Realization of the Tradeoff Between Internal and External Entanglement

PHYSICAL REVIEW RESEARCH 2, 043068 (2020)

Realization of the tradeoff between internal and external entanglement

Jie Zhu,1,3 Meng-Jun Hu,1,3 Yue Dai, 2,6 Yan-Kui Bai,4 S. Camalet,5 Chengjie Zhang,2,6,* Chuan-Feng Li,1,3 Guang-Can Guo,1,3 and Yong-Sheng Zhang1,3,† 1Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei 230026, China 2School of Physical Science and Technology, Ningbo University, Ningbo 315211, China 3CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei 230026, China 4College of Physics and Hebei Key Laboratory of Photophysics Research and Application, Hebei Normal University, Shijiazhuang, Hebei 050024, China 5Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France 6School of Physical Science and Technology, Soochow University, Suzhou 215006, China

(Received 9 January 2019; revised 20 August 2020; accepted 25 August 2020; published 13 October 2020)

We experimentally realize the internal and external entanglement tradeoff, which is a kind of entanglement monogamy relation different from that usually discussed. Using a source of twin photons, we ﬁnd that the external entanglement in the 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, which illustrates 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 relations.

DOI: 10.1103/PhysRevResearch.2.043068

I. INTRODUCTION to Gaussian states. Moreover, other entanglement measures, such as the squashed entanglement [10,11], the negativity Entanglement monogamy is one of the most fundamental [12–15], and the squared entanglement of formation [16–18], properties for multipartite quantum states, which means that were also employed to derive the corresponding entanglement if two qubits, A and B, are maximally entangled, then A or B monogamy inequalities. On the other hand, many experimen- cannot be entangled with the third qubit C [1,2]. The quan- tal results on entanglement and nonlocality have also been titative entanglement monogamy inequality was first proved reported [44,45], including entanglement monogamy in exper- by Coffman, Kundu, and Wootters (CKW) for three-qubit iments [46,47]. states [3], Recently, new kinds of monogamy relations have been 2 + 2 2 , CA|B CA|C CA|BC (1) derived by Camalet [48–51], i.e., internal entanglement (or local quantum resource) and external entanglement have a where C2 denotes the squared concurrence for quantifying bi- tradeoff. The usually discussed entanglement monogamy in- partite entanglement [4]. From Eq. (1), one can easily find that equalities in Refs. [5–7,11–14,16–18] indicate the tradeoff there is a consequent tradeoff between the amount of entangle- relation between E(AB ) and E(AC ) (or its extension to the ment shared by qubits A and B and the entanglement shared N-partite case), where E is one kind of entanglement mea- by qubits A and C. For three-qubit pure states, the difference sure and AB and AC are reduced density matrices from a between the right-hand side and the left-hand side of Eq. (1)is three-qubit state. Unlike these previously derived inequalities, defined as the so-called “three-tangle” [3], which is a genuine Camalet has proposed a different entanglement monogamy three-qubit entanglement measure. After the CKW inequality, inequality [48]. This monogamy relation shows that the local several entanglement monogamy inequalities [5–41] and even resource can influence the entanglement between subsystems monogamy equalities [42,43] were introduced. Osborne and and other external systems. Besides there is no limitation Verstraete proved the CKW monogamy inequality for N-qubit about local resources, which can be entanglement between states [5]. In Refs. [6,7], the CKW inequality was generalized different degrees, entanglement between different particles of the subsystems, local coherence, and so on. Consider the tripartite quantum state A1A2B illustrated in Fig. 1, where A1 *[email protected] and A2 come from the same physical system, but have been †[email protected] encoded in different degrees of freedom, and B is encoded in another physical system. This inequality shows the tradeoff ˜ Published by the American Physical Society under the terms of the relation between the internal entanglement EA1|A2 and the ex- ˜ Creative Commons Attribution 4.0 International license. Further ternal entanglement EA1A2|B, where E and E are two different ˜ distribution of this work must maintain attribution to the author(s) but related entanglement measures, and EA1A2|B (EA1|A2 ) de- and the published article’s title, journal citation, and DOI. notes the entanglement of A1A2B ( A1A2 ) under the bipartition

2643-1564/2020/2(4)/043068(10) 043068-1 Published by the American Physical Society JIE ZHU et al. PHYSICAL REVIEW RESEARCH 2, 043068 (2020)

between A1 and A2, the inequality (2) for a general three-qubit |ψ pure state A1A2B becomes + |ψ , EF ( A1A2 ) EF ( A1A2|B ) 1 (3)

˜ | = where the intraparticle entanglement EA1 A2 is EF ( A1A2 ) / + − 2 / H(1 2 1 C ( A1A2 ) 2), H is the binary entropy =− − − − = H(x): x log2 x (1 x)log2(1 x), and C( ) max{0,σ1 − σ2 − σ3 − σ4} is the concurrence of with {σi} ∗ being the square roots of eigenvalues of σy ⊗ σy σy ⊗ σy

in decreasing order [4]. The external entanglement EA1A2|B is |ψ FIG. 1. For the tripartite quantum state A1A2B, the subsystems EF ( A1A2|B ), as defined by A and A are in the same physical system but they are encoded in 1 2 |ψ = − † EF ( A1A2|B ): 1 max EF (U A1A2U ) different degrees of freedom, and B is encoded in another physical U system. E˜ | and E | represent the internal entanglement be- A1 A2 A1A2 B = − { ,λ − λ − λ λ } , tween A1 and A2 and the external entanglement between A1A2 and B, 1 f (max 0 1 3 2 2 4 ) (4) respectively. where U√denotes the unitary operators of A, f (x):= / + − 2/ {λ } H(1 2 1 x 2), and i are the eigenvalues of A1A2 A1A2|B (A1|A2). The inequality holds whatever is the under- in nonascending order [48,53]. In Ref. [53], the maximum lying physical nature of the internal entanglement. It can be entanglement for a given spectrum {λi} measured by the neg- entanglement between two long-distance-separated particles, ativity and the relative entropy of entanglement have also two short-distance-separated particles, or even 2 degrees of been provided. Thus, one can obtain the inequality (2) with freedom of a single particle. This last kind of internal entan- the intraparticle entanglement measure being the negativ- glement is considered here and for precision we refer to it as ity and the relative entropy of entanglement as well (see intraparticle entanglement. Appendix A). Here we experimentally demonstrate the entanglement For simplicity, we first consider a class of three-qubit pure monogamy relation between the intraparticle and external states with one parameter φ as an example: entanglement, with a source of twin photons. As shown in |01+|10 Fig. 1, there are two qubits (the polarization qubit A1 and the |= cos φ|110+sin φ √ |1 . (5) path qubit A2) encoded in photon A, but only one qubit (the 2 A1A2B polarization qubit B) is encoded in photon B. We provide a BasedonEqs.(3) and (4), one can obtain its intraparticle direct experimental observation of the tradeoff between the and external entanglement by using the entanglement of for- | 2 intraparticle entanglement in A1 A2 and the external entangle- mation, i.e., E ( ) = f (sin φ) and E (|ψ | ) = 1 − | F A1A2 F A1A2 B ment in A1A2 B. f (max{cos2 φ,sin2 φ}). Moreover, the states such as 1 cos θ|01+sin θ|10 II. THEORETICAL FRAMEWORK |= √ |110+ √ |1 (6) 2 2 A1A2B Let us focus on the tripartite state , where A and A A1A2B 1 2 = are also analyzed. For these states, we find EF ( A1A2 ) are encoded in the same physical system A by using different 1 f ( sin 2θ ), which depends on θ, and E (|ψ | ) = 1 − degrees of freedom (see Fig. 1). The third party is encoded in 2 F A1A2 B / ≈ . system B. Camalet’s entanglement monogamy inequality is f (1 2) 0 645 is a constant. The theoretical results are shown in Fig. 2 as the solid and dotted lines. We can see ˜ + ˜ , EA1|A2 EA1A2|B Emax (2) that all the results are bounded by the green dashed line, i.e., the monogamy inequality (3) always holds. Moreover, ˜ | where EA1 A2 denotes the intraparticle entanglement measure the inequality (3) is valid for any finite-dimensional subsys- | between A1 and A2, EA1A2 B is the external entanglement mea- tems. However, not all the general allowed region can be ˜ ˜ | sure between A1A2 and B, and Emax is the value of EA1 A2 when reached with three-qubit states. For all the three-qubit pure A1 and A2 are maximally entangled [48]. It is worth noting that states, since λ1 1/2 and there are at most only two nonzero E˜ and E are strongly related, although they are two different {λ ,λ } = − { ,λ } eigenvalues 1 2 for A1A2 , EF 1 f (max 0 1 ) entanglement measures. From the inequality (2), one can see 1 − f (1/2) ≈ 0.645, and this bound also holds for all three- | ˜ that EA1A2 B is also bounded by Emax. When the state A1A2B qubit mixed states. Therefore, the allowed region specific to is pure and the reduced density operator A1A2 is absolutely all three-qubit states is given in Fig. 2, and our experimental | separable [52,53], the external entanglement EA1A2 B is equal points can reach the corresponding borders. The GHZ state is ˜ to the maximum value Emax. On the other hand, when the a three-qubit state, so it cannot reach the maximum external ˜ | intraparticle entanglement EA1 A2 is maximal, the external en- entanglement 1; it can only reach the maximum 0.645 for | tanglement EA1A2 B must be vanishing. three-qubit states. ˜ Inequality (2) holds for any convex measure EA1|A2 which can thus be, for instance, the entanglement of formation E F III. EXPERIMENTAL REALIZATION [4], the negativity EN [54,55], and the relative entropy of entanglement ER [56]. When we choose the entanglement In order to demonstrate this entanglement monogamy re- of formation EF to quantify the intraparticle entanglement lation, we prepare some quantum states where the quantity

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As shown in Fig. 3, we introduce three parts of the setup: (i) state preparation, (ii) qubits A1 and A2 (owned by Al- ice), and (iii) qubit B (owned by Bob). First, the source of twin photons is realized via a type-I spontaneous parametric down-conversion process in which the crystal is a joint β-barium borate (BBO) whose size is 8.0 × 8.0 × 0.3mm ◦ and whose optic axis is cut at θpm = 29.3 [57]. The source |ψ= φ| | + of the two-qubit entangled state is cos H A H B φ| | φ sin V A V B, where the parameter is modulated by H1, a half-wave plate (HWP) put in front of the BBO crystal to adjust the polarization of the pump. Here the pump is a continuous-wave diode laser with 140 mW and the wave- length is 404 nm. The fidelity between the experimental state and the theoretical state is beyond 98.80%. The computational bases |0 and |1 are encoded in the horizontal polarization |H and the vertical polarization |V of the photons, respectively. The photon pair is separated spatially via a single mode fiber. One photon is sent to Alice and the other one is sent to Bob. In Alice part, as illustrated in Fig. 3,weuse three beam displacers (BDs) in which the vertical-polarized photon remains on its path while the horizontal-polarized photon shifts down. The intraparticle entanglement is realized ˜ between the path and the polarization degrees of freedom FIG. 2. The intraparticle entanglement EA1|A2 quantified by of Alice’s photon. The upper path state is encoded into |0, EF ( A1A2 ) and the external entanglement EA1A2|B quantified by |ψ and the down path state corresponds to |1. After BD1, the EF ( A1A2|B ) are bounded by the green dashed straight line for all tripartite states with finite dimension. Meanwhile, for all the photons in different polarization states travel two paths. Thus |ψ the polarization and the path are entangled. Due to the HWP at three-qubit states EF ( A1A2|B ) is bounded by the dotted line, i.e., |ψ − / ≈ . 45◦ (H7), the horizontal and vertical polarizations exchange, EF ( A1A2|B ) 1 f (1 2) 0 645. The states given by Eqs. (5) and (6) and the corresponding experimental states are shown here. whereafter the HWP in the upper path (H8) rotates the po- The blue dots correspond to Eq. (5) and the red ones to Eq. (6). All larization. The angle θ modulated by H8 is the controllable blue dots should be on the solid line and all red ones on the dotted parameter of the path-polarization coupling. Right after H8, line according to the theoretical predictions but some dots are not. there is another beam displacer, BD2, to fulfill the preparation The deviation is from the visibility of interferometers, and the error of the intraparticle entanglement of Alice’s photon. On the bars are from the Poissonian distribution of photon counts and the other hand, in Bob’s part, the photons are measured directly. uncertainty of wave plates. Finally we get the three-qubit states which contain intraparticle and external entanglement and can be described as of intraparticle and external entanglement can be controlled. |ψ= φ| | | + φ|ϕ | , cos 1 A1 1 A2 0 B sin A1A2 1 B (7) We use the polarization and the path degrees of freedom to |ϕ = θ| | + θ| | produce the target three-qubit states in Eqs. (5) and (6). where A1A2 cos 0 A1 1 A2 sin 1 A1 0 A2 .

FIG. 3. Experimental setup. The polarization-entangled photon pairs are generated by the spontaneous parametric down-conversion process. In the Alice part, the polarization and path states are entangled. In each mode, half-wave plates, quarter-wave plates, and polarization beam splitters are set for state tomography. In the experiment, the photons are collected by two single-photon counting modules and identiﬁed by the coincidence counter. H, half-wave plate, H1–H11; Q, quarter-wave plate, Q1–Q3; P, polarization beam splitter, P1–P3; BD, beam displacer, BD1–BD3.

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the experimental results coincide with the theoretical predictions. Although the different values of θ and φ represent the ˜ + different states, the sum of EA1|A2 EA1A2|B will not exceed 1, which experimentally demonstrates the monogamy relation

(3). We remark that EA1A2|B is evaluated using Eq. (4), which is valid only for genuine pure states. However, the actual

entanglement EA1A2|B of the experimental state, which is not exactly pure, is lower than the value obtained from Eq. (4) ˜ [48,50]. On the other hand, we choose EA1|A2 as the horizontal

ordinate and EA1A2|B as the vertical ordinate to plot Fig. 2.

We find that for the states (5) EA1A2|B first increases and then ˜ decreases as EA1|A2 increases; but for the states (6) EA1A2|B is almost invariant. All the experimental results agree with the theoretical results. Moreover, these values are in the area = − ˜ below the straight line EA1A2|B 1 EA1|A2 . In this experiment, the visibility of the Mach-Zehnder interferometer is about 100:1 and the average fidelity [59]

FIG. 4. E˜ | + E | as a function of θ for the states (6) [panel between the experimental states and theoretical states is A1 A2 A1A2 B . ± . (a)], and as a function of φ for the states (5) [panel (b)]. The dots are 98 86 0 41%. Moreover, the negativity in inequality (8) has the experimental results and the curves are the theoretical predic- been evaluated for the mixed experimental states. Besides, tions. The error bars are from the Poissonian distribution of photon the single-photon avalanche photon-diode (SPCM-3369) is counts and the uncertainty of the wave plates. 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 ±3ns. In addition, the coincidence counts are about 1000 s−1 and In order to reconstruct the density matrices of these three- we record clicks for 10 s. There are many sources of the qubit states, we perform quantum state tomography for these measurement uncertainty, such as counting statistics, detector states. According to the maximum likelihood estimation, the efficiency, the detector’s dead time, timing uncertainty, and density matrices are reconstructed [58]. The project measure- the alignment error of wave plates. However, the resulting ment of polarization is realized by a standard polarization uncertainty is dominated by counting statistics [60], which we tomography setup (SPTS), which consists of a quarter-wave have calculated via the Poissonian distribution and the errors plate (QWP), a half-wave plate, and a polarization beam introduced by wave plates cannot be ignored. Both of them splitter (PBS). As shown in Fig. 3, there are three such se- are shown in the figure. tups: (Q1,H5,P1), (Q2,H9,P2), and (Q3,H11,P3). The first two setups measure the polarization states of Bob and Alice, IV. OTHER MONOGAMY INEQUALITIES respectively. As for the last one, it is used to measure the path state of Alice. The following demonstrates how it works. Now we show that the tradeoff between intraparticle Consider BD2, it is the last element in the state preparation. and external entanglement is not restricted to some spe- If a photon is in the upper path after BD2, its polarization is cific measures of entanglement, and it can be shown by horizontal when it arrives at Q3, i.e., the last SPTS, and if it other entanglement measures as well, such as the negativ- is in the down path, it will be vertical-polarized. Therefore, ity and concurrence. In Ref. [48], Camalet also presented the last SPTS measures the path state via the polarization a monogamy inequality involving only one entanglement state tomography. The external entanglement is calculated monotone, the negativity EN . For the bipartite state AB,the = TB − / from the density matrix of the complete system, i.e., the negativity is defined by EN ( AB ) ( AB 1) 2[54,55], · three-qubit state, and the intraparticle entanglement is calcu- where is the trace norm and TB is the partial transpose with respect to system B. Contrary to other entanglement lated independently using reduced density matrix tomography in Alice’s part and simple photon detection in Bob’s part measures, such as EF or EF , EN is readily computable for for coincidence counting. Besides it is necessary to men- any state. For the three-qubit state A1A2B, the monogamy tion that the HWP right after P2 in the down path (H10) is inequality is at 45◦. EN (A A ) + g[EN (A A |B )] EN,max, (8) We first fix the value of θ at 45◦ and adjust H1 to change 1 2 1 2 ◦ ◦ the values of φ from 0 to 90 . Thus, the experimental state where EN,max, the maximum value of EN , is equal to 1/2 ◦ in Eq. (7) becomes the states in Eq. (5). If φ = 90 , the intra- for the two-qubit states,√ and the nondecreasing function g is particle entanglement of system A is maximal, and if φ = 0◦, given by g(x) = (3/2 − 1 − 2x2 − 1/4 − x2 )/2[48]. The the state is |ψ=|110, which is separable. Furthermore, the monogamy inequality (8) has been calculated for our experi- states with a fixed value of φ = 45◦ and adjustable values of mentally realized states in Appendix B. θ from 0◦ to 90◦ are tested, which are the states in Eq. (6). Many familiar entanglement monotones satisfy monogamy We reconstruct the density matrices of all these states and inequalities of the form of Eq. (8)[51], but determining ˜ then calculate their entanglements EA1|A2 and EA1A2|B based on explicitly the corresponding function g may not be always EF using the expressions given above. As shown in Fig. 4, possible. Now we present another case for which this can

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|φ be achieved. For the qubit-qudit√ pure state AB , the con- with |φ = − 2 currence is defined by C( AB ) 2(1 Tr B )[3,4], where = † EN A A max EN U A1A2U = Tr (|φ φ |) is the reduced density operator of sys- 1 2 U B A AB AB tem B. It is generalized to mixed states via the convex roof 1 2 2 extension [3,4]. For the three-qubit state , the following = max 0, (λ1 − λ3) + (λ2 − λ4) A1A2B 2 monogamy inequality holds: λ λ − 2 − 4 , (A3) + | , C( A1A2 ) g˜[C( A1A2 B )] Cmax (9) 2 2 where U denotes the unitary operators of A, is the where Cmax, the maximum value of C, is equal to 1 for the A1A2 two-qubit states,√ and the nondecreasing functiong ˜ is given by density operator corresponding to the maximum over U, and 2 {λ } g˜(x) = (1 − 1 − x )/2 . The monogamy inequality (9) has i are the eigenvalues of A1A2 in nonascending order [53]. also been calculated for our experimentally realized states in Now we consider a class of three-qubit pure states with one Appendix C. parameter φ, |01+|10 |=cos φ|110+sin φ √ |1. (A4) V. CONCLUSION 2 We have experimentally observed the tradeoff relation be- BasedonEqs.(A1)–(A3), one can obtain its internal and tween intraparticle and external entanglement in a photonic external entanglement measured by the negativity: system. Although the experimental states are not exactly pure E ( ) = 1 3 + cos(4φ) − 1 cos2 φ, (A5) states, the monogamy inequality (3) still holds for experimen- N A1A2 4 2 tal mixed states. This realization has verified the theoretical 1 1 2 2 E (|ψA A |B ) = + min{cos φ,sin φ} prediction that the entanglement between different degrees of N 1 2 2 2 freedom of a quantum single particle can restrict its entangle- − 1 3 + cos(4φ). (A6) ment with other particles. This property may have applications 4 θ in quantum information, such as the construction of quan- Moreover, we also consider the pure state with parameter , tum communication networks. Our experiment opens the door 1 cos θ|01+sin θ|10 to experimentally investigating the distribution of different |=√ |110+ √ |1. (A7) 2 2 kinds of entanglement in multipartite systems and points a way to generate external entanglement by decreasing internal BasedonEqs.(A1)–(A3), one can obtain its internal and entanglement. Furthermore, our experiment will be greatly external entanglement measured by the negativity: helpful for further research on the other monogamy relations E ( ) = 1 ( − 2 + 6 − 2 cos(4θ )), (A8) in multipartite systems. For instance, one can demonstrate the N A1A2 8 √ inequality between local coherence and entanglement [48,61] 1 E (|ψ | ) = (3 − 2). (A9) and the inequality between intraparticle entanglement and N A1A2 B 4 external correlations [51]. The theoretical and experimental results have been shown in Fig. 5. We can see that all the sums of internal and external entanglement are bounded by 1/2; i.e., the inequality (A1) ACKNOWLEDGMENTS always holds. This work is funded by the National Natural Science Foun- dation of China (Grants No. 11504253, No. 11575051, No. APPENDIX B: THE MONOGAMY INEQUALITY (8) 11674306, No. 61590932, and No. 11734015), the National INVOLVING ONLY EN Key R&D Program (Grants No. 2016YFA0301300 and No. 2016A0301700), the Anhui Initiative in Quantum Information In Ref. [48], the author also presented a monogamy in- Technologies, the K.C. Wong Magna Fund in Ningbo Univer- equality involving only one entanglement monotone, the sity, and the Hebei NSF (Grant No. A2016205215). negativity EN . For the bipartite state AB, the negativity is = TB − / · defined by EN ( AB ) ( AB 1) 2, where is the trace norm and TB is the partial transpose with respect to system APPENDIX A: INEQUALITY (2) FOR THE NEGATIVITY EN |ψ B. For the three-qubit pure state A1A2B, the monogamy inequality is If we use the negativity EN to quantify the internal entanglement between A and A , the inequality (2) in the main text 1 1 2 EN (A A ) + g[EN (|ψA A |B )] , (B1) becomes 1 2 1 2 2 where EN,max, the maximum value of EN ( A1A2 ), is equal to 1 + |ψ | , / EN ( A1A2 ) EN ( A1A2 B ) 2 (A1) 1 2 for the two-qubit state A1A2 , and the nondecreasing function g is given by TA2 √ √ where EN (A A ) = ( −1)/2, and EN,max, the maxi- 1 2 A1A2 3 1 − 2x2 1 − 4x2 mum value of EN (A A ), is equal to 1/2 for the two-qubit g(x) = − − , (B2) 1 2 4 2 4 state A A . The external entanglement E (|ψA A |B )is 1 2 N 1 2 when the number of nonzero eigenvalues of A1A2 is equal to 1 E (|ψ | ) = − E ( ), (A2) 2[48]. N A1A2 B 2 N A1A2

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|ψ FIG. 5. We use the negativities EN ( A1A2 )andEN ( A1A2|B ) to quantify the entanglement among these three qubits. The red and blue dots are experimental results and the lines are theoretical predictions.

Now we consider a class of three-qubit pure states with one where 1 is the maximal value of C for two-qubit states and parameter φ, √ − − 2 = 1 1 x . |01+|10 g˜(x) (C2) |=cos φ|110+sin φ √ |1. (B3) 2 2 Proof. For a pure three-qubit state, the external entangle- Based on Eqs. (B1) and (B2), one can obtain its internal and ment is given by external entanglement measured by the negativity: 2 C(|ψA A |B ) = 2 1 − Tr = 2 λ1(1 − λ1), (C3) 1 1 1 2 A1A2 = + φ − 2 φ, EN ( A1A2 ) 3 cos(4 ) cos (B4) λ 4 2 with 1 being the maximal eigenvalue of A1A2 . Since the above function of is concave and the concurrence is − 2 φ 2 φ A1A2 |ψ = 3 − 1 2 cos sin defined via the convex roof extension for mixed states, the g[EN ( A1A2|B )] 4 2 external entanglement obeys 1 − 4 cos2 φ sin2 φ − . (B5) C(A A |B ) 2 1(1 − 1), (C4) 4 1 2 for a general three-qubit state = p |ψ ψ | with θ A1A2B i i i i Moreover, we also consider the pure state with parameter , = λ λ f ( 1) i pi f ( 1,i ) and 1,i being the maximal eigen- θ| + θ| value of Tr |ψ ψ |. The internal entanglement can be |=√1 | +cos 01 √ sin 10 | . B i i 110 1 (B6) obtained by the formula C( ) = max{0,σ − σ − σ − 2 2 A1A2 1 2 3 σ4}, with {σi} being the square roots of eigenvalues of Based on Eqs. (B1) and (B2), one can obtain its internal and σ ⊗ σ ∗ σ ⊗ σ in decreasing order [4]. In the bi- A1A2 y y A1A2 y y external entanglement measured by the negativity: partite partition A1A2|B, the internal entanglement can be maximized via two-qubit unitary transformations on A1A2, E ( ) = 1 ( − 2 + 6 − 2 cos(4θ )), (B7) N A1A2 8 and the following relation holds: √ 1 † g[E (|ψ | )] = (3 − 2). (B8) C A A maxC UA A U , (C5) N A1A2 B 4 1 2 U 1 2 The theoretical and experimental results have been shown in where the equality is satisfied for the so-called maximally Fig. 6. We can see that all the sums of internal and external entangled mixed state (MEMS) [53]. In the case A1A2 entanglement are bounded by 1/2; i.e., the inequality (B1) of two-qubit MEMSs [53], its concurrence is C( ) = √ A1A2 always holds. { ,λ − λ − λ λ } λ max 0 1 3 2 2 4 1, and hence C( A1A2 ) λ1 1. Therefore, we have APPENDIX C: THE MONOGAMY INEQUALITY (9) C(A A ) + g˜[C(A A |B )] 1 + g˜(2 1(1 − 1)) INVOLVING ONLY C 1 2 1 2 √ − − − = + 1 1 4 1(1 1) Proposition 1. For the three-qubit state A1A2B,thein- 1 ternal entanglement C( ) and the external entanglement 2 A1A2 C( | ), as quantified by the concurrence, obey the − − A1A2 B = + 1 (2 1 1) monogamy relation 1 2 + , = , C A1A2 g˜ C A1A2|B 1 (C1) 1 (C6)

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|ψ FIG. 6. In order to use the same measure, EN ( A1A2 )andEN ( A1A2|B ), to quantify the entanglement, we employ the function g. Note that g is not shown in the ﬁgure. The red and blue dots are experimental results and the lines are theoretical predictions. where 1/2 λ1,i 1 and the monotonicity and convexity of consider the pure state with parameter θ, g˜ have been used. Thus, we obtain the monogamy relation 1 cos θ|01+sin θ|10 (C1). |=√ |110+ √ |1. (C11) Now we consider a class of three-qubit pure states with one 2 2 parameter φ, One can obtain its internal and external entanglement mea- | +| sured by concurrence: |= φ| + φ 01 √ 10 | . cos 110 sin 1 (C7) = 1 | θ |, 2 C( A1A2 ) 2 sin(2 ) (C12)

1 One can obtain its internal and external entanglement mea- |ψ | = − λ = , g˜[C( A1A2 B )] 1 1 2 (C13) sured by concurrence: with λ1 = 1/2. 2 The theoretical and experimental results have been shown C(A A ) = sin φ, (C8) 1 2 in Fig. 7. We can see that all the sums of internal and external entanglement are bounded by 1; i.e., the inequality (C1) g˜[C(|ψA A |B )] = 1 − λ1, (C9) 1 2 always holds. with

2 2 APPENDIX D: QUANTUM STATE TOMOGRAPHY λ1 = max{cos φ,sin φ}. (C10) We performed tomography on all the states we prepared. 2 2 2 One can see that if λ1 = sin φ (i.e., sin φ cos φ), then Their density matrices are described in Eq. (7)ofthemain + |ψ = θ ◦ φ C( A1A2 ) g˜[C( A1A2|B )] 1 holds. Moreover, we also text. First, is ﬁxed at 45 ,but is changing. Then we change

FIG. 7. We use the concurrence C to quantify the internal and external entanglement with the functiong ˜. The red and blue dots are experimental results and the lines are theoretical predictions.

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TABLE I. Fidelities for the quantum state Eq. (7) of the main text. (i) We change the value of θ with the fixed φ at 45◦; (ii) θ is fixed at 45◦ but φ is changing from 0◦ to 90◦. The average fidelity of all these states is 98.86 ± 0.41%.

θ 0◦ 10◦ 15◦ 20◦ 30◦ Fidelity 98.17 ± 0.13% 99.17 ± 0.12% 98.77 ± 0.02% 99.28 ± 0.03% 98.39 ± 0.11% θ 40◦ 45◦ 60◦ 75◦ 90◦ Fidelity 98.85 ± 0.03% 98.15 ± 0.01% 98.52 ± 0.04% 98.72 ± 0.08% 98.96 ± 0.09% φ 0◦ 15◦ 30◦ 45◦ 50◦ Fidelity 99.05 ± 0.11% 99.16 ± 0.07% 99.51 ± 0.02% 99.16 ± 0.05% 98.94 ± 0.05% φ 60◦ 70◦ 80◦ 90◦ Average Fidelity 99.10 ± 0.06% 98.77 ± 0.03% 98.20 ± 0.12% 99.39 ± 0.05% 98.86 ± 0.41%

◦ the value of θ with the fixed φ at 45 . We can get their density + − † EF A1A2 1 p jEF V j,A1A2V matrices that are obtained by maximum likelihood estimation. j In Table I, we list the fidelity of all the states which only φ θ change the value of or . † E + 1 − E V p , V Although the experimental states are not exactly pure F A1A2 F j j A1A2 j states, the monogamy inequality (3) in the main text still holds for experimental mixed states. Suppose that the exper- = + − EF A1A2 1 EF A A imentally realized tripartite state is , thus the external 1 2 A1A2B 1, (D2) entanglement is defined by the convex roof, = |ψ . EF ( A1A2|B ) inf piEF ( i A1A2|B ) (D1) {p ,|ψ } i i A1A2B i = |ψ ψ | |ψ = − where j,A1A2 TrB j j , EF ( j A1A2|B ) 1 We assume that A A B = p j|ψ j ψ j| is the optimal † 1 2 j EF ( , ), EF ( , ) = maxU EF (U j,A A U ) = j A1A2 j A1A2 1 2 decomposition for A1A2B to achieve the above infimum. † † EF (Uj j,A A U ), EF ( ) = maxU EF (UA A U ) = Therefore, 1 2 j A1A2 1 2 † EF (V A1A2V ), the first inequality holds since E + E | † † † F A1A2 F A1A2 B E (U , U ) = max E (U , U ) E (V , V ), F j j A1A2 j U F j A1A2 F j A1A2 = + |ψ and the second inequality is from the convex property of EF A1A2 p jEF j A1A2|B j EF . This result also follows from the fact that EF is a concave function of A1A2, as shown by Proposition 2 of the † = + − , EF A1A2 1 p jEF Uj j A1A2Uj Supplemental Material of Ref. [48]. j

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