Realization of the tradeoff between internal and external entanglement
Jie Zhu, Meng-Jun Hu, Yue Dai, Yan-Kui Bai, S. Camalet, Chengjie Zhang, Chuan-Feng Li, Guang-Can Guo, Yong-Sheng Zhang
aa r X i v : . [ qu a n t - ph ] J a n Realization of the tradeoff between internal and external entanglement
Jie Zhu,
1, 2
Meng-Jun Hu,
1, 2
Yue Dai, Yan-Kui Bai, S. Camalet, ChengjieZhang, ∗ Chuan-Feng Li,
1, 2
Guang-Can Guo,
1, 2 and Yong-Sheng Zhang
1, 2, † Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China School of Physical Science and Technology, Soochow University, Suzhou, 215006, China College of Physics Science and Information Engineering and Hebei Advanced Thin Films Laboratory,Hebei Normal University, Shijiazhuang, Hebei 050024, China Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France (Dated: January 10, 2019)We experimentally realize the internal and external entanglement tradeoff, which is a new kindof entanglement monogamy relation different from that usually discussed. Using a source of twinphotons, 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 theinternal state is maximally entangled, the external entanglement must be vanishing, that illustrateentanglement monogamy. Our results of the experiment coincide with the theoretical predictions,and therefore provide a direct experimental observation of the internal and external entanglementmonogamy relation.
Entanglement monogamy is one of the most funda-mental properties for multipartite quantum states, whichmeans that if two qubits A and B are maximally entan-gled, then A or B cannot be entangled with the thirdqubit C [1, 2]. The quantitative entanglement monogamyinequality was first proved by Coffman, Kundu, andWootters (CKW) for three-qubit states [3], C A | B + C A | C ≤ C A | BC , (1)where C denotes the squared concurrence for quanti-fying bipartite entanglement [4]. From Eq. (1), one caneasily find that there is a consequent tradeoff between theamount of entanglement shared by qubits A and B , andthe entanglement shared by qubits A and C . For three-qubit pure states, the difference between the right handside and left hand side of Eq. (1) is defined as the so-called “three-tangle” [3], which is a genuine three-qubitentanglement measure. After the CKW inequality, sev-eral entanglement monogamy inequalities [5–43] and evenmonogamy equalities [44, 45] were introduced. Osborneand Verstraete proved the CKW monogamy inequalityfor N -qubit states [5]. In Refs. [6, 7], the CKW inequal-ity was generalized to Gaussian states. Moreover, otherentanglement measures, such as the squashed entangle-ment [10, 11], the negativity [12–15], and the squaredentanglement of formation [16–18], were also employedto derive the corresponding entanglement monogamy in-equalities.Recently, new kinds of monogamy relation have beenderived by Camalet [46–49], i.e. internal entanglment(or local quantum resource) and external entanglementhave a tradeoff. The usually discussed entanglementmonogamy inequalities in Refs. [5–7, 11–14, 16–18] indi-cate the trade-off relation between E ( ̺ AB ) and E ( ̺ AC )(or its extension to N -partite case), where E is one kindof entanglement measure, ̺ AB and ̺ AC are reduced den- FIG. 1: For a tripartite quantum state ̺ A A B , the subsys-tems A and A are in the same physical system but theyare encoded in different degrees of freedom, and B is encodedin another physical system. ˜ E A | A and E A A | B representthe internal entanglement between A and A and externalentanglement between A A and B , respectively. sity matrices from a three-qubit state. Unlike these pre-viously derived inequalities, Camalet has proposed a newentanglement monogamy inequality [46]. Consider a tri-partite quantum state ̺ A A B illustrated in Fig. 1, where A and A come from the same physical system but havebeen encoded in different degrees of freedom, and B is en-coded in another physical system. This inequality showsthe tradeoff relation between the internal entanglement˜ E A | A and the external entanglement E A A | B , where ˜ E and E are two different but related entanglement mea-sures, and E A A | B ( ˜ E A | A ) denotes the entanglement of ̺ A A B ( ̺ A A ) under the bipartition A A | B ( A | A ).Here we experimentally demonstrate the entanglementmonogamy relation between the internal and external en-tanglement, with a source of twin photons. As shown inFig. 1, there are two qubits (the polarization qubit A and the path qubit A ) encoded in photon A ; but onlyone qubit (the polarization qubit B ) is encoded in photon B . Here we provide a direct experimental observation of (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:7)(cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) FIG. 2: The internal entanglement ˜ E A | A quantified by E F ( ̺ A A ) and the external entanglement E A A | B quantifiedby E ′ F ( | ψ i A A | B ) are bounded by the dashed blue straightline. We calculated the states in Eq. (5) and the correspond-ing nine different experimental states, the results are shownas the black curve and red dots, respectively. The left sixred dots should be on the curve according to the theoreti-cal prediction but some dots do not. The deviation is fromthe visibility of interferometers, and the error bar is from thePoissonian distribution of photon counts. the tradeoff between the internal entanglement in A | A and the external entanglement in A A | B . Theoretical framework.—
Let us focus on a tripartitestate ̺ A A B where A and A are encoded in the samephysical 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˜ E A | A + E A A | B ≤ ˜ E max , (2)where ˜ E A | A denotes the internal entanglement measurebetween A and A , E A A | B is the external entangle-ment measure between A A and B , and ˜ E max is thevalue of ˜ E A | A when A and A are maximally entangled[46]. It is worth noting that ˜ E and E are strongly related,although they are two different entanglement measures.From the inequality (2), one can see that E A A | B is alsobounded by ˜ E max . When the state ̺ A A B is pure andthe reduced density operator ̺ A A is absolutely separa-ble [50, 51], the external entanglement E A A | B is equalto the maximum value ˜ E max . On the other hand, whenthe internal entanglement ˜ E A | A is maximal, the exter-nal entanglement E A A | B must be vanishing.The internal entanglement measure ˜ E A | A in Eq. (2)can be arbitrary entanglement measures, such as theentanglement of formation E F [4], the negativity E N [52, 53], and the relative entropy of entanglement E R [54]. When we choose the entanglement of formation E F to quantify the internal entanglement between A and A , the inequality (2) for a general three-qubit pure state | ψ i A A B becomes to E F ( ̺ A A ) + E ′ F ( | ψ i A A | B ) ≤ , (3)where the internal entanglement ˜ E A | A is E F ( ̺ A A ) = H (cid:0) / p − C ( ̺ A A ) / (cid:1) , H is the binary entropy H ( x ) := − x log x − (1 − x ) log (1 − x ), and C ( ̺ ) =max { , σ − σ − σ − σ } is the concurrence of ̺ with { σ i } being the square roots of eigenvalues of ̺σ y ⊗ σ y ̺ ∗ σ y ⊗ σ y in decreasing order [4]. The external entanglement E A A | B is E ′ F ( | ψ i A A | B ), as defined by E ′ F ( | ψ i A A | B ) := 1 − max U E F ( U ̺ A A U † ) (4)= 1 − f (max { , λ − λ − p λ λ } )where U denotes the unitary operators of A , f ( x ) := H (1 / √ − x / { λ i } are the eigenvalues of ̺ A A in nonascending order [46, 51]. In Ref. [51], themaximum entanglement for a given spectrum { λ i } mea-sured by the negativity and the relative entropy of en-tanglement have also been provided. Thus, one can ob-tain the inequality (2) with the internal entanglementmeasure being the negativity and the relative entropy ofentanglement as well [55].Now we consider a class of three-qubit pure states withone parameter φ , | ψ i = cos φ | i + sin φ | i + | i√ | i . (5)Based on Eqs. (3)-(4), one can obtain its internal andexternal entanglement by using the entanglement of for-mation, E F ( ̺ A A ) = f (sin φ ) , (6) E ′ F ( | ψ i A A | B ) = 1 − f (max { cos φ, sin φ } ) . (7)The theoretical results have been shown in Fig. 2 by thesolid curve. We can see that all the results are boundedby the dashed line, i.e., the monogamy inequality (3)always holds. Experimental realization.—
In order to demonstratethis new entanglement monogamy relation, we preparesome quantum states where the quantity of internal andexternal entanglement can be controlled. We use the po-larization and the path degrees of freedom to produce thetarget three-qubit state in Eq. (5).As shown in Fig. 3, we will introduce three parts ofthe setup: (i) state preparation, (ii) qubits A and A (owned by Alice), (iii) qubit B (owned by Bob). First,the source of twin photons is realized via a type-I spon-taneous parametric down-conversion (SPDC) process in FIG. 3:
Experimental setup
The polarization entangled photon pairs are generated by the spontaneous parameteric down-conversion process. In the Alice part, the polarization and path states are entangled. In each mode, half-wave plate (HWP),quarter-wave plate (QWP) and polarization beam splitter (PBS) are set for state tomography. In the experiment, the photonsare 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. which the crystal is a joint β -barium-borate ( β -BBO)[56]. The source of the two-qubit entangled state is | ψ i = cos φ | H i A | H i B + sin φ | V i A | V i B , where the pa-rameter φ is modulated by H1, a half-wave plate (HWP)put in front of the BBO crystal to adjust the polarizationof pump. Here the pump is a continuous-wave diode laserwith 140mW and the wavelength is 404nm. The fidelitybetween the experimental state and the theoretical stateis beyond 99%. The computational basis | i and | i areencoded in the horizontal polarization | H i and verticalpolarization | V i of the photons, respectively. The photonpair is separated spatially via a single mode fiber (SMF).One photon is sent to Alice and the other one is sent toBob. In Alice part, as illustrated in Fig. 3, we use threebeam displacers (BDs) in which the vertical-polarizedphoton remains on its path while the horizontal-polarizedphoton shifts down. The internal entanglement is real-ized between the path and the polarization degrees offreedom of Alice’s photon. The upper path state is en-coded into | i , and the down path state is correspondingto | i . After BD1, the photons in different polarizationstates travel two paths. Thus the polarization and pathare entangled. Due to the HWP at 45 ◦ (H7), the hori-zontal and vertical polarization exchange, whereafter theHWP in the upper path (H8) rotates the polarization.The angle θ modulated by H8 is the controllable pa-rameter of the internal entanglement. Right after theH8, there is another beam displacer, BD2, to fulfill thepreparation of the internal entanglement of the Alice’sphoton. On the other hand, in Bob’s part, the photonsare measured directly. Finally we get the three-qubitstates which contain internal and external entanglementand can be described as | ψ i = cos φ | i A | i A | i B + sin φ | ϕ i A A | i B , (8) where | ϕ i A A = cos θ | i A | i A + sin θ | i A | i A .In order to reconstruct the density matrices of thesethree-qubit states, we perform quantum state tomogra-phy for these states. According to the maximum likeli-hood estimation, the density matrices are reconstructed[57]. The project measurement of polarization is realizedby a standard polarization tomography setup (SPTS),which consists of a quart-wave plate (QWP), a half-waveplate, and a polarization beam splitter (PBS). As shownin the Fig. 3, there are three such setups, (Q1,H5,P1),(Q2,H9,P2), and (Q3,H11,P3). The first two setups mea-sure the polarization states of Bob and Alice respectively.As for the last one, it is used to measure the path stateof Alice. The following is demonstrated how it works.Consider the BD2, it is the last element in the statepreparation. 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 bevertical-polarized. Therefore, the last SPTS measure thepath state via the polarization state tomography. Be-sides it is necessary to mention that the HWP right afterP2 in down path (H10) is at 45 ◦ .We fix the value of θ at 45 ◦ and adjust the H1 to changethe values of φ from 0 ◦ to 90 ◦ . Thus, the experimentalstate in Eq. (8) becomes to Eq. (5). If φ = 90 ◦ , the inter-nal entanglement of system A is maximal; and if φ = 0 ◦ ,the state is | ψ i = | i which is separable. Without lossof generality, we choose nine typically states to analyze.We reconstruct the density matrices of these nine states,then calculate their entanglements ˜ E A | A and E A A | B based on E F using the expressions given above. As shownin Fig. 4, the experimental results coincide with the the-oretical prediction within the margin of error. Althoughthe different value of the angle φ represent the differentstates, the sum of ˜ E A | A + E A A | B will not exceed 1, (cid:1)(cid:2)(cid:1) (cid:1)(cid:0)(cid:2) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:4)(cid:2)(cid:1) ϕ (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) + (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) FIG. 4: We adjusted the degree of H1 to change the values of φ and chose nine states to calculate ˜ E A | A + E A A | B . Thered dots are the experimental results and the black curve isthe theoretical values. The error bar is from the Poissioniandistribution of photon counts. which experimentally demonstrates the monogamy rela-tion (2). We remark that E A A | B is evaluated using Eq.(4) which is strictly speaking valid only for genuine purestates. However, the actual entanglement E A A | B of theexperimental state, which is not exactly pure, is lowerthan the value obtained from Eq. (4) [46, 48, 55]. Onthe other hand, we choose ˜ E A | A as the horizontal ordi-nate and E A A | B as the vertical ordinate to plot Fig. 2.We find that E A A | B firstly increases and then decreasesas ˜ E A | A increases, which also agrees with the theoreti-cal results. Moreover, these values are in the area belowthe straight line E A A | B = 1 − ˜ E A | A . Other monogamy inequalities.—
In Ref. [46], Camaletalso presented a monogamy inequality involving only oneentanglement monotone, the negativity E N . For a bipar-tite state ̺ AB , the negativity is defined by E N ( ̺ AB ) =( k ̺ T B AB k − / k · k 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 ′ F , E N is readily computable for any state. For athree-qubit state ̺ A A B , the monogamy inequality is E N ( ̺ A A ) + g [ E N ( ̺ A A | B )] ≤ E N, max , (9)where E N, max , the maximum value of E N , is equal to 1 / g is given by g ( x ) = (3 / − √ − x − p / − x ) / g may not be always possible. Now we presentanother case for which this can be achieved. Fora qubit-qudit pure state | φ AB i , the concurrence isdefined by C ( | φ AB i ) = p − Tr ̺ B ) [3, 4], where ̺ B = Tr A ( | φ AB ih φ AB | ) is the reduced density operatorof system B . It is generalized to mixed states via theconvex roof extension [3, 4]. For a three-qubit state ̺ A A B , the following monogamy inequality holds C ( ̺ A A ) + ˜ g [ C ( ̺ A A | B )] ≤ C max , (10)where C max , the maximum value of C , is equal to 1 forthe two-qubit states, and the nondecreasing function ˜ g isgiven by ˜ g ( x ) = (1 − √ − x ) / Discussion and conclusion.—
In this experiment, thevisibility of the MZ interferometer is about 100:1 and theaverage fidelity [58] between the experimental states andtheoretical states is 99 . ± . ± − andwe record clicks for 10s. There are many sources of themeasurement uncertainty, such as counting statistics, de-tector efficiency, detector’s dead time, timing uncertaintyand alignment error of wave plates. However, the result-ing uncertainty is dominated by counting statistics [59],which we have calculated via the Poissonian distributionand shown in the figures.In summary, we have demonstrated the internal andexternal entanglement tradeoff in a photonic system withtunable entangled sources. This realization verifies thetheoretical prediction that the entanglement between dif-ferent degrees of freedom of a quantum single particle re-stricts its entanglement with other particles. This prop-erty may have applications in quantum information, suchas the construction of quantum communication network.On the other hand, this realization also can be gener-alized into other physical systems, such as NV centers,atoms, trapped ions, superconductor and so on. Particu-larly realization in a hybrid system, such as a photon andan atom, is more expected. For future research, one mayexperimentally demonstrate other monogamy inequali-ties, such as the inequality between local coherence andentanglement [46, 60], and the inequality between inter-nal entanglement and external correlations [49].This work is funded by the National Natural Sci-ence Foundation of China (Grants Nos. 11504253,11575051, 11674306, 61590932 and 11734015), NationalKey R&D Program (No. 2016YFA0301300 and No.2016A0301700), Anhui Initiative in Quantum Informa-tion Technologies, the startup funding from SoochowUniversity (Grant No. Q410800215) and the Hebei NSF(Grant No. A2016205215). ∗ Electronic address: [email protected] † Electronic address: [email protected][1] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki,
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Quantum coherence and corre-lations in quantum system , Sci. Rep. , 10922 (2015). SUPPLEMENTAL MATERIALI. Inequality (3) for the negativity E N If we use the negativity E N to quantify the internal entanglement between A and A , the inequality (3) in themain text becomes to E N ( ̺ A A ) + E ′ N ( | ψ i A A | B ) ≤ , (S1)where E N ( ̺ A A ) = ( k ̺ T A A A k − /
2, and E N, max , the maximum value of E N ( ̺ A A ), is equal to 1 / ̺ A A . The external entanglement E ′ N ( | ψ i A A | B ) is E ′ N ( | ψ i A A | B ) = 12 − E N ( ̺ ′ A A ) , (S2)with E N ( ̺ ′ A A ) = max U E N ( U ̺ A A U † )= max { , p ( λ − λ ) + ( λ − λ ) − λ − λ } (S3)where U denotes the unitary operators of A , ̺ ′ A A the density operator corresponding to the maximum over U , and { λ i } are the eigenvalues of ̺ A A in nonascending order [1].Now we consider a class of three-qubit pure states with one parameter φ , | ψ i = cos φ | i + sin φ | i + | i√ | i . (S4)upplemental Material –2/5Based on Eqs. (S1)-(S3), one can obtain its internal and external entanglement measured by the negativity, E N ( ̺ A A ) = 14 p φ ) −
12 cos φ, (S5) E ′ N ( | ψ i A A | B ) = 12 + 12 min { cos φ, sin φ } − p φ ) . (S6)The theoretical and experimental results have been shown in Fig. S1. We can see that all the sums of internal andexternal entanglement are bounded by 1 /
2, i.e., the inequality (S1) always holds. (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:3) ϕ (cid:1) (cid:1) ( (cid:2) (cid:1) (cid:2) (cid:2) ) + (cid:1) (cid:1) ( | ψ 〉 (cid:2) (cid:1) (cid:2) (cid:2) (cid:3) ) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6) (cid:1) (cid:1) ( ρ (cid:2) (cid:1) (cid:2) (cid:2) ) (cid:1) (cid:1) ψ 〉 (cid:2) (cid:1) (cid:2) (cid:2) (cid:3) ) FIG. S1: We use the negativity E N ( ̺ A A ) and E ′ N ( | ψ i A A | B ) to quantify the entanglement among these three qubits. Thered dots are experimental results and the lines are theoretical predictions. II. The monogamy inequality (9) involving only E N In Ref. [2], the author also presented a monogamy inequality involving only one entanglement monotone, thenegativity E N . For a bipartite state ̺ AB , the negativity is defined by E N ( ̺ AB ) = ( k ̺ T B AB k − /
2, where k · k is thetrace norm and T B is the partial transpose with respect to system B . For a three qubit pure state | ψ i A A B , themonogamy inequality is E N ( ̺ A A ) + g [ E N ( | ψ i A A | B )] ≤ , (S7)where E N, max , the maximum value of E N ( ̺ A A ), is equal to 1 / ̺ A A , and the nondecreasingfunction g is given by g ( x ) = 34 − √ − x − √ − x , (S8)when the number of nonzero eigenvalues of ̺ A A is equal to 2 [2].Now we consider a class of three-qubit pure states with one parameter φ , | ψ i = cos φ | i + sin φ | i + | i√ | i . (S9)Based on Eqs. (S7) and (S8), one can obtain its internal and external entanglement measured by the negativity, E N ( ̺ A A ) = 14 p φ ) −
12 cos φ, (S10) g [ E N ( | ψ i A A | B )] = 34 − p − φ sin φ − p − φ sin φ . (S11)The theoretical and experimental results have been shown in Fig. S2. We can see that all the sums of internal andexternal entanglement are bounded by 1 /
2, i.e., the inequality (S7) always holds.upplemental Material –3/5 (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:3) ϕ (cid:1) (cid:1) ( ρ (cid:2) (cid:1) (cid:2) (cid:2) ) + (cid:1) (cid:1) (cid:1) ( | ψ 〉 (cid:2) (cid:1) (cid:2) (cid:2) (cid:3) ) ] (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6) (cid:1) (cid:1) ( (cid:2) (cid:1) (cid:2) (cid:2) ) (cid:1) (cid:1) (cid:1) ( | (cid:2) (cid:1) (cid:2) (cid:2) (cid:3) ) ] FIG. S2: In order to use the same measure, E N ( ̺ A A ) and E N ( | ψ i A A | B ), to quantify the entanglement, we employ thefunction g . Note that g is not shown in the Figure. The red dots are experimental results and the lines are theoreticalpredictions. III. The monogamy inequality (10) involving only C Proposition 1.
For a three-qubit state ̺ A A B , the internal entanglement C ( ̺ A A ) and the external entanglement C ( ̺ A A | B ), as quantified by the concurrence, obey the monogamy relation C ( ̺ A A ) + ˜ g [ C ( ̺ A A | B )] ≤ , (S12)where 1 is the maximal value of C for two-qubit states and˜ g ( x ) = 1 − √ − x . (S13) Proof.
For a pure three-qubit state, the external entanglement is given by C ( | ψ i A A | B ) = q − Tr ̺ A A ) = 2 p λ (1 − λ ) , (S14)with λ being the maximal eigenvalue of ̺ A A . Since the above function of ̺ A A is concave and the concurrence isdefined via the convex roof extension for mixed states, the external entanglement obeys C ( ̺ A A | B ) ≤ p λ (1 − λ ) , (S15)in the general case. The internal entanglement can be obtained by the formula C ( ̺ A A ) = max { , σ − σ − σ − σ } with { σ i } being the square roots of eigenvalues of ̺ A A σ y ⊗ σ y ̺ ∗ A A σ y ⊗ σ y in decreasing order [3]. In the bipartitepartition A A | B , the internal entanglement can be maximized via two-qubit unitary transformations on A A , andthe following relation holds C ( ̺ A A ) ≤ max U C ( U ̺ A A U † ) , (S16)where the equality is satisfied for the so-called maximally entangled mixed state (MEMS) ̺ ′ A A [1]. In the case oftwo-qubit MEMSs [1], its concurrence is C ( ̺ ′ A A ) = max { , λ − λ − √ λ λ } ≤ λ , and hence C ( ̺ A A ) ≤ λ .Therefore, we have C ( ̺ A A ) + ˜ g [ C ( ̺ A A | B )] ≤ λ + ˜ g (cid:16) p λ (1 − λ ) (cid:17) = λ + 1 − p − λ (1 − λ )2= λ + 1 − (2 λ − , (S17)upplemental Material –4/5where 1 / ≤ λ ≤ (cid:3) Now we consider a class of three-qubit pure states with one parameter φ , | ψ i = cos φ | i + sin φ | i + | i√ | i . (S18)One can obtain its internal and external entanglement measured by concurrence, C ( ̺ A A ) = sin φ, (S19)˜ g [ C ( | ψ i A A | B )] = 1 − λ , (S20)with λ = max { cos φ, sin φ } . (S21)One can see that if λ = sin φ (i.e., sin φ ≥ cos φ ), then C ( ̺ A A ) + ˜ g [ C ( | ψ i A A | B )] = 1 holds. The theoretical andexperimental results have been shown in Fig. S3. We can see that all the sums of internal and external entanglementare bounded by 1, i.e., the inequality (S12) always holds. (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4)(cid:2)(cid:1) (cid:4)(cid:2)(cid:3)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:4)(cid:2)(cid:1) ϕ (cid:1) ( (cid:1) (cid:1) (cid:1) (cid:2) (cid:1) [ (cid:1) ( | ψ 〉 (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) ) ] (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:3) (cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:4) (cid:6)(cid:7)(cid:8) (cid:1) ( (cid:1) (cid:1) (cid:1) (cid:2) ) (cid:1) [ (cid:1) ( | ψ 〉 (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) ) ] FIG. S3: We use the concurrence C to quantify the internal and external entanglement simultaneously so we introduce thefunction ˜ g which is not shown in the Figure. The red dots are experimental results and the lines are theoretical predictions. IV. Quantum state tomography
We performed tomography to nine states we prepared. Their density matrices can be described in Eq. (8) of themain text. The θ is fixed at 45 ◦ but the φ is changing. Following are their density matrices that are obtained bymaximum likelihood estimation. We only labeled the values of φ to distinguish different states and also listed thefidelity of them. TABLE I: Fidelities for the quantum state Eq. (8) of the main text. The θ is fixed at 45 ◦ but the φ is changing from 0 ◦ to90 ◦ . The average fidelity of these states is 99 . ± . φ ◦ ◦ ◦ ◦ ◦ Fidelity 99 . ± .
11% 99 . ± .
07% 99 . ± .
02% 99 . ± .
05% 98 . ± . φ ◦ ◦ ◦ ◦ averageFidelity 99 . ± .
06% 98 . ± .
03% 98 . ± .
12% 99 . ± .
05% 99 . ± . Although the experimental states are not exactly pure states, the monogamy inequality (3) in the main text stillholds for experimental mixed states. Suppose that the experimentally realized tripartite state is ̺ A A B , thus theexternal entanglement is defined by the convex roof, E ′ F ( ̺ A A | B ) = inf { p i , | ψ i i A A B } X i p i E ′ F ( | ψ i i A A | B ) . (S22)upplemental Material –5/5We assume that ̺ A A B = P j p j | ψ j ih ψ j | is the optimal decomposition for ̺ A A B to achieve the above infimum.Therefore, E F ( ̺ A A ) + E ′ F ( ̺ A A | B ) = E F ( ̺ A A ) + X j p j E ′ F ( | ψ j i A A | B )= E F ( ̺ A A ) + 1 − X j p j E F ( U j ̺ j,A A U † j ) ≤ E F ( ̺ A A ) + 1 − X j p j E F ( V ̺ j,A A V † ) ≤ E F ( ̺ A A ) + 1 − E F ( V X j p j ̺ j,A A V † )= E F ( ̺ A A ) + 1 − E F ( ̺ ′ A A ) ≤ , (S23)where ̺ j,A A = Tr B | ψ j ih ψ j | , E ′ F ( | ψ j i A A | B ) = 1 − E F ( ̺ ′ j,A A ), E F ( ̺ ′ j,A A ) = max U E F ( U ̺ j,A A U † ) = E F ( U j ̺ j,A A U † j ), E F ( ̺ ′ A A ) = max U E F ( U ̺ A A U † ) = E F ( V ̺ A A V † ), the first inequality holds since E F ( U j ̺ j,A A U † j ) = max U E F ( U ̺ j,A A U † ) ≥ E F ( V ̺ j,A A V † ), and the second inequality is from the convex prop-erty of E F . This result also follows from the fact that E ′ F is a concave function of ̺ A A , as shown by the Proposition2 of the supplemental material of Ref. [2]. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] F. Verstraete, K. Audenaert, and B. De Moor,
Maximally entangled mixed states of two qubits , Phys. Rev. A , 012316(2001).[2] S. Camalet, Monogamy Inequality for Any Local Quantum Resource and Entanglement , Phys. Rev. Lett. , 110503 (2017).[3] W. K. Wootters,
Entanglement of formation of an arbitrary state of two qubits , Phys. Rev. Lett.80