Local discrimination of mixed states
aa r X i v : . [ qu a n t - ph ] A p r Local discrimination of mixed states
J. Calsamiglia, J. I. de Vicente, R. Mu˜noz-Tapia, E. Bagan
Grup d’Informaci´o Qu`antica (GIQ), Departament de F´ısica,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
We provide rigorous, efficiently computable and tight bounds on the average error probability ofmultiple-copy discrimination between qubit mixed states by Local Operations assisted with ClassicalCommunication (LOCC). In contrast to the pure-state case, these experimentally feasible protocolsperform strictly worse than the general collective ones. Our numerical results indicate that the gapbetween LOCC and collective error rates persists in the asymptotic limit. In order for LOCC andcollective protocols to achieve the same accuracy, the former requires up to twice the number ofcopies of the latter. Our techniques can be used to bound the power of LOCC strategies in othersimilar settings, which is still one of the most elusive questions in quantum communication.
Quantum communication and computation tasks in-volve, broadly speaking, transforming an input state andreading the corresponding output state. One of the mostprominent features of quantum mechanics is that, hardas one may try, the readout will be unavoidably imper-fect unless the various output states are orthogonal. Thishas both fundamental and practical implications that lieat the heart of quantum mechanics and its applications.The most simple scenario where this deep fact manifestitself is what is known as quantum state discriminationor, more generically, quantum hypothesis testing [1]. Inits simplest form, given one of two possible sources thatprovide N independent copies of a state ρ or ρ (i. e., σ = ρ ⊗ N or σ = ρ ⊗ N ), we ask ourselves what is in av-erage the minimum error probability P e of making a con-clusive guess of the identity of the source/state. Hereafterwe will refer to this minimum average simply as the er-ror probability, and for simplicity we will further assumeequal prior probabilities for ρ a , a = 0 ,
1. State discrimi-nation is an essential primitive for many quantum infor-mation tasks, such as quantum cryptography [2], or evenquantum algorithms [3]. Moreover, with the remarkableexperimental advances in the preparation and measure-ment of quantum states it becomes essential to have atheory to assess the performance of state discriminationprotocols.The error probability for two arbitrary states was givenalready decades ago by Helmstrom [4], who provided aformal expression for P e in terms of the trace distancebetween the two density matrices σ and σ . An explicitsimple expression can be given for single qubit states( N = 1), and only very recently the asymptotic errorrate when N → ∞ for qudits has been found throughthe quantum Chernoff bound [5, 6]. For finite (moder-ately large) number of qubits the permutation invarianceof the multiple-copy states { σ a } , of size ∼ N , enablesus to write them in block-diagonal form. Thus, we cannumerically compute the error probability in terms ofthe trace-distance between small-sized ( ∼ N ) blocks, andthe difficulty of the problem (required memory size) be-comes polynomial in N despite the exponentially grow-ing dimension of the Hilbert space we are dealing with. This collection of results constitute a fairly completetheory regarding the optimal (unconstrained/collective-measurement) multiple-copy discrimination of quantumstates.The picture changes completely when it comes to dis-crimination of states using Local Operations (on indi-vidual copies) and Classical Communication (LOCC) in-stead of general (collective) measurements. This sce-nario is interesting from the fundamental point of view,as it sheds light on the role of quantum correlations inquantum information tasks, but it is of paramount in-terest from a practical point of view since it puts underscrutiny the attainability of previous bounds in imple-mentations, where collective measurements are usuallyunfeasible. For pure states it has been shown [7] that theminimum collective error probability can be attained by aLOCC one-way adaptive protocol consisting in perform-ing a different von Neumann measurement on each copy,where each measurement is chosen according to the out-come(s) of the previous one(s). Furthermore, it is shownthat an even simpler fixed local strategy, where the verysame particular von Neumann measurement is repeatedon every copy, though not optimal for finite N , does pro-vide the collective asymptotic error rate as N → ∞ . Thelatter also holds when only one of the states is pure [6].For mixed states, recently, Hayashi [8] has proven that,as far as the asymptotic error rate is concerned, one-way adaptive strategies are not advantageous over fixedstrategies, which do not make use of classical communi-cation between measurements. In the last months Hig-gins et al. [9] have studied theoretically and experimen-tally the performance of various local strategies includingadaptive von Neumann measurements. These adaptivestrategies, which are optimized using dynamic program-ming (DP) techniques [10], outperform the others un-der consideration, but there still remain the fundamentalopen questions of whether or not these strategies are thebest one can achieve by LOCC (which include general-ized local measurements and unlimited communicationrounds), and whether or not those can attain the collec-tive bounds.In this letter we show how to efficiently computebounds on LOCC protocols. This enables us to comparesuch protocols to their collective counterparts and bench-mark the performance of particular state discriminationstrategies or experiments. Before presenting our analy-sis, let us note that there has been some recent interest inthe related problem of LOCC discrimination of bipartitestates (see [11] and references therein), where each of theparties has joint access to a share of all copies. We findthat in our scenario, although the states are disentan-gled, quantum correlations play an essential role, whichis, in this sense, a manifestation of non-locality withoutentanglement.At this point we need to go into a more technical dis-cussion. The error probability for the case under consid-eration can be written as [4] P e = 12 (cid:26) ≤ E ≤ tr [( σ − σ ) E ] (cid:27) . (1)The matrix E , together with − E , are the elementsof the Positive Operator Valued Measure (POVM) thatrepresents mathematically the measuring protocol. Notethat they can be taken to be symmetric under permuta-tions of the individual systems (invariant under the ac-tion of the symmetric group S N ), because so are σ and σ , and can thus be put in block-diagonal form.We realize that Eq. (1) defines a Semidefinite Program-ming (SDP) problem, for which very efficient numericalalgorithms have been recently developed [12]. Bounds onLOCC strategies could be obtained if in addition to thepositivity constraint 0 ≤ E ≤ E and − E one would further impose, e. g., Positive Par-tial Transposition (PPT), i. e., 0 ≤ E Γ ≤ not preserve the block-diagonal formof E (just note that Γ breaks permutation invariance).Hence, the size of the matrices (in particular, E Γ ) oneneeds to deal with remains ∼ N and the error probabil-ity cannot be computed but for very small values of N .We next show how to go around this problem.The main observation is that for (two) qubit-state dis-crimination the POVM element E can always be cho-sen to be PT invariant, E Γ = E (for any bipartite splitof the N qubits). This follows from the fact that withthe appropriate choice of basis, one can always cancel allphases in ρ and ρ simultaneously, so that they have onlyreal elements and are, therefore, symmetric: ρ T a = ρ a .Obviously, this implies PT invariance, σ Γ a = σ a , for anybipartite split. Hence, for a given E that satisfies PPT,the PT invariant operator E ′ = ( E + E Γ ) /
2, which alsosatisfies 0 ≤ E ′ = E ′ Γ ≤
1, provides the exact same er-ror probability P e . Therefore, we can restrict ourselvesto PT invariant operators without any loss of generality.Since E can be put in block-diagonal form, and there is no need to apply Γ to check PPT, the sizes of the matriceswe have to deal with grow polynomially (quadratically)in N .Applying the procedure sketched above requires,nonetheless, finding an efficient parametrization of PTinvariant matrices in block-diagonal form. The first steptowards this end is identifying the independent matrixelements in the computational basis | i i . . . i N i . For N qubits, the operator E can be written as E = X { i p } X { i ′ s } E i ′ i ′ ...i ′ N i i ...i N | i i . . . i N ih i ′ i ′ . . . i ′ N | , (2)where all i p , i ′ s ( p, s = 1 , . . . N ) are either 0 or 1 andeach sum runs over the 2 N possible binary lists (num-bers) of N digits. Invoking permutation invariance andhermiticity the independent components of E can be cho-sen to be E ... ... ... ... ... ... ... ... ≡ ˜ E Q,Q ′ R , where the first R digits in the subscript are ones and the Q ( Q ′ ) first dig-its on top of them (the remaining N − R zeros) are alsoones. We further impose that R ≤ Q + Q ′ and notethat Q ≤ R . We next use PT invariance to exchange thelast R − Q ones in the subscript with the zeros on topof them by raising (lowering) the corresponding i p ( i ′ p ).This proves that the PT invariant matrices we are dealingwith have ( N + 1)( N + 2) / E ... ... ... ... ... ... ≡ E qr , r ≤ q, (3)where q and r are the number of ones in the superscriptand the subscript respectively.We next wish to write E in block-diagonal form. Tothis end, we map each qubit to a spin 1 / | i p i → | m p i = | ( − i p / i , where m p is the magnetic number of the p -thspin, and change from the uncoupled (computational) ba-sis to the total spin eigenbasis {| j, m i} jm = − j , which spanthe irreducible representations (irreps) of SU (2). In thisbasis E becomes block-diagonal and the matrix elementsof each block E ( j ) , i. e., [ E ( j ) ] m ′ m = h j, m | E | j, m ′ i , are ex-pressed as linear combinations of the independent param-eters E qr . We write (cid:2) E ( j ) (cid:3) m ′ m = P r,q (cid:2) M ( j ) (cid:3) m ′ rm q E qr , whichfacilitates the SDP implementation of the optimization.Some comments are in order. For given j and m , thestate | j, m i is degenerate. Note, though, that all blockswith the same j are identical, as E is fully symmetric.Therefore, the contribution of E ( j ) to the error probabil-ity will have to be multiplied by the corresponding de-generacy, n j = (cid:0) NN/ − j (cid:1) (2 j + 1) / ( N/ j + 1). The ma-trices M ( j ) turn out to be (see supplementary material) (cid:2) M ( j ) (cid:3) m ′ rm q = X k [∆ ( j ) k ] m ′ m (cid:18) N − j q − r + m ′ − m − k (cid:19) × ( − q − r + m ′ − m − k δ q + r,N − m − m ′ , (4)where we have defined (cid:2) ∆ ( j ) k (cid:3) m ′ m = p ( j − m )!( j + m )!( j − m ′ )!( j + m ′ )!( j − m − k )!( j + m ′ − k )!( m − m ′ + k )! k ! . (5)In the above expressions the sums run over all integervalues for which the factorials make sense. We note inpassing that the very same coefficient (20) appears inthe expression of the Wigner d-matrices, which give thedifferent irreps of a rotation about the y axis (see sup-plementary material).Now that we have a minimal parameterization of theoperators that are invariant under permutations andpartial-transpostions we can compute the error proba-bility by the following SDP instance: P PPT e = 12 n
1+ min { E qr } X j n j tr h ( σ ( j )0 − σ ( j )1 ) E ( j ) i o . (6)Here the minimization is constrained by 0 ≤ E ( j ) ≤ j , and the matrixblocks σ ( j ) a are computed to be[ σ ( j ) a ] m ′ m = (cid:0) − r a (cid:1) N − j N X k [∆ ( j ) k ] m ′ m (cid:20) ( − a r a sin θ (cid:21) m − m ′ +2 k × (cid:18) r a cos θ (cid:19) j + m ′ − k (cid:18) − r a cos θ (cid:19) j − m − k , (7)where r a , often referred to as purity or degree of mixed-ness, is the length of the Bloch vector ~r a of the singlequbit state ρ a , and θ is the relative angle between ~r and ~r .As argued above, P PPT e provides a lower bound to theerror probability attainable by the most general LOCCstrategy, which includes weak generalized local measure-ments interlaced with an unlimited number of classicalcommunication rounds. In what follows we will com-pare this bound to the error probability of the optimalcollective strategy P col e and to that of two LOCC strate-gies that use rank one projective measurements on theindividual copies. More precisely: i) Repeated strategy,where the very same two-outcome measurement is per-formed on every copy. The error probability, P rep e , isobtained by minimizing over the azimuthal angle Θ thatspecifies the unit Bloch vector of the two measurementprojectors. The corresponding asymptotic error rate canalso be obtained from the classical Chernoff bound [6].ii) Adaptive strategy, where copies are measured sequen-tially and the choice of the azimuthal angle Θ s , corre-sponding to the measurement on the s -th copy, dependson the outcomes obtained upon measuring the preced-ing s − N , is knownbeforehand it is possible to find the optimal adaptivestrategy very efficiently using DP [9], as detailed in thesupplementary material.Figure 1 shows the error probability of discrimina-tion between two states with equal purity ( r = r = . θ = π/ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - P e N N ∆ FIG. 1: (color online) LOCC lower-bound P PPT e (solidline) and error probability for collective (squares), adaptive(crosses) and repeated (dashed) strategies for r = r = . θ = π/
2. Inset: Gap [Eq. (8)] vs. N for θ = π/ r = . r = 1 (solid), r = . r = . r = . between the collective strategy and the LOCC lower-bound ( P PPT e ). In addition we note that the error proba-bility for repeated and adaptive strategies fall almost ontop of the LOCC lower-bound. This shows that P PPT e isa very tight bound and that it can be taken as a goodestimate of the minimal LOCC error probability for mostpractical purposes. The figure clearly shows that, as ex-pected, the error probability falls exponentially with N : P e ∼ e − CN . By fitting the data in the figure to anerror rate of the form C = C + C log N/N + C /N for 25 ≤ N ≤
35, we can obtain its asymptotic value C for the various strategies. For the collective and repeatedstrategies the results agree up to the third significantdigit with the analytical results provided by the quan-tum and classical Chernoff bounds respectively [6]. Moreinterestingly, within numerical accuracy the fits indicatethat the gap between collective and LOCC error ratespersists in the asymptotic limit. The results are also con-sistent with a convergence of the asymptotic error ratesfor the two LOCC strategies and that of the LOCC lower-bound ( P PPT e ), although here, due to the already smalldifferences, it is harder to exclude the existence of a (tiny)non-vanishing gap.In the inset of Figure 1 we plot the gap betweenthe collective error rate and that of the LOCC lower-bound ( P PPT e ), i. e.:∆ = C col − C PPT = − N log P col e P PPT e . (8)We notice that the gap reaches its asymptotic value al-ready for a small number of copies. This is so for allvalues of r but for r = 1 (solid line), for which ∆, aftergrowing to a maximun at N ≈
4, decreases to zero asit should, according to Ref. [6]. There, as mentioned atthe beginning of this letter, it is shown that when oneof the states, say ρ , is pure the collective error rate isasymptotically attainable by a repeated strategy. Moreprecisely, by one consisting in performing the measure-ment defined by E = ρ on each copy. The unknownstate is claimed to be ρ if the N outcomes of the mea-surements correspond to E (none to − E ), and it isclaimed to be ρ otherwise ( unanimity vote ). The asymp-totic error rate of this strategy attains the upper-bound C ≤ − log F ( ρ , ρ ), where F ( ρ , ρ ) is the fidelity, de-fined as F ( ρ , ρ ) = (tr |√ ρ √ ρ | ) [6]. For the collectivestrategy it also holds that − (1 /
2) log F ( ρ , ρ ) ≤ C col0 .Figure 2 shows the error rate C = − (1 /N ) log P e ,for two equally mixed states, θ = π/ N = 25,as one varies their degree of mixedness r . We identifyfour parameter regions in this plot: i) For very mixedstates ( r . .
5) collective and repeated local strategieshave essentially the same performance. ii) As the pu-rity increases ( . . r . . P PPT e (upper-bound on the error rate). The measurements of this re-peated strategy have Θ = π/
2, which means that theirBloch vector is proportional to ~r − ~r [14]. The decisionis taken by “ majority vote ”, i.e., the most frequent out-come determines the decision. iii) At very high purities( . . r ≤ r ∗ ) the three LOCC curves start to split. iv) Atpurities larger than a critical one, r ∗ , all LOCC curvesstart to rapidly converge to the collective one. At r = r ∗ the measurement angle of the repeated strategy startsto change from Θ = π/ π/ r → θ , one has Θ → θ/
2, i.e., the Bloch vec-tor of the measurements goes to either ~r or ~r ) andthe decision rule gradually shifts from a majority vote tothe unanimity vote described above. In the asymptoticlimit N → ∞ , r ∗ can be computed to arbitrary accuracyas a solution of a transcendental equation; for θ = π/ r ∗ ≈ . r < r ∗ the error rate of the re-peated strategy, C rep0 , saturates the fidelity lower-boundintroduced above: C rep0 = − (1 /
2) log F ( ρ , ρ ).As mentioned in the introductory part, for asymp-totically large N repeated and one-way adaptive errorrates coincide [8]. The tiny gap between the correspond-ing (dotted) curve and the crosses in Fig. 2 is explainedby the relatively small number of copies ( N = 25) usedin the plot. It is plausible, and consistent with our data,that also the gap between the LOCC error rate bound(dashed line) and C rep0 vanishes as N → ∞ .Note that the smaller the error rate, the more copieswe need to achieve the same error probability. The ra-tio f = C col /C PPT tells us that we need f N copies in or-der for the best LOCC strategy to discriminate with theaccuracy of the collective one. The general features of f as a function of r can be immediately grasped from Fig. 2.For very mixed states LOCC and collective strategies re-quire a similar number of copies ( f ≈
1) to discriminatewith the same error probability, but as the states becomemore pure the LOCC strategies demand up to twice thisnumber (i. e., f .
2, which asymptotically is the ratio of the two fidelity bounds). In the limit of very pure states( r & r ∗ ) the factor f drops back down again to one. ‰ ‰ ‰ ‰ ‰‰‰‰ C r ∗ r FIG. 2: (b) Error rate vs. r = r = r for N = 25 and θ = π/ In summary, we have lower-bounded the error proba-bility of LOCC discrimination between two qubit mixedstates. To do so, we have characterized all permutation-and PT-invariant operators, which may find applica-tion in other quantum communication problems involv-ing LOCC. Our results indicate an error rate gap betweenthe best LOCC and collective discrimination protocolsthat persists as the number of copies goes to infinity. Thisgap takes its largest value in the region of nearly pure,but strictly mixed, states. Excluding this region, thereare no significant differences in performance between thesimplest (repeated) and optimal LOCC strategies.We thank E. Ronco and G. Via for their contributionsin the earlier stages of this work. We acknowledge finan-cial support from: the Spanish MICINN, through theRam´on y Cajal program (JC), contract FIS2008-01236,and project QOIT (CONSOLIDER2006-00019); from theGeneralitat de Catalunya CIRIT, contract 2009SGR-0985; and from Alianza 4 Universidades program (JIdV). [1] Two introductory reviews are J. A. Bergou et al. , Lect.Notes Phys. , 417 (2004) and A. Chefles, Contemp.Phys. , 401 (2000).[2] A. Acin et al. , Phys. Rev. A , 012327 (2006); see alsothe review N. Gisin et al. , Rev. Mod. Phys , 145 (2002).[3] D. Bacon et al. , Proc. 46th IEEE FOCS 2005, pp. 469-478. E-print quant-ph/0504083.[4] C.W. Helstrom, Quantum Detection and EstimationTheory . (Academic Press, New York, 1976); A. S. Holevo,Theor. Prob. Appl.
411 (1979).[5] K. Audenaert et al. , Phys. Rev. Lett. , 160501 (2007);M. Nussbaum and A. Szkola, Ann. Statist. et al. , Phys. Rev. A , 032311 (2008).[7] D. Brody and B. Meister, Phys. Rev. Lett. , 1 (1996);A. Acin et al. , Phys. Rev. A , 032338 (2005); S. Vir-mani, et al. , Phys. Lett. A , 62 (2001).[8] M. Hayashi, IEEE Trans. Inf. Theory , 3807 (2009). [9] B.L. Higgins et al. , Phys. Rev. Lett. , 220503 (2009).[10] G. L. Nemhauser, Introduction to Dynamic Program-ming (John Wiley and Sons, New York, 1966).[11] M. Hayashi et al. , Phys. Rev. Lett. 96, 040501 (2006).[12] L. Vandenberghe and S. Boyd. SIAM Rev. , 49 (1996);We used the matlab toolbox: J. L¨ofberg, In Proc. of theCACSD Conference, Taipei, Taiwan, 2004.[13] E. Rains, IEEE Trans. Inf. Theor. , 2921 (2001)[14] This holds for N even, but it is only an approximationfor N odd, becoming exact as N → ∞ . SUPPLEMENTARY MATERIAL
Section I of this supplementary material contains de-tails on the derivation of the block-diagonal form of per-mutation and PT invariant operators on ( C ) ⊗ N . Sec-tion II provides an example of dynamic programmingapplied to state discrimination with one-way LOCC mea-suremens I. Permutation and PT invariant operators inblock-diagonal form
Let us consider the permutation invariant operator E = X { i p } X { i ′ s } E i ′ i ′ ...i ′ N i i ...i N | i i . . . i N ih i ′ i ′ . . . i ′ N | , (9)where {| i i . . . i N i} is the (qubit) computational basisand each sum runs over the 2 N possible sequences of N bits. Permutation invariance implies that E can be writ-ten in block-diagonal form (the blocks corresponding toirreducible tensors) by an appropriate change of basis.This is more familiarly done in the context of spin and/orangular momentum, so we map each qubit to a spin 1 / | i p i → | m p i = | ( − i p / i , (10)where m p is the magnetic number of the p -th spin.In this, more physical, picture block-diagonalization isachieved by the total spin ( j ) eigenbasis {| j, m, α i} jm = − j .The index α labels the bases that span the various equiv-alent irreducible representations (irreps) of SU (2), eachcorresponding to a particular way of coupling the indi-vidual spins to give the same total spin j . Permuta-tion invariance also implies that the projections of E onthese irreducible subspaces, span( {| j, m, α i} jm = − j ), areidentical for all values of α (they depend only on j ).Hence, from hereafter, we drop this index and, e. g.,write [ E ( j ) ] m ′ m = h j, m | E | j, m ′ i for the matrix elementsof the blocks. It follows that,[ E ( j ) ] m ′ m = X { m p } X { m ′ s } h j, m | m m · · · m N i× h m ′ m ′ · · · m ′ N | j, m ′ i E i ′ i ′ ··· i ′ N i i ··· i N , (11) where we implicitly use (10) and the sum runs over allpossible configurations of the magnetic numbers suchthat N X p =1 m p = m ; N X s =1 m ′ s = m ′ . (12)Since all the matrix blocks with the same j are identi-cal, to compute the values of the (generalized) Clebsch-Gordan coefficients h m m . . . m N | j, m i in (11) it sufficesto couple the individual spins in the way that leads to thesimplest calculation. For a given j , it proved useful topair in singlets the N − j spins/qubits that are at thebeginning of the sequence and couple the remaining 2 j to give total spin j . Thus, we take the state | j, m i to beof the form | j, m i = | ψ − i ⊗ ( N/ − j ) | j, m i sym , (13)where | ψ − i is the spin singlet [ | ψ − i = ( | i − | i ) / √ | j, m i sym is the fully symmetricstate of 2 j spins with total spin j . With this choice,the only non-vanishing Clebsch-Gordan coefficients arethose for which the first N − j spins in h m m . . . m N | consist of l pairs of the form h − | ≡ h σ | and p pairsof the form h−
12 12 | ≡ h µ | (01 ≡ σ and 10 ≡ µ in bi-nary notation) with l + p = N/ − j . In this case, andfurther assuming that conditions (12) hold, a straightfor-ward calculation yields h m m . . . m N | j, m i = ( − p (cid:18) √ (cid:19) N − j (cid:18) jj + m (cid:19) − , (14)where we have used that the fully symmetricstate | j, m i sym is an equal superposition of (cid:0) jj + m (cid:1) vectorsof the form ⊗ Np = N − j | m p i .We can use again permutation invariance to cast theindependent components of E into the form E l z}|{ σ...σ p z}|{ µ...µ Q z}|{ ... ... Q ′ z}|{ ... ... σ...σ σ...σ | {z } l + p ... ... | {z } R ... ... ≡ e E l,p ; Q,Q ′ R , (15)where l , p , R , Q and Q ′ stand for the number of σ -pairs, µ -pairs and ones indicated by the braces. For a fixedchoice of l and p , the unpaired bits are arranged so thatin the subscript sequence (the row index of E ) no zeroprecedes a one. The first R bits on the right of the µ -pairsin the superscript sequence (column index) are arrangedalong the same pattern, and so are the bits placed on topof the N − l − p − R zeroes of the subscript sequence.Note that Q ≤ R and e E l,p ; Q,Q ′ R = e E , Q + l,Q ′ + pR + l + p ≡ e E Q + l,Q ′ + pR + l + p . (16)It is a straightforward exercise in combinatorics to com-pute the number of matrix elements of E whose valueis e E l,p ; Q,Q ′ R . It is given by2 l + p (cid:18) l + pl (cid:19)(cid:18) N − l − pR (cid:19)(cid:18) RQ (cid:19)(cid:18) N − l − p − RQ ′ (cid:19) . (17)The first factor takes into account that by applying apermutation (thus leaving E unaltered) we can replaceany of the l + p σ -pairs in the subscript by a µ -pair (withan exchange µ ↔ σ at the exact same position of thesuperscript sequence). The second factor is the numberof ways the l σ -pairs (and the p µ -pairs) can be placed inthe first l + p positions of the superscript sequence. Thethird factor gives the different ways the R ones in thesubscript sequence can be placed in N − l − p positions(with a corresponding repositioning of the bits right ontop of them). Similarly, the forth (fifth) factor is thenumber of ways Q ( Q ′ ) ones can be placed on top ofthe R ones ( N − l − p − R zeroes) of the subscriptsequence.With the Clebsch-Gordan coefficients (14), the nota-tion introduced in (15), and the combinatorics in (17),we can give an explicit expression for the matrix-blocks E ( j ) as[ E ( j ) ] m ′ m = 2 j − N s(cid:18) jj + m (cid:19) (cid:18) jj + m ′ (cid:19) × N − j X l =0 j − m X k =0 ( − N − j − l e E l, N − j − l ; k,j − m ′ − kj − m × N − j (cid:18) N − jl (cid:19)(cid:18) jj − m (cid:19)(cid:18) j − mk (cid:19)(cid:18) j + mj − m ′ − k (cid:19) , (18)where we have used that p = N/ − j − l , R = j − m , Q ′ = j − m ′ − Q , and finally replaced the dummy index Q by k . Using Eq. (16) and making the changes k → j − m − k and l → N/ − j − l , this last expression reduces to[ E ( j ) ] m ′ m = X k [∆ ( j ) k ] m ′ m X l (cid:18) N − jl (cid:19) × ( − l e E N − m − k − l,m − m ′ + k + lN − m , (19)where we have defined (cid:2) ∆ ( j ) k (cid:3) m ′ m = p ( j − m )!( j + m )!( j − m ′ )!( j + m ′ )!( j − m − k )!( j + m ′ − k )!( m − m ′ + k )! k ! , (20)and the sums run over all integer values for which thefactorials make sense.Partial transposition invariance implies that e E Q,Q ′ R = e E ,Q ′ + RQ ≡ E Q ′ + RQ , (21)and we can further simplify the expression of the matrix blocks E ( j ) ,[ E ( j ) ] m ′ m = X k [∆ ( j ) k ] m ′ m X l (cid:18) N − jl (cid:19) × ( − l E N − m ′ + k + lN − m − k − l , (22)from which Eq. (4) in the main text follows immediately.Eqs. (19) provides a useful parameterization of gen-eral permutation invariant operators. As an example, wecompute the block decomposition of the product state σ = ρ ⊗ N , which is also used in the main text [Eq. (7)].The independent components of σ are easily found us-ing (15) [here we do not use superscripts; we write ρ ij ( i, j = 0 ,
1) instead of ρ ji for the matrix elements of ρ ] e E Q,Q ′ R = ρ Q ρ R − Q ρ Q ′ ρ N − R − Q ′ . (23)Substituting in Eq. (19) we obtain[ σ ( j ) ] m ′ m = X k [∆ ( j ) k ] m ′ m X l (cid:18) N − jl (cid:19) × ( − l ρ k ρ m − m ′ + k ρ j − m − k ρ j + m ′ − k × ( ρ ρ ) l ( ρ ρ ) N − j − l . (24)Note that the sum over l can be readily performed to give( ρ ρ − ρ ρ ) N/ − j = (det ρ ) N/ − j , and[ σ ( j ) ] m ′ m = (det ρ ) N/ − j X k [∆ ( j ) k ] m ′ m × ρ k ρ m − m ′ + k ρ j − m − k ρ j + m ′ − k (25)Eq. (7) in the main text follows from the Bloch parame-terization of a rebit used there ρ = 1 + r cos( θ/ , ρ = 1 − r cos( θ/ ,ρ = ρ = r sin( θ/ . (26)Finally, Eq. (19) can also be used to provide an alter-native derivation of the well-known Wigner d-matrices.Consider the block decomposition of the product repre-sentation d ⊗ N of the SU (2) “rotation” about the y -axis: d = cos β , d = cos β , d = − d = − sin β . (27)Proceeding as above [Eqs. (23) to (25)] one obtains[ d ( j ) ( β )] m ′ m = X k [∆ ( j ) k ] m ′ m ( − m − m ′ + k × (cid:16) cos β (cid:17) j − m + m ′ − k (cid:16) sin β (cid:17) m − m ′ +2 k , (28)which are the standard Wigner d-matrices. II. One-way LOCC State discrimination usingdynamic programming