Maximum Violation of Monogamy of Entanglement for Indistinguishable Particles by Measures that are Monogamous for Distinguishable Particles
aa r X i v : . [ qu a n t - ph ] F e b Maximum Violation of Monogamy of Entanglement for Indistinguishable Particles byMeasures that are Monogamous for Distinguishable Particles
Goutam Paul, ∗ Soumya Das, † and Anindya Banerji ‡ Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security,Indian Statistical Institute, Kolkata 700108, India Quantum Science and Technology Laboratory, Physical Research Laboratory, Ahmedabad 380009, India
Two important results of quantum physics are the no-cloning theorem and the monogamy ofentanglement . The former forbids the creation of an independent and identical copy of an arbitraryunknown quantum state and the latter restricts the shareability of quantum entanglement amongmultiple quantum systems. For distinguishable particles, one of these results imply the other. In thisLetter, we show that in qubit systems with indistinguishable particles (where each particle cannot beaddressed individually), a maximum violation of the monogamy of entanglement is possible by themeasures that are monogamous for distinguishable particles. To derive this result, we formulate thedegree of freedom trace-out rule for indistinguishable particles corresponding to a spatial locationwhere each degree of freedom might be entangled with the other degrees of freedom. Our resultremoves the restriction on the shareability of quantum entanglement for indistinguishable particles,without contradicting the no-cloning theorem.
Introduction. — An interesting feature of quantum physics is the presence of identical particles and their distin-guishability. Throughout this Letter, by identical particles [1, 2] we mean a set of particles with the same physicalproperties, except possibly their spatial locations; and by indistinguishable particles [3], we mean a set of identi-cal particles that cannot be labeled separately even by their spatial locations. Such particles find applications inBose-Einstein condensate [4, 5], quantum metrology [6, 7], quantum dots [8, 9], ultracold atomic gases [10] and as aresource [11] for tasks like teleportation [12, 13], entanglement swapping [14, 15] etc.One method of producing indistinguishable particles starting from identical ones is known as particle exchange[16, 17]. Figure 1 shows a conceptual representation of this process.
FIG. 1. Creation of indistinguishable particles. (a) Two identical particles A and B with two degrees of freedom (DoFs) each,denoted by square and triangle shapes, are present in distinct spatial locations s and s in such a way that their wave-functionsdo not overlap. Though identical, they are distinguishable via their spatial locations. (b) The particles are brought close toeach other so that their wave-functions overlap and they become indistinguishable. If the measurement is done in any DoF inthe overlapped region, i.e., s , then it is not possible to detect which particle is measured. Even if the particles are again movedapart, they can no longer be labeled. The information about which of A and B appears at s or s is lost. A composite system of indistinguishable particles cannot be decomposed in terms of distinguishable subsystems,as the underlying Hilbert space structure is fundamentally different [3]. This difference becomes crucial when oneconsiders entanglement of distinguishable vs. indistinguishable particles [18–29]. While the notion of entanglementfor distinguishable particles corresponds to the tensor product of the corresponding Hilbert spaces [30, 31], for indis-tinguishable case one must consider either the symmetric (bosonic) or the antisymmetric (fermionic) subspace of thewhole Hilbert space [32, Sec. III]. This leads to either particle-based first quantization [3, 18–23] or mode-based secondquantization [24–27] approaches giving contradictory results by representing some uncorrelated states as entangledones [23, 33]. To settle this issue, a new approach was proposed in [12, 34] which we use as a definition of entanglementof indistinguishable particles throughout this Letter (see Supplemental Material [35, Appendix A]).One important feature of quantum entanglement of distinguishable particles [30] is its restriction upon the share-
FIG. 2. Consider three particles A , B , C , and a bipartite entanglement measure E where E X | Y measures the entanglementbetween the subsystems X and Y of the composite system XY and E max denotes its maximum value. Now consider thefollowing five scenarios: (a) Particle-based MoE obeying Eq. (1). (b) Here, A is maximally entangled with B in DoF 1 (i.e., E A | B = E max ) and with C in DoF 2 (i.e., E A | C = E max ). In particle view, apparently MoE is violated; but in DoF view, itis not. (c) Inter-DoF MoE proposed in Eq. (3) which resolves the previous apparent violation. (d) & (e) Two-particle inter-DoFMoE, where E A | A measures entanglement between the two DoFs A , A of A and E A | B between A , B (in (d)); and E A | B j between A of A and B j of B , j ∈ { , } (in (e)). ability among composite systems (consisting of particles or degrees of freedom (DoFs)), known as monogamy ofentanglement (MoE). It has applications in key distribution [36], quantum games [37, 38], state classification [39],etc. [40–46].A bipartite entanglement measure E that obeys the relation E A | B ( ρ AB ) + E A | C ( ρ AC ) ≤ E A | BC ( ρ ABC ) , (1)for all ρ ABC where ρ AB = Tr C ( ρ ABC ), ρ AC = Tr B ( ρ ABC ), E X | Y measures the entanglement between the systems X and Y of the composite system XY , and the vertical bar represents bipartite splitting, is called monogamous . Suchinequality was first shown for squared concurrence ( C ) [47, 48] by Coffman, Kundu and Wootters (CKW) for threeparties [49] and later generalized for n parties [50].Some commonly used monogamous entanglement measures for qubit systems are the entanglement of formation [51–55], log-negativity [56, 57], Tsallis-q entropy [58, 59], R´enyi- α entanglement [60, 61], Unified-(q, s) entropy [62, 63],etc. [64, 65]. For higher dimensional systems, squared concurrence is known to violate [66] Eq. (1), and only a fewentanglement measures are monogamous like one-way distillable entanglement [67] and squashed entanglement [68, 69].Throughout this Letter, we focus on MoE and its violation using bipartite entanglement measures in qubit systemsonly.Suppose a bipartite entanglement measure E attains the maximum value E max for maximally entangled states.Consider a situation when E A | B ( ρ AB ) < E max , E A | C ( ρ AC ) < E max , but E A | B ( ρ AB ) + E A | C ( ρ AC ) > E max . Obviously,this causes a violation of MoE which we call a non-maximal violation . Consider another situation, when E A | B ( ρ AB ) = E max and E A | C ( ρ AC ) = E max , (2)i.e., when A is maximally entangled with both B and C , we call the corresponding violation as the maximal violation of MoE. For qubit systems with distinguishable particles, the first situation above would not lead to a violation of theno-cloning theorem [70, 71], but the second situation would do (see Supplemental Material [35, Appendix B]).The above result holds irrespective of whether A , B and C are single-DoF particles or DoFs of the same/differentparticles as shown in Fig. 2 (a) and (b). The entanglement measures which are monogamous for distinguishableparticles are also so for systems of indistinguishable particles, where A , B , and C are distinct spatial locations [22,24, 72] with one particle each. However, interesting scenarios might arise when the involved particles are entangled inmultiple DoFs which we investigate here.In this Letter, we establish a generalized DoF trace-out rule that covers single or multiple DoF scenarios for bothdistinguishable and indistinguishable systems. Partial trace-out operation [30, 31] is a typical method of finding thereduced density matrix of a subsystem which can be either one whole particle or a single DoF for distinguishablesystems. However, for indistinguishable systems, applying the above method results in a contradiction in identifyingentanglement [23, 33]. Experimental works on such systems [8, 9, 73–77] existed earlier, but a common mathematicalframework for a consistent theoretical interpretation was first attempted in [12, 34], by providing a method of partialtrace-out for a whole indistinguishable particle. One may be tempted to think that the same rule can trace out asingle DoF also. However, this is not so straightforward. When particles become indistinguishable, performing thepartial trace-out of a particular DoF is challenging, because a DoF cannot be associated with a specific particle. Wepropose a method to perform partial trace-out of a DoF when the particles are indistinguishable by suitably modifyingthe framework of [12, 34]. Our generalized method covers in a unified manner DoF or particle trace-out for single ormultiple DoF scenarios for both distinguishable and indistinguishable particles. Using this generalized DoF trace-out,we show that MoE can be violated maximally by indistinguishable particles in qubit systems for measures (such assquared concurrence, log-negativity, etc.) that are monogamous for distinguishable particles. This result establishes anew fundamental difference between distinguishable and indistinguishable systems. In the former case, the no-cloningtheorem and non-sharability of maximal entanglement (i.e., MoE) are equivalent. However, in the latter case, theno-cloning theorem remains valid, but maximal entanglement can be shared violating MoE.
Inter-DoF MoE .— Here we reformulate Eq. (1) in a more general framework to include multiple DoFs of thesame/different particles/entities. Although this is not a contribution, we include it here to establish the backgroundfor subsequent analysis.Consider three entities A , B , and C , each with n DoFs, numbered 1 to n . If the joint state of the i -th, j -th and k -th DoFs of A , B , and C respectively is represented by ρ A i B j C k , then the inter-DoF MoE can be formulated as E A i | B j ( ρ A i B j ) + E A i | C k ( ρ A i C k ) ≤ E A i | B j C k ( ρ A i B j C k ) , (3)where ρ A i B j = Tr C k ( ρ A i B j C k ), ρ A i C k = Tr B j ( ρ A i B j C k ), and E X i | Y j measures the entanglement between subsystems X i and Y j of the composite system X i Y j as is shown in Fig. 2 (c). It means that if the i -th DoF of A is maximallyentangled with the j -th DoF of B , then it cannot share any correlation with the k -th DoF of C .The inter-DoF MoE of Eq. (3) is more general than the particle-based MoE of Eq. (1). The former includes thelatter when the three DoFs i , j , and k belong to three different particles A , B , and C respectively. However, theinter-DoF MoE can capture many other scenarios that are illustrated in Fig. 2 (d) and (e).Two interesting types of MoE involving only two particles can also be explained using the inter-DoF formulation.(i) Type I : Here, MoE is calculated using E A i | A j and E A i | B k , as shown in Fig. 2 (d). Equation (3) can capture thisscenario by setting A = B . The recent analysis for distinguishable particles in [78, 79] is a specific example of thistype.(ii) Type II : Here, MoE is calculated using E A i | B j and E A i | B k , as shown in Fig. 2 (e). Equation (3) can capture thisscenario by setting B = C .This formulation also includes the case of single-particle entanglement [21, 80, 81], when all the three DoFs comefrom a single particle. Equation (3) can capture this scenario by setting A = B = C . Further, inter-DoF MoE is alsovalid for indistinguishable particles where the labels A , B , and C denote spatial locations with each mode containingexactly one particle and i , j , and k represents the DoFs at each spatial mode. DoF trace-out for indistinguishable particles and its physical significance. — We have already discussed that the trace-out operation of [12, 34] for indistinguishable particles is not readily applicable to trace out DoFs of such particles,particularly when the particles are entangled in multiple DoFs. In order to treat the cases of both distinguishable andindistinguishable particles under a uniform mathematical framework, we define trace-out of DoFs, rather than thatof whole particles, by suitably modifying the formulation of [12, 34].Assume two indistinguishable particles each having two DoFs are associated with spatial labels α and β . The i -thand the j -th DoFs are represented by a i and b j respectively, where i, j ∈ N = { , } . The general state of such asystem is written as | Ψ (2) i = X α,a ,a ,β,b ,b κ αa a βb b | αa a , βb b i , (4)where α, β ranges over S = { s , s , · · · , s p } which refers to distinct spatial locations with p ≥
2. Each of a i , b i rangesover D i = (cid:8) D i , D i , · · · , D i qi (cid:9) which refers to the eigenvalues of the i -th DoF, where q i ≥
2, since each DoF musthave at least two distinct eigenvalues.The value of q i may vary with i . For example, consider two DoFs: polarization and optical orbital angular mo-mentum (OAM), associated with a system of indistinguishable photons. Generally, the polarization belongs to atwo-dimensional Hilbert space, whereas the OAM lies in an infinite-dimensional Hilbert space governed by the az-imuthal index l . In practical implementations, this mismatch in Hilbert space dimensions between the two DoFs istaken care of by mapping the larger dimensional space to the lower dimensional one [80, 82]. For OAM, the infinite-dimensional Hilbert space is generally mapped into a two-dimensional one with the eigenvalues { l, l + 1 } or { + l, − l } .Also, the Hilbert space is sometimes restricted to smaller dimensions by proper state engineering in which case onlycertain chosen values of l are allowed.The general density matrix is expressed as ρ (2) = X α,β,γ,δ,a ,a ,b ,b ,c ,c ,d ,d κ αa a βb b κ γc c ∗ δd d | αa a ,βb b ih γc c ,δd d | , (5)where α, β, γ, δ span S and a i , b i , c i , d i span D i . To perform DoF trace-out of the i -th DoF, i ∈ N , of spatial region s x ∈ S , we define the reduced density matrix as ρ s x ¯ i ≡ Tr s xi (cid:16) ρ (2) (cid:17) ≡ X m i ∈D i h s x m i | ρ (2) | s x m i i := X m i (cid:26) X α,β,a i ,a ¯ i ,b ,b ,γ,δ,c i ,c ¯ i ,d ,d κ αa i a ¯ i βb b κ γc i c ¯ i ∗ δd d h s x m i | αa i ih γc i | s x m i i| αa ¯ i ,βb b ih γc ¯ i ,δd d | + η X α,β,a ,a ,b i ,b ¯ i ,γ,δ,c i ,c ¯ i ,d ,d κ αa a βb i b ¯ i κ γc i c ¯ i ∗ δd d h s x m i | βb i ih γc i | s x m i i| αa a ,βb ¯ i ih γc ¯ i ,δd d | + η X α,β,a i ,a ¯ i ,b ,b ,γ,δ,c ,c ,d i ,d ¯ i κ αa i a ¯ i βb b κ γc c ∗ δd i d ¯ i h s x m i | αa i ih δd i | s x m i i| αa ¯ i ,βb b ih γc c ,δd ¯ i | + X α,β,a ,a ,b i ,b ¯ i ,γ,δ,c ,c ,d i ,d ¯ i κ αa a βb i b ¯ i κ γc c ∗ δd i d ¯ i h s x m i | βb i ih δd i | s x m i i| αa a ,βb ¯ i ih γc c ,δd ¯ i | (cid:27) , (6)where ¯ i := (3 − i ). The parameter η is +1 ( −
1) for bosons (fermions). Equation (6) can be generalized for n DoFs andit includes particle trace-out as a special case for n = 1 (see Supplemental Material [35, Appendix C]).Our DoF trace-out rule plays a very critical role with respect to the recently introduced complex systems with inter-DoF entanglements [78, 79, 83]. When such entanglement exists, measuring or non-measuring one of the participatingDoFs would influence the measurement results of the other participating DoFs. The statistics so obtained cannot bepredicted using the existing trace-out rules in either the first or second quantization notations. Equation (6), on theother hand, can deal with all such systems with inter-DoF correlations in indistinguishable particles, leading to theprediction of perfect measurement statistics. Further, it generalizes the standard existing trace-out rule and is thereforesuitable for such entanglement structures of distinguishable particles as well. More specifically, for distinguishableparticles, tracing out a single DoF of a particle is analogous to tracing out a whole particle; for indistinguishableparticles, on the other hand, tracing out a single DoF is performed for a specific spatial location where wave-functionsof multiple particles might be overlapping. These overlaps are taken care of in the inner-product terms in the expressionof Eq. (6). Violation of MoE by indistinguishable particles.—
The inter-DoF MoE is not absolute and can be violated maximallyby indistinguishable particles. For illustration, consider two-particle inter-DoF entanglement [83, Eq. (4)] between spinand path. It can be represented as | Ψ (2) i of Eq. (4) with the parameters α, β ∈ { s , s } , a , b ∈ { L, D, R, U } , a , b ∈ {↑ , ↓} . The coefficients κ s L ↓ s R ↓ = − κ s D ↑ s U ↑ = ( κ + κ ), κ s D ↑ s R ↓ = κ s L ↓ s U ↑ = i ( κ − κ ), κ s R ↓ s R ↓ = κ s U ↑ s U ↑ = iκ , κ s D ↑ s D ↑ = κ s L ↓ s L ↓ = iκ and the rest are 0, where κ = e i ( φ R + φ L ) and κ = e i ( φ D + φ U ) (see Supplemental Material [35, Appendix D]).Next we show maximal violation of MoE through squared concurrence measure as follows. First, we apply a projectorΠ s s on | Ψ (2) i as in [12, 85], so that Alice and Bob have exactly one particle each, whereΠ s s := X ζ ∈{ L,D } ,ζ ∈{ R,U } ,τ ∈{↓ , ↑} | s ζ τ,s ζ τ ih s ζ τ,s ζ τ | . (7)This results in the normalized density matrix ρ (2) s s = | Ψ (2) i s s h Ψ (2) | where | Ψ (2) i s s = Π s s | Ψ (2) i √ h Ψ (2) | Π s s | Ψ (2) i . Now, wehave to trace-out the path DoF of each particle from ρ (2) s s using Eq. (6) resulting in the reduced density matrix ρ s a s b .Calculations show that the concurrence is C s a | s b ( ρ s a s b ) = 1, when Alice and Bob both measure the particlesin spin DoF (see Supplemental Material [35, Appendix D]). Similarly, the concurrence C s a | s b ( ρ s a s b ) when Alicemeasures in spin DoF and Bob measures in path DoF is also 1. Thus we clearly get a maximum violation of Eq. (3).Interestingly, this violation is irrespective of any particular entanglement measure like squared concurrence. It canbe shown that such a violation happens in indistinguishable particles by any monogamous bipartite entanglementmeasure for qubit systems. This leads to the following result. Theorem 1.
In qubit systems, indistinguishability is a necessary criterion for maximum violation of monogamy ofentanglement by the same measures that are monogamous for distinguishable particles.
For a formal proof of this theorem, please see Supplemental Material [35, Appendix E].
Discussion.–
MoE is widely regarded as one of the basic principles of quantum physics [86]. Qualitatively, it isalways expected to hold, as a maximal violation will have consequences for the no-cloning theorem. So much so, thata quantitative violation is interpreted as the non-monogamistic nature of the entanglement measure and not of thesystem of particles itself [87]. Some of those non-monogamous measures can be elevated to be monogamous throughconvex roof extension [88].Through Theorem 1, we establish a qualitative violation of MoE which was hitherto unheard of. We show thatusing quantum indistinguishability, it is possible to maximally violate the MoE for all such entanglement measureswhich are known to be monogamous for distinguishable systems. To establish this theorem, we first needed to modifythe qualitative definition of MoE itself, transiting from the particle-view to the DoF-view and had to introduce theDoF trace-out rule for indistinguishable particles. Thus, this is a non-trivial extension of the well-known MoE.Further, our framework also takes into account the recently introduced inter-DoF entanglement [83]. Quantumphysics dictates the measurement results of a particular DoF when it is correlated with another DoF. Taking a partialtrace while keeping the rule of quantum physics intact is extremely non-trivial and requires a rigorous mathematicaltreatment. Our framework, therefore, captures these nuances of quantum physics better than any other existingframework.Theorem 1 unveils a non-trivial difference between distinguishable and indistinguishable systems. For distinguish-able systems, MoE and no-cloning theorem imply one another (see Supplemental Material [35, Appendix B]). Thesignificance of our result is that for indistinguishable systems, the no-cloning theorem remains more fundamental thanMoE and the former does not necessarily imply the latter. In fact, no-cloning is derived from the linearity of quantummechanics [84] and hence even indistinguishable particles are also bound to follow it. It appears that the only wayto reconcile the co-existence of MoE violation and no-cloning for indistinguishable particles is to consider that suchparticles do not yield unit fidelity in quantum teleportation [12–14].Moreover, indistinguishability is not a sufficient condition for violation of MoE. There can be scenarios whereindistinguishable subsystems may be maximally entangled, respecting monogamy. Only specific entanglement struc-tures (such as the circuit discussed in this work) can lead to a maximum violation of MoE. That is why we callindistinguishability a necessary criterion for maximum violation of MoE.Theorem 1 raises a few fundamental questions on the properties of entanglement for indistinguishable particles.(i) There are several applications of MoE for distinguishable particles such as [36–40, 42, 43]. In particular, forcryptographic applications [36–38], MoE provides security in the distinguishable scenario. What happens to suchapplications in the indistinguishable case?(ii) Can there be a new application of sharability of maximal entanglement among indistinguishable particles thatare not possible for distinguishable ones?(iii) Do indistinguishable particles also exhibit maximum violation of monogamy for general quantum correla-tions [89] such as discord [90–93], coherence [94, 95], steering [96–98], etc.?All the above remains a matter of thorough investigation and can be part of interesting future works.
Conclusion.–
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Rudolph, New J. Phys. , 042105 (2016).[99] P. G. Kwiat, J. R. Mitchell, P. D. Schwindt, and A. G. White, J. Mod. Opt. , 257 (2000).[100] W. Pauli, Z. Phys. , 765 (1925).[101] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895 (1993).[102] S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, Nat. Photonics , 641 (2015).[103] N. L¨utkenhaus, J. Calsamiglia, and K.-A. Suominen, Phys. Rev. A , 3295 (1999). SUPPLEMENTAL MATERIALAppendix A: Revisiting the notation of Lo Franco et al. [34] for indistinguishable particles
If the state vector of two indistinguishable particles are labeled by φ and ψ , then the two-particle state is representedby a single entity | φ, ψ i . The two-particle probability amplitudes is represented by h ϕ, ζ | φ, ψ i := h ϕ | φ i h ζ | ψ i + η h ϕ | ψ i h ζ | φ i , (8)where ϕ, ζ are one-particle states of another global two-particle state vector and η = 1 for bosons and η = − | φ, ψ i = η | ψ, φ i . From Eq. (8), the probability of finding two particles in the same state | ϕ i is h ϕ, ϕ | φ, ψ i = (1 + η ) h ϕ | φ i h ϕ | ψ i which is zero for fermions due to Pauli exclusion principle [100] and maximum for bosons. As Eq. (8) follows symmetryand linearity property, the symmetric inner product of states with spaces of different dimensionality is defined as h ψ k | · | ϕ , ϕ i ≡ h ψ k | ϕ , ϕ i = h ψ k | ϕ i | ϕ i + η h ψ k | ϕ i | ϕ i , (9)where | ˜Φ i = | ϕ , ϕ i is the un-normalized state of two indistinguishable particles and | ψ k i is a single-particle state.Equation (9) can be interpreted as a projective measurement where the two-particle un-normalized state | ˜Φ i isprojected into a single particle state | ψ k i . Thus, the resulting normalized pure-state of a single particle after theprojective measurement can be written as | φ k i = h ψ k | Φ i q h Π (1) k i Φ , (10)where | Φ i := √ N | ˜Φ i with N = 1 + η | h ϕ | ϕ i | and Π (1) k = | ψ k i h ψ k | is the one-particle projection operator. The one-particle identity operator can be defined as I (1) := P k Π (1) k . So, using the linearity property of projection operators,one can write similar to Eq. (9): | ψ k i h ψ k | · | ϕ , ϕ i = h ψ k | ϕ i | ψ k , ϕ i + η h ψ k | ϕ i | ϕ , ψ k i . (11)Note that I (1) | Φ i = 2 | Φ i , (12)where the probability of resulting the state | ψ k i is p k = h Π (1) k i Φ /
2. The partial trace in this method is can be writtenas ρ (1) = 12 Tr (1) | Φ i h Φ | = 12 X k h ψ k | Φ i h Φ | ψ k i = X k p k | φ k i h φ k | , (13)where the factor 1 / localized partial trace [34], which means that local measurements are being performedon a region of space M where the particle has a non-zero probability of being found. So, performing the localizedpartial trace on a region M , we get ρ (1) M = 1 N M Tr (1) M | Φ i h Φ | , (14)where N M is a normalization constant such that Tr (1) ρ (1) M = 1. The entanglement entropy can be calculated as E M ( | Φ i ) := S ( ρ (1) M ) = − X i λ i ln λ i , (15)where S ( ρ ) = − Tr( ρ ln ρ ) is the von Neumann entropy and λ i are the eigenvalues of ρ (1) M . We will call the state asentangled state if we get a non-zero value of Eq. (15). Appendix B: Equivalence of the monogamy of entanglement and the no-cloning theorem for distinguishableparticles
To show that no-cloning implies monogamy of entanglement (MoE), let us prove its contrapositive. When MoEis violated maximally, one can achieve quantum cloning [70, 71] of any unknown quantum state using standardteleportation protocol [101, 102] as follows. Assume a particle A is maximally entangled with the particles B and C and their joint state is denoted by | ψ i ABC and the particle X is in unknown quantum state | φ i C . To achieve cloningof the state | φ i , one has to perform Bell state measurements (BSM) [103] jointly on the particles A and X . Based onthe measurement result denoted by y , suitable unitary operations U y have to be performed on the particles B and C so that the state | φ i appears on each of them, where U y ∈ {I , σ x , σ y , σ z } , I being the identity operation and σ i ’s( i = x, y, z ) the Pauli matrices. Thus we can have two copies of the unknown state | φ i as | φ i B and | φ i C .Next, to show that MoE implies no-cloning, again we prove its contrapositive. Let two particles A and B share amaximally entangled state | ψ i AB . If possible, suppose one of them, say, B is cloned and we get a copy B of B , thenin the tripartite state | ψ i ABB , A is maximally entangled with both B and B simultaneously, thus violating the MoEmaximally. FIG. 3. Circuit to get violation of the no-cloning theorem from the maximum violation of MoE.
Appendix C: Degree of freedom (DoF) trace-out rule for two indistinguishable particles
Lo Franco et al. [34] have defined the partial trace-out rule for indistinguishable particles where each particle has aspatial label and a single DoF. By a non-trivial extension of their concept, We define DoF trace-out rule when eachindistinguishable particle has n DoFs and a spatial label. Lets us consider two indistinguishable particles each having n DoFs, having spatial labels α and β whose i -th and j -th DoF are represented by a i and b j respectively, where i, j ∈ N = { , , · · · , n } . The general state of two particles can be represented by | Ψ ( n ) i = X α,β,a ,a , ··· ,a n ,b ,b , ··· ,b n κ αa a ··· a n βb b ··· b n | αa a · · · a n , βb b · · · b n i . (16)Here, α, β ranges over S = { s , s , · · · , s p } which refers to distinct spatial modes with p ≥
2. Each of a i , b i ranges over D i = (cid:8) D i , D i , · · · , D i qi (cid:9) which refers to the eigenvalues of the i -th DoF, where i ∈ N . Note that q i ≥ q i may be different for different i .The general density matrix of two indistinguishable particles can be described as ρ ( n ) = X α,β,a ,a , ··· ,a n ,b ,b , ··· ,b n X γ,δ,c ,c , ··· ,c n ,d ,d , ··· ,d n κ αa a ··· a n βb b ··· b n κ γc c ··· c n ∗ δd d ··· d n | αa a ··· a n ,βb b ··· b n ih γc c ··· c n ,δd d ··· d n | , (17)where α, β, γ, δ span S and a i , b i , c i , d i span D i with i ∈ N . If we want to perform partial trace in only one region, say s x ∈ S , then the non-normalized density matrix can be written as˜ ρ (1) M = Tr M (cid:16) ρ ( n ) (cid:17) = X m ,m , ··· ,m n h s x m m · · · m n | ρ ( n ) | s x m m · · · m n i , (18)where m i span D i , where i ∈ N . Equations (16) to (18) can trivially be generalized for n particles.We define the probability of the joint measurement on | Ψ ( n ) i , where one particle is measured in the h -th eigenvalueof the i -th DoF denoted by D i h in localized region s x and another particle is measured in the k -th eigen value of j -thDoF denoted by D j k in localized region s y where x, y ∈ { , , · · · , p } , i, j ∈ N , h ∈ { , , · · · , q i } , and k ∈ { , , · · · , q j } , asbelow: X α,β,a ,a , ··· ,a n ,b ,b , ··· ,b n h s x a a · · · a i − D i h a i +1 · · · , a n , s y b b · · · b j − D j k b j − · · · b n | Ψ ( n ) i . (19)The joint measurement rule defined in Eq. (19) is the generalized version of the joint measurement rule defined in (8).Now we define the DoF trace-out rule. First, we consider distinguishable particles, and then we will extend it forindistinguishable particles.0If the particles A and B were distinguishable such that the particle A and B are in the spatial region α and β respectively, then Eq. (16) could be represented as | Ψ ( n ) i AB = X a ,a , ··· ,a n ,b ,b , ··· ,b n κ a a ··· a n b b ··· b n | a a · · · a n i ⊗ | b b · · · b n i , (20)and the density matrix of Eq. (17) would take the form ρ ( n ) AB = X a ,a , ··· ,a n ,b ,b , ··· ,b n X c ,c , ··· ,c n ,d ,d , ··· ,d n κ a a ··· a n b b ··· b n κ c c ··· c n ∗ d d ··· d n | a a · · · a n i | b b · · · b n i ⊗ h c c · · · c n | h d d · · · d n | . (21)If we want to trace-out the i -th DoF of particle A , then from Eq. (21), the reduced density matrix can be writtenas ρ a ¯ i ≡ Tr a i (cid:16) ρ ( n ) AB (cid:17) := X a i ,a ¯ i ,b ,b , ··· ,b n X c i c ¯ i ,d , ··· ,d n κ a ¯ i b b ··· b n κ c ¯ i ∗ d d ··· d n | a ¯ i i | b b · · · b n i h c ¯ i | h d d · · · d n | {h a i | c i i} , (22)where a ¯ i = a a · · · a i − a i +1 · · · a n and similar meaning for c ¯ i . One can show that when the DoF trace-out rule inEq. (22) is applied to the same particle for n times, it becomes equivalent to our familiar particle trace-out rule [31,Eq. 2.178].Next we define DoF trace-out rule for indistinguishable particles from the general density matrix of two particlesas defined in Eq. (17). Suppose we want to trace-out the i -th DoF of location s x ∈ S . Then the DoF reduced densitymatrix is ρ s x ¯ i ≡ Tr s xi (cid:16) ρ ( n ) (cid:17) ≡ X m i h s x m i | ρ ( n ) | s x m i i := X m i (cid:26) X α,β,a i ,a ¯ i ,b ,b , ··· ,b n ,γ,δ,c i ,c ¯ i ,d ,d , ··· ,d n κ αa i a ¯ i βb b ··· b n κ γc i c ¯ i ∗ δd d ··· d n h s x m i | αa i ih γc i | s x m i i| αa ¯ i ,βb b ··· b n ih γc ¯ i ,δd d ··· d n | + η X α,β,a ,a , ··· ,a n ,b i ,b ¯ i ,γ,δ,c i ,c ¯ i ,d ,d , ··· ,d n κ αa a ··· a n βb i b ¯ i κ γc i c ¯ i ∗ δd d ··· d n h s x m i | βb i ih γc i | s x m i i| αa a ··· a n ,βb ¯ i ih γc ¯ i ,δd d ··· d n | + η X α,β,a i ,a ¯ i ,b ,b , ··· ,b n ,γ,δ,c ,c , ··· ,c n ,d i ,d ¯ i κ αa i a ¯ i βb b ··· b n κ γc c ··· c n ∗ δd i d ¯ i h s x m i | αa i ih δd i | s x m i i| αa ¯ i ,βb b ··· b n ih γc c ··· c n ,δd ¯ i | + X α,β,a ,a , ··· ,a n ,b i ,b ¯ i ,γ,δ,c ,c , ··· ,c n ,d i ,d ¯ i κ αa a ··· a n βb i b ¯ i κ γc c ··· c n ∗ δd i d ¯ i h s x m i | βb i ih δd i | s x m i i| αa a ··· a n ,βb ¯ i ih γc c ··· c n ,δd ¯ i | (cid:27) , (23)where a ¯ i := a a · · · a i − a i +1 · · · a n and similar meaning for b ¯ i , c ¯ i and d ¯ i .It may be noted that for n = 2, the DoF trace-out rule defined in Eq. (23) reduces to Eq. (6) in the main text. For n = 1, this becomes equivalent to the particle trace-out rule as defined Eq. (13). On the other hand, for n >
1, if weapply DoF trace-out rule of Eq. (23) n times, the effect will not be the same as the particle trace-out in Eq. (13). Thereason behind this is as follows. For indistinguishable particles, the particle trace-out operation vanishes all the DoFstogether for one particle; whereas each DoF trace-out operation leaves an expression with many terms each of whichvanishes the corresponding DoF from one particle at a time and retains the same DoF in the remaining particles. Appendix D: Description of the circuit of Li et al. [83]
The circuit of Yurke et al. [16, 17] to generate quantum entanglement between the same DoFs of two indistinguish-able particles (bosons and fermions) is extended by Li et al. [83] to generate inter-DoF entanglement between twoindistinguishable bosons. Details of their generation scheme are as follows.For bosons, the second quantization formulation deals with bosonic operators b i, p with | i, p i = b † i, p | i , where | i is the vacuum and | i, p i describes a particle with spin | i i and momentum p . These operators satisfy the canonicalcommutation relations:1 FIG. 4. Circuit to generate hyper-hybrid entangled state as proposed by Li et al. [83]. Here the bi-directional arrow representsthe measurement is done either in spin DoF or in Path DoF. h b i, p i , b j, p j i = 0 , h b i, p i , b † j, p j i = δ ( p i − p j ) δ ij . (24)Analysis of the circuit of Li et al. [83] for bosons involves an array of hybrid beam splitters (HBS) [83, Fig. 3], phaseshifts, four orthogonal external modes L , D , R and U and two orthogonal internal modes ↑ and ↓ as shown in Fig. 4.Here, particles exiting through the modes L and D are received by Alice (A) who can control the phases ϕ L and ϕ D ,whereas particles exiting through the modes R and U are received by Bob (B) who can control the phases ϕ R and ϕ U .In this circuit, two particles, each with spin |↓i , enter the set up in the mode R and L for Alice and Bob respectively.The initial state of the two particles is | Ψ i = b †↓ ,R b †↓ ,L | i . Now, the particles are sent to HBS such that one outputport of HBS is sent to other party ( R or L ) and the other port remains locally accessible ( D or U ). Next, each partyapplies state-dependent (or spin-dependent) phase shifts. Lastly, the output of local mode and that received from theother party is mixed with HBS and then the measurement is performed in either external or internal modes. The finalstate can be written as | Ψ i = 14 h e iϕ R (cid:16) b †↓ ,R + ib †↑ ,U (cid:17) + ie iϕ D (cid:16) b †↑ ,D + ib †↓ ,L (cid:17)i ⊗ h e iϕ L (cid:16) b †↓ ,L + ib †↑ ,D (cid:17) + ie iϕ U (cid:16) b †↑ ,U + ib †↓ ,R (cid:17)i | i . (25)Now we represent the circuit of Li et al. using our proposed extended version of the notation of Lo Franco et al. asdescribed in Appendix C. The final state in Eq. (25) can be represented by Eq. (16) as | Ψ (2) i = X α,β ∈{ s ,s } ,a ,b ∈{ L,D,R,U } ,a ,b {↑ , ↓} κ αa a βb b | αa a , βb b i , (26)where the coefficients are κ s L ↓ s R ↓ = − κ s D ↑ s U ↑ = 14 ( κ + κ ) , κ s D ↑ s R ↓ = κ s L ↓ s U ↑ = i κ − κ ) , κ s R ↓ s R ↓ = κ s U ↑ s U ↑ = iκ , κ s D ↑ s D ↑ = κ s L ↓ s L ↓ = iκ , and the rest are 0, where κ = e i ( φ R + φ L ) and κ = e i ( φ D + φ U ) . Here, we denote the specialized location where Alice andBob have performed the measurement as s and s respectively.2 Appendix E: Proof of Theorem 1
In this Section, we will prove our Theorem 1 of the main text. We will give details calculations that maximumviolation of MoE happens using squared concurrence measure and log-negativity as entanglement measure. Thesimilar calculation will follow for other monogamous entanglement measures also.First, we calculate concurrence of the state | Ψ (2) i as described in Eq. (26) of Appendix D.Projecting ρ (2) = | Ψ (2) i h Ψ (2) | onto the (operational) subspace spanned by the computational basisΩ s s = { | s L ↓ , s R ↓i , | s L ↓ , s U ↓i , | s L ↓ , s R ↑i , | s L ↓ , s U ↑i , | s L ↑ , s R ↓i , | s L ↑ , s U ↓i , | s L ↑ , s R ↑i , | s L ↑ , s U ↑i , | s D ↓ , s R ↓i , | s D ↓ , s U ↓i , | s D ↓ , s R ↑i , | s D ↓ , s U ↑i , | s D ↑ , s R ↓i , | s D ↑ , s U ↓i , | s D ↑ , s R ↑i , | s D ↑ , s U ↑i} , (27)by the projector Π s s = X σ,τ = {↑ , ↓} ,ς = { L,D } ,υ = { R,U } | s ςσ, s υτ i h s ςσ, s υτ | , (28)one gets the distributed resource state where each localized region s and s have exactly one particle as | Ψ (2) i s s = Π s s | Ψ (2) i p h Ψ (2) | Π s s | Ψ (2) i = X a ∈{ L,D } ,b ∈{ R,U } ,a ,b ∈{↑ , ↓} κ s a a s b b | s a a , s b b i , (29)where the non-zero coefficients are κ s L ↓ s R ↓ = − κ s D ↑ s U ↑ = 12 √ κ + κ ) , κ s D ↑ s R ↓ = κ s L ↓ s U ↑ = i √ κ − κ ) . (30)The density matrix ρ (2) s s = | Ψ (2) i s s h Ψ (2) | can also be calculated as ρ (2) s s = Π s s ρ (2) Π s s Tr (cid:0) Π s s ρ (2) (cid:1) , (31)where ρ (2) = | Ψ (2) i h Ψ (2) | .Now from Eq. (31), if we trace-out the path DoFs of location s and s using Eq. (23) (the order does not matter),we get the reduced density matrix as ρ s a s b = Tr s a s b (cid:16) ρ (2) s s (cid:17) = X a ,b ,c ,d ∈{↑ , ↓} κ s a s b κ s c ∗ s d | s a , s b i h s c , s d | , (32)where κ s ↓ s ↓ = − κ s ↑ s ↑ = 12 √ κ + κ ) , κ s ↑ s ↓ = κ s ↓ s ↑ = i √ κ − κ ) , and rest are zero where complex conjugates are calculated accordingly.To calculate the maximum violation using squared concurrence, first we calculate the following: e ρ s a s b = σ s y ⊗ σ s y ρ ∗ s a s b σ s y ⊗ σ s y , (33)where σ Xy = | X ih X | ⊗ σ y , X ∈ { s , s } , σ y is Pauli matrix and the asterisk denotes complex conjugation. So, the ex-pression becomes e ρ s a s b = X a ,b ,c ,d ∈{↑ , ↓} ˜ κ s a s b ˜ κ s c ∗ s d | s a , s b i h s c , s d | , (34)where˜ κ s ↓ s ↓ ˜ κ s ↓∗ s ↓ = − ˜ κ s ↓ s ↓ ˜ κ s ↑∗ s ↑ = − ˜ κ s ↑ s ↑ ˜ κ s ↓∗ s ↓ = ˜ κ s ↑ s ↑ ˜ κ s ↑∗ s ↑ = 12 cos φ, ˜ κ s ↓ s ↓ ˜ κ s ↑∗ s ↓ = ˜ κ s ↓ s ↓ ˜ κ s ↓∗ s ↑ = − ˜ κ s ↑ s ↑ ˜ κ s ↑∗ s ↓ = − ˜ κ s ↑ s ↑ ˜ κ s ↓∗ s ↑ = ˜ κ s ↑ s ↓ ˜ κ s ↓∗ s ↓ = ˜ κ s ↓ s ↑ ˜ κ s ↓∗ s ↓ = ˜ κ s ↑ s ↓ ˜ κ s ↑∗ s ↑ = ˜ κ s ↓ s ↑ ˜ κ s ↑∗ s ↑ = 12 cos φ sin φ, ˜ κ s ↑ s ↓ ˜ κ s ↑∗ s ↓ = ˜ κ s ↑ s ↓ ˜ κ s ↓∗ s ↑ = ˜ κ s ↓ s ↑ ˜ κ s ↑∗ s ↓ = ˜ κ s ↓ s ↑ ˜ κ s ↓∗ s ↑ = 12 sin φ, φ = { φ D + φ U − φ R − φ L } . Now we calculate concurrence as C s a | s b (cid:16) ρ s a s b (cid:17) = max n , p λ − p λ − p λ − p λ o , (35)where λ i are the eigenvalues, in decreasing order, of the non-Hermitian matrix R = ρ s a s b e ρ s a s b = X a ,b ,c ,d ∈{↑ , ↓} ¯ κ s a s b ¯ κ s c ∗ s d | s a , s b i h s c , s d | , (36)where¯ κ s ↓ s ↓ ¯ κ s ↓∗ s ↓ = − ¯ κ s ↓ s ↓ ¯ κ s ↑∗ s ↑ = − ¯ κ s ↑ s ↑ ¯ κ s ↓∗ s ↓ = ¯ κ s ↑ s ↑ ¯ κ s ↑∗ s ↑ = 14 cos φ, ¯ κ s ↓ s ↓ ¯ κ s ↑∗ s ↓ = ¯ κ s ↓ s ↓ ¯ κ s ↓∗ s ↑ = − ¯ κ s ↑ s ↑ ¯ κ s ↑∗ s ↓ = − ¯ κ s ↑ s ↑ ¯ κ s ↓∗ s ↑ = ¯ κ s ↑ s ↓ ¯ κ s ↓∗ s ↓ = ¯ κ s ↓↑ ¯ κ ↓∗↓ = ¯ κ ↑ s ↓ ¯ κ s ↑∗ s ↑ = ¯ κ s ↓ s ↑ ¯ κ s ↑∗ s ↑ = 14 cos φ sin φ, ¯ κ s ↑ s ↓ ¯ κ s ↑∗ s ↓ = ¯ κ s ↑ s ↓ ¯ κ s ↓∗ s ↑ = ¯ κ s ↓ s ↑ ¯ κ s ↑∗ s ↓ = ¯ κ s ↓ s ↑ ¯ κ s ↓∗ s ↑ = 14 sin φ. So, the eigenvalues of R are { , , , } . Thus C s a | s b (cid:16) ρ s a s b (cid:17) = 1 . (37)Similar calculations follows that, C s a | s b (cid:16) ρ s a s b (cid:17) = 1.Likewise, we can also calculate the log-negativity [56, 57] for the density matrix ρ s a s b in Eq. (32). For that, weneed the eigenvalues of the density matrix ρ s a after taking the partial transpose with respect to s b . The eigenvaluesare found to be {− , , , } . Thus the value of negativity is and so the log-negativity is given by E N (cid:16) ρ s a s b (cid:17) = 1 . (38)Similar calculations give E N (cid:16) ρ s a s b (cid:17) = 1 . All other monogamous measures of entanglement for qubit systems such as entanglement of formation [52–55], log-negativity [56, 57], Tsallis-q entropy [58, 59], R´enyi- α entanglement [60, 61], Unified-(q, s) entropy [62, 63], one-waydistillable entanglement [67], squashed entanglement [68, 69] etc. [64] are calculable from the reduced density matrix.If one starts with the same reduced density matrix as in Eq. (32), one can easily show that all the above measuresattain their respective maximum value for both the subsystems { s a , s b } and { s a , s b } simultaneously, therebyviolating the MoE. Thus, for any bipartite monogamous entanglement measure E , we get E s a | s b (cid:16) ρ s a s b (cid:17) = E s a | s b (cid:16) ρ s a s b (cid:17) = 1 ..