_{A_{1}A_{2}})}/2\big{)}, $H$ is the binary entropy $H(x):=-x\log_{2}x-(1-x)\log_{2}(1-x)$, and $C(\varrho)=\max\{0,\sigma_{1}-\sigma_{2}-\sigma_{3}-\sigma_{4}\}$ is the concurrence of $\varrho$ with $\{\sigma_{i}\}$ being the square roots of eigenvalues of $\varrho\sigma_{y}\otimes\sigma_{y}\varrho^{*}\sigma_{y}\otimes\sigma_{y}$ in decreasing order Wootters . The external entanglement $E_{A_{1}A_{2}|B}$ is $E_{F}^{\prime}(|\psi\rangle_{A_{1}A_{2}|B})$, as defined by

[TABLE]

where $U$ denotes the unitary operators of $A$, $f(x):=H(1/2+\sqrt{1-x^{2}}/2)$, and $\{\lambda_{i}\}$ are the eigenvalues of $\varrho_{A_{1}A_{2}}$ in nonascending order Camalet ; MEMS . In Ref. MEMS , the maximum entanglement for a given spectrum $\{\lambda_{i}\}$ measured by the negativity and the relative entropy of entanglement have also been provided. Thus, one can obtain the inequality (2) with the internal entanglement measure being the negativity and the relative entropy of entanglement as well SM .

Now we consider a class of three-qubit pure states with one parameter $\phi$,

[TABLE]

Based on Eqs. (3)-(Realization of the tradeoff between internal and external entanglement), one can obtain its internal and external entanglement by using the entanglement of formation,

[TABLE]

The theoretical results have been shown in Fig. 2 by the solid curve. We can see that all the results are bounded by the dashed line, i.e., the monogamy inequality (3) always holds.

Experimental realization.— In order to demonstrate this new entanglement monogamy relation, we prepare some quantum states where the quantity of internal and external entanglement can be controlled. We use the polarization and the path degrees of freedom to produce the target three-qubit state in Eq. (5).

As shown in Fig. 3, we will introduce three parts of the setup: (i) state preparation, (ii) qubits $A_{1}$ and $A_{2}$ (owned by Alice), (iii) qubit $B$ (owned by Bob). First, the source of twin photons is realized via a type-I spontaneous parametric down-conversion (SPDC) process in which the crystal is a joint $\beta$-barium-borate ($\beta$-BBO) P.G.Kwiat1999 . The source of the two-qubit entangled state is $|\psi\rangle=\cos\phi\left|H\right\rangle_{A}\left|H\right\rangle_{B}+\sin\phi\left|V\right\rangle_{A}\left|V\right\rangle_{B}$, where the parameter $\phi$ is modulated by H1, a half-wave plate (HWP) put in front of the BBO crystal to adjust the polarization of pump. Here the pump is a continuous-wave diode laser with 140mW and the wavelength is 404nm. The fidelity between the experimental state and the theoretical state is beyond 99$\%$. The computational basis $\left|0\right\rangle$ and $\left|1\right\rangle$ are encoded in the horizontal polarization $\left|H\right\rangle$ and vertical polarization $\left|V\right\rangle$ of the photons, respectively. The photon pair is separated spatially via a single mode fiber (SMF). One photon is sent to Alice and the other one is sent to Bob. In Alice part, as illustrated in Fig. 3, we use three beam displacers (BDs) in which the vertical-polarized photon remains on its path while the horizontal-polarized photon shifts down. The internal entanglement is realized between the path and the polarization degrees of freedom of Alice’s photon. The upper path state is encoded into $\left|0\right\rangle$, and the down path state is corresponding to $\left|1\right\rangle$. After BD1, the photons in different polarization states travel two paths. Thus the polarization and path are entangled. Due to the HWP at $45^{\circ}$ (H7), the horizontal and vertical polarization exchange, whereafter the HWP in the upper path (H8) rotates the polarization. The angle $\theta$ modulated by H8 is the controllable parameter of the internal entanglement. Right after the H8, there is another beam displacer, BD2, to fulfill the preparation of the internal entanglement of the Alice’s photon. On the other hand, in Bob’s part, the photons are measured directly. Finally we get the three-qubit states which contain internal and external entanglement and can be described as

[TABLE]

where $|\varphi\rangle_{A_{1}A_{2}}=\cos\theta|0\rangle_{A_{1}}|1\rangle_{A_{2}}+\sin\theta|1\rangle_{A_{1}}|0\rangle_{A_{2}}$.

In order to reconstruct the density matrices of these three-qubit states, we perform quantum state tomography for these states. According to the maximum likelihood estimation, the density matrices are reconstructed Daniel F.V.James2001 . The project measurement of polarization is realized by a standard polarization tomography setup (SPTS), which consists of a quart-wave plate (QWP), a half-wave plate, and a polarization beam splitter (PBS). As shown in the Fig. 3, there are three such setups, (Q1,H5,P1), (Q2,H9,P2), and (Q3,H11,P3). The first two setups measure the polarization states of Bob and Alice respectively. As for the last one, it is used to measure the path state of Alice. The following is demonstrated how it works. Consider the BD2, it is the last element in the state preparation. If a photon is in the upper path after BD2, its polarization is horizontal when it arrives at Q3, i.e. the last SPTS; and if it is in the down path, it will be vertical-polarized. Therefore, the last SPTS measure the path state via the polarization state tomography. Besides it is necessary to mention that the HWP right after P2 in down path (H10) is at $45^{\circ}$.

We fix the value of $\theta$ at $45^{\circ}$ and adjust the H1 to change the values of $\phi$ from $0^{\circ}$ to $90^{\circ}$. Thus, the experimental state in Eq. (8) becomes to Eq. (5). If $\phi=90^{\circ}$, the internal entanglement of system $A$ is maximal; and if $\phi=0^{\circ}$, the state is $|\psi\rangle=|110\rangle$ which is separable. Without loss of generality, we choose nine typically states to analyze. We reconstruct the density matrices of these nine states, then calculate their entanglements $\tilde{E}_{A_{1}|A_{2}}$ and $E_{A_{1}A_{2}|B}$ based on $E_{F}$ using the expressions given above. As shown in Fig. 4, the experimental results coincide with the theoretical prediction within the margin of error. Although the different value of the angle $\phi$ represent the different states, the sum of $\tilde{E}_{A_{1}|A_{2}}+E_{A_{1}A_{2}|B}$ will not exceed 1, which experimentally demonstrates the monogamy relation (2). We remark that $E_{A_{1}A_{2}|B}$ is evaluated using Eq. (Realization of the tradeoff between internal and external entanglement) which is strictly speaking valid only for genuine pure states. However, the actual entanglement $E_{A_{1}A_{2}|B}$ of the experimental state, which is not exactly pure, is lower than the value obtained from Eq. (Realization of the tradeoff between internal and external entanglement) Camalet ; Camalet3 ; SM . On the other hand, we choose $\tilde{E}_{A_{1}|A_{2}}$ as the horizontal ordinate and $E_{A_{1}A_{2}|B}$ as the vertical ordinate to plot Fig. 2. We find that $E_{A_{1}A_{2}|B}$ firstly increases and then decreases as $\tilde{E}_{A_{1}|A_{2}}$ increases, which also agrees with the theoretical results. Moreover, these values are in the area below the straight line $E_{A_{1}A_{2}|B}=1-\tilde{E}_{A_{1}|A_{2}}$.

Other monogamy inequalities.— In Ref. Camalet , Camalet also presented a monogamy inequality involving only one entanglement monotone, the negativity $E_{N}$. For a bipartite state $\varrho_{AB}$, the negativity is defined by $E_{N}(\varrho_{AB})=(\|\varrho_{AB}^{T_{B}}\|-1)/2$ neg1 ; neg2 , where $\|\cdot\|$ is the trace norm and $T_{B}$ is the partial transpose with respect to system $B$. Contrary to other entanglement measures, such as $E_{F}$ or $E^{\prime}_{F}$, $E_{N}$ is readily computable for any state. For a three-qubit state $\varrho_{A_{1}A_{2}B}$, the monogamy inequality is

[TABLE]

where $E_{N,\mathrm{max}}$, the maximum value of $E_{N}$, is equal to $1/2$ for the two-qubit states, and the nondecreasing function $g$ is given by $g(x)=(3/2-\sqrt{1-2x^{2}}-\sqrt{1/4-x^{2}})/2$ Camalet . The monogamy inequality (9) has been calculated for our experimentally realized states in the Supplemental Material SM .

Many familiar entanglement monotones satisfy monogamy inequalities of the form of Eq. (9) Camalet4 , but determining explicitly the corresponding function $g$ may not be always possible. Now we present another case for which this can be achieved. For a qubit-qudit pure state $|\phi_{AB}\rangle$, the concurrence is defined by $C(|\phi_{AB}\rangle)=\sqrt{2(1-{\rm Tr}\varrho_{B}^{2})}$ Wootters ; CKW , where $\varrho_{B}={\rm Tr}_{A}(|\phi_{AB}\rangle\langle\phi_{AB}|)$ is the reduced density operator of system $B$. It is generalized to mixed states via the convex roof extension Wootters ; CKW . For a three-qubit state $\varrho_{A_{1}A_{2}B}$, the following monogamy inequality holds

[TABLE]

where $C_{\mathrm{max}}$, the maximum value of $C$, is equal to $1$ for the two-qubit states, and the nondecreasing function $\tilde{g}$ is given by $\tilde{g}(x)=(1-\sqrt{1-x^{2}})/2$ SM . The monogamy inequality (10) has also been calculated for our experimentally realized states in the Supplemental Material SM .

Discussion and conclusion.— In this experiment, the visibility of the MZ interferometer is about 100:1 and the average fidelity Nielsen between the experimental states and theoretical states is $99.11\pm 0.04\%$. Although the experimental states are not exactly pure states, the monogamy inequality (3) still holds for experimental mixed states. Moreover, the negativity in inequality (9) has been evaluated for the mixed experimental states. Besides the single-photon avalanche photon-diode is used to detect photons whose detection efficiency is 68%. The detection events from the same pair are identified by a coincidence counter as long as they arrive within $\pm 3$ns. In addition, the coincidence counts are about 1000$\text{s}^{-1}$ and we record clicks for 10s. There are many sources of the measurement uncertainty, such as counting statistics, detector efficiency, detector’s dead time, timing uncertainty and alignment error of wave plates. However, the resulting uncertainty is dominated by counting statistics Hou Shun Poh2015 , which we have calculated via the Poissonian distribution and shown in the figures.

In summary, we have demonstrated the internal and external entanglement tradeoff in a photonic system with tunable entangled sources. This realization verifies the theoretical prediction that the entanglement between different degrees of freedom of a quantum single particle restricts its entanglement with other particles. This property may have applications in quantum information, such as the construction of quantum communication network. On the other hand, this realization also can be generalized into other physical systems, such as NV centers, atoms, trapped ions, superconductor and so on. Particularly realization in a hybrid system, such as a photon and an atom, is more expected. For future research, one may experimentally demonstrate other monogamy inequalities, such as the inequality between local coherence and entanglement Fan ; Camalet , and the inequality between internal entanglement and external correlations Camalet4 .

This work is funded by the National Natural Science Foundation of China (Grants Nos. 11504253, 11575051, 11674306, 61590932 and 11734015), National Key R$\&$D Program (No. 2016YFA0301300 and No. 2016A0301700), Anhui Initiative in Quantum Information Technologies, the startup funding from Soochow University (Grant No. Q410800215) and the Hebei NSF (Grant No. A2016205215).

I SUPPLEMENTAL MATERIAL

I.1 I. Inequality (3) for the negativity $E_{N}$

If we use the negativity $E_{N}$ to quantify the internal entanglement between $A_{1}$ and $A_{2}$, the inequality (3) in the main text becomes to

[TABLE]

where $E_{N}(\varrho_{A_{1}A_{2}})=(\|\varrho_{A_{1}A_{2}}^{T_{A_{2}}}\|-1)/2$, and $E_{N,\mathrm{max}}$, the maximum value of $E_{N}(\varrho_{A_{1}A_{2}})$, is equal to $1/2$ for two-qubit state $\varrho_{A_{1}A_{2}}$. The external entanglement $E_{N}^{\prime}(|\psi\rangle_{A_{1}A_{2}|B})$ is

[TABLE]

with

[TABLE]

where $U$ denotes the unitary operators of $A$, $\varrho_{A_{1}A_{2}}^{\prime}$ the density operator corresponding to the maximum over $U$, and $\{\lambda_{i}\}$ are the eigenvalues of $\varrho_{A_{1}A_{2}}$ in nonascending order SMEMS .

Now we consider a class of three-qubit pure states with one parameter $\phi$,

[TABLE]

Based on Eqs. (S1)-(S3), one can obtain its internal and external entanglement measured by the negativity,

[TABLE]

The theoretical and experimental results have been shown in Fig. S1. We can see that all the sums of internal and external entanglement are bounded by $1/2$, i.e., the inequality (S1) always holds.

I.2 II. The monogamy inequality (9) involving only $E_{N}$

In Ref. SCamalet , the author also presented a monogamy inequality involving only one entanglement monotone, the negativity $E_{N}$. For a bipartite state $\varrho_{AB}$, the negativity is defined by $E_{N}(\varrho_{AB})=(\|\varrho_{AB}^{T_{B}}\|-1)/2$, where $\|\cdot\|$ is the trace norm and $T_{B}$ is the partial transpose with respect to system $B$. For a three qubit pure state $|\psi\rangle_{A_{1}A_{2}B}$, the monogamy inequality is

[TABLE]

where $E_{N,\mathrm{max}}$, the maximum value of $E_{N}(\varrho_{A_{1}A_{2}})$, is equal to $1/2$ for two-qubit state $\varrho_{A_{1}A_{2}}$, and the nondecreasing function $g$ is given by

[TABLE]

when the number of nonzero eigenvalues of $\varrho_{A_{1}A_{2}}$ is equal to 2 SCamalet .

Now we consider a class of three-qubit pure states with one parameter $\phi$,

[TABLE]

Based on Eqs. (S7) and (S8), one can obtain its internal and external entanglement measured by the negativity,

[TABLE]

The theoretical and experimental results have been shown in Fig. S2. We can see that all the sums of internal and external entanglement are bounded by $1/2$, i.e., the inequality (S7) always holds.

I.3 III. The monogamy inequality (10) involving only $C$

Proposition 1. For a three-qubit state $\varrho_{A_{1}A_{2}B}$, the internal entanglement $C(\varrho_{A_{1}A_{2}})$ and the external entanglement $C(\varrho_{A_{1}A_{2}|B})$, as quantified by the concurrence, obey the monogamy relation

[TABLE]

where $1$ is the maximal value of $C$ for two-qubit states and

[TABLE]

Proof. For a pure three-qubit state, the external entanglement is given by

[TABLE]

with $\lambda_{1}$ being the maximal eigenvalue of $\varrho_{A_{1}A_{2}}$. Since the above function of $\varrho_{A_{1}A_{2}}$ is concave and the concurrence is defined via the convex roof extension for mixed states, the external entanglement obeys

[TABLE]

in the general case. The internal entanglement can be obtained by the formula $C(\varrho_{A_{1}A_{2}})=\max\{0,\sigma_{1}-\sigma_{2}-\sigma_{3}-\sigma_{4}\}$ with $\{\sigma_{i}\}$ being the square roots of eigenvalues of $\varrho_{A_{1}A_{2}}\sigma_{y}\otimes\sigma_{y}\varrho_{A_{1}A_{2}}^{*}\sigma_{y}\otimes\sigma_{y}$ in decreasing order SWootters . In the bipartite partition $A_{1}A_{2}|B$, the internal entanglement can be maximized via two-qubit unitary transformations on $A_{1}A_{2}$, and the following relation holds

[TABLE]

where the equality is satisfied for the so-called maximally entangled mixed state (MEMS) $\varrho^{\prime}_{A_{1}A_{2}}$ SMEMS . In the case of two-qubit MEMSs SMEMS , its concurrence is $C(\varrho^{\prime}_{A_{1}A_{2}})=\max\{0,\lambda_{1}-\lambda_{3}-2\sqrt{\lambda_{2}\lambda_{4}}\}\leq\lambda_{1}$, and hence $C(\varrho_{A_{1}A_{2}})\leq\lambda_{1}$. Therefore, we have

[TABLE]

where $1/2\leq\lambda_{1}\leq 1$ has been used. Thus, we obtain the monogamy relation (S12). $\square$

Now we consider a class of three-qubit pure states with one parameter $\phi$,

[TABLE]

One can obtain its internal and external entanglement measured by concurrence,

[TABLE]

with

[TABLE]

One can see that if $\lambda_{1}=\sin^{2}\phi$ (i.e., $\sin^{2}\phi\geq\cos^{2}\phi$), then $C(\varrho_{A_{1}A_{2}})+\tilde{g}[C(|\psi\rangle_{A_{1}A_{2}|B})]=1$ holds. The theoretical and experimental results have been shown in Fig. S3. We can see that all the sums of internal and external entanglement are bounded by 1, i.e., the inequality (S12) always holds.

I.4 IV. Quantum state tomography

We performed tomography to nine states we prepared. Their density matrices can be described in Eq. (8) of the main text. The $\theta$ is fixed at $45^{\circ}$ but the $\phi$ is changing. Following are their density matrices that are obtained by maximum likelihood estimation. We only labeled the values of $\phi$ to distinguish different states and also listed the fidelity of them.

Although the experimental states are not exactly pure states, the monogamy inequality (3) in the main text still holds for experimental mixed states. Suppose that the experimentally realized tripartite state is $\varrho_{A_{1}A_{2}B}$, thus the external entanglement is defined by the convex roof,

[TABLE]

We assume that $\varrho_{A_{1}A_{2}B}=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|$ is the optimal decomposition for $\varrho_{A_{1}A_{2}B}$ to achieve the above infimum. Therefore,

[TABLE]

where $\varrho_{j,A_{1}A_{2}}={\rm Tr}_{B}|\psi_{j}\rangle\langle\psi_{j}|$, $E_{F}^{\prime}(|\psi_{j}\rangle_{A_{1}A_{2}|B})=1-E_{F}(\varrho_{j,A_{1}A_{2}}^{\prime})$, $E_{F}(\varrho_{j,A_{1}A_{2}}^{\prime})=\max_{U}E_{F}(U\varrho_{j,A_{1}A_{2}}U^{\dagger})=E_{F}(U_{j}\varrho_{j,A_{1}A_{2}}U_{j}^{\dagger})$, $E_{F}(\varrho_{A_{1}A_{2}}^{\prime})=\max_{U}E_{F}(U\varrho_{A_{1}A_{2}}U^{\dagger})=E_{F}(V\varrho_{A_{1}A_{2}}V^{\dagger})$, the first inequality holds since $E_{F}(U_{j}\varrho_{j,A_{1}A_{2}}U_{j}^{\dagger})=\max_{U}E_{F}(U\varrho_{j,A_{1}A_{2}}U^{\dagger})\geq E_{F}(V\varrho_{j,A_{1}A_{2}}V^{\dagger})$, and the second inequality is from the convex property of $E_{F}$. This result also follows from the fact that $E^{\prime}_{F}$ is a concave function of $\varrho_{A1A2}$, as shown by the Proposition 2 of the supplemental material of Ref. SCamalet .