Requirement of Dissonance in Assisted Optimal State Discrimination

OPEN Requirement of Dissonance in Assisted

SUBJECT AREAS: Optimal State Discrimination QUANTUM Fu-Lin Zhang1, Jing-Ling Chen2,3, L. C. Kwek3,4,5 & Vlatko Vedral3,6,7 INFORMATION THEORETICAL PHYSICS 1Physics Department, School of Science, Tianjin University, Tianjin 300072, China, 2Theoretical Physics Division, Chern Institute of QUANTUM MECHANICS Mathematics, Nankai University, Tianjin, 300071, China, 3Centre for Quantum Technologies, National University of Singapore, 3 QUBITS Science Drive 2, Singapore 117543, 4National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, 5Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, Singapore 639673, 6Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom, 7Department of Physics, National Received University of Singapore, 2 Science Drive 3, Singapore 117542. 3 January 2013 Accepted A fundamental problem in quantum information is to explore what kind of quantum correlations is 13 June 2013 responsible for successful completion of a quantum information procedure. Here we study the roles of entanglement, discord, and dissonance needed for optimal quantum state discrimination when the latter is Published assisted with an auxiliary system. In such process, we present a more general joint unitary transformation 4 July 2013 than the existing results. The quantum entanglement between a principal qubit and an ancilla is found to be completely unnecessary, as it can be set to zero in the arbitrary case by adjusting the parameters in the general unitary without affecting the success probability. This result also shows that it is quantum dissonance that plays as a key role in assisted optimal state discrimination and not quantum entanglement. Correspondence and A necessary criterion for the necessity of quantum dissonance based on the linear entropy is also presented. requests for materials PACS numbers: 03.65.Ta, 03.67.Mn, 42.50.Dv. should be addressed to F.-L.Z. (flzhang@tju. edu.cn) or J.-L.C. n important distinctive feature of quantum mechanics is that quantum coherent superposition can lead to quantum correlations in composite quantum systems like quantum entanglement1, Bell nonlocality2 and ([email protected]) A quantum discord3,4. Quantum entanglement has been extensively studied from various perspectives, and it has served as a useful resource for demonstrating the superiority of quantum information processing. For instance, entangled quantum states are regarded as key resources for some quantum information tasks, such as teleportation, superdense coding and quantum cryptography5. In contrast to quantum entanglement, quantum discord measures the amount of nonclassical correlations between two subsystems of a bipartite quantum system. A recent report regarding the deterministic quantum computation with one qubit (DQC1)6,7 demonstrates that a quantum algorithm to determine the trace of a unitary matrix can surpass the performance of the corresponding classical algorithm in terms of computational speedup even in the absence of quantum entanglement between the the control qubit and a completely mixed state. However, the quantum discord is never zero. This result is somewhat surprising and it has engendered much interest in quantum discord in recent years. In particular, it has led to further studies on the relation of quantum discord with other measures of correlations. Moreover, it has been shown that it is possible to formulate an operational interpretation in the context of a quantum state merging protocol8,9 where it can be regarded as the amount of entanglement generated in an activation protocol10 or in a measurement process11. Also, a unified view of quantum correlations based on the relative entropy12 introduces a new measure called quantum dissonance which can be regarded as the nonclassical correlations in which quantum entanglement has been totally excluded. For a separable state (with zero entanglement), its quantum dissonance is exactly equal to its discord. It is always interesting to uncover non-trivial roles of nonclassical correlations in quantum information processing. The quantum algorithm in DQC1 has been widely regarded as the first example for which quantum discord, rather than quantum entanglement, plays a key role in the computational process. Moreover, a careful consideration of the natural bipartite split between the control qubit and the input state reveals that the quantum discord is nothing but the quantum dissonance of the system. This simple observation naturally leads to an interesting question: Can quantum dissonance serve as a similar key resource in some quantum information tasks? The affirmative answer was shown in an interesting piece of work by Roa, Retamal and Alid-Vaccarezza13 where the roles of entanglement, discord, and dissonance needed for performing unambiguous quantum state discrimination assisted by an auxiliary qubit14,15 was studied. This protocol for assisted optimal state discrimina- tion (AOSD) in general requires both quantum entanglement and discord. However, for the case in which there

SCIENTIFIC REPORTS | 3 : 2134 | DOI: 10.1038/srep02134 1 www.nature.com/scientificreports exist equal a priori probabilities, the entanglement of the state of being two arbitrary states orthogonal to the right hands of Eq. (1) and system-ancilla qubits is absent even though its discord is nonzero, U{ Uz ~ ~ hij 0, and a kkj a 0. Obviously, only the terms with and hence the unambiguous state discrimination protocol is imple- hy~ j hjk have effect on the initial state jy æjkæ . mented successfully only with quantum dissonance. This protocol + a 6 a The state of the system-ancilla qubits is given by therefore provides an example for which dissonance, and not entan- ÀÁ glement, plays as a key role in a quantum information processing ~ 6 { rSA pzU yz yz jik ahjk U task. ÀÁð2Þ z 6 { In this work, we show more generally that quantum entanglement p{U jiy{ hjy{ jik ahjk U , is not even necessary for AOSD. Moreover, we look at the roles of correlations in the AOSD under the most general settings by con- which depends on b and d, and it is generally not equivalent to the sidering a generic AOSD protocol. We also show that only disson- corresponding one in13 under local unitary transformations unless ance in general is required for AOSD and quantum entanglement is jWæ 5 j1æ. The state discrimination is successful if the ancilla col- never needed. lapses to j0æa. This occurs with success probability given by ÂÃÀÁ ~ 6 Psuc Tr s ji0 ahj0 rSA Results ÀÁÀÁ ð3Þ ~ { 2 z { 2 The general AOSD protocol. Suppose Alice and Bob share an pz 1 jjaz p{ 1 jja{ , pffiffiffiffiffiffi pffiffiffiffiffiffi entangled two-qubit state jif ~ pz y ji0 z p{jiy ji1 (see z c { c where is the unit matrix for the system qubit. Without loss of Fig. 1), where p g [0, 1] and p 1 p 5 1, jy æ are two s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 1 2 6 ~ nonorthogonal states of the qubit of Alice (system qubit S), and generality, let us assume that p1 # p2 and denote a pz=p{. The analysis of the optimal success probability can be divided into {j0æc, j1æc} are the orthonormal bases for the one of Bob (qubit C). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi v ~ 4 The reduced state of system qubit r 5 p1jy1æÆy1j 1 p2jy2æÆy2j is two cases: (i) jja a, Psuc is attained for jjaz p{=pz jja ; (ii) 13 a realization of the model in in which a qubit is prepared in the two aƒjja ƒ1, Psuc is attained for ja1j 5 1 (or equivalently ja2j 5 jaj). nonorthogonal states jy6æ with a priori probabilities p6.To One has discriminate the two states jy æ or jy æ unambiguously, the ffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ~ { p system is coupled to an auxiliary qubit A, prepared in a known Psuc,max 1 2 pzp{jja , for case(i) , ð4aÞ initial pure state jkæ . Under a joint unitary transformation U ÀÁ a ~ { 2 between the system and the ancilla, one obtains Psuc,max 1 jja p{, for case(ii): ð4bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ { 2 z Before proceeding further to explore the roles of correlations in the U yz jik a 1 jjaz ji0 ji0 a azjiW ji1 a, ð1aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AOSD, we make the following remarks. ~ { 2 z U jiy{ jik a 1 jja{ ji1 ji0 a a{jiW ji1 a, ð1bÞ Remark 1. State discrimination of a subsystem in a reduced mixed state has practical interest in conclusive quantum teleportation id where jWæ 5 cos bj0æ 1 sin be j1æ,{j0æ, j1æ} and {j0æa, j1æa} are the where the resource is not prepared in a maximally entangled state bases for the system and the ancilla, respectively. The probability (see Refs. 16–18). In the conclusive teleportation protocol, the sender ~ amplitudes a1 and a2 satisfy aza{ a, where a 5 Æy1jy2æ 5 Alice possesses an arbitrary one-qubit state jQæAlice 5 aj0æ 1 bj1æ, and ih jaje is the priori overlap between the two nonorthogonal states. she shares a non-maximally entangled state jY1(h)æ 5 cos hj00æ 1 The unitary transformation can be constructedÀÁ by performing an sin hj11æ with the receiver Bob. Under the protocol, one has operation W~zðÞjihj0 z{jihj1 6ji0 hj0 z jiW hj0 z W hj1 6 a pffiffiffi ~ jiYtel ~jiQ 6jiYzðÞh ji1 ahj 1 on the original one in Ref. 13, where ji+ ðÞji0 +ji1 2 Alice and W ~sin bji0 {cos beidji1 . It has the form as 1 ~ fjiYzðÞh 6jiQ zjiY{ðÞh 6szjiQ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Bob Bob ~ 1 { 2 z ÀÁ U 2 1 jjaz ji0 ji0 a azjiW ji1 a 6 z 6 { 1{jja jiWzðÞh sxjiQ Bob jiW{ðÞh isy jiQ Bobg, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ~ z { 2 z ~ z hyzjahjk 1 jja{ ji1 ji0 a a{jiW ji1 a hy{jahjk V, where jY6(h)æ 5 cos hj00æ 6 sin hj11æ, jW6(h)æ 5 sin hj01æ 6 cos 19 hj10æ, and sx, sy, sz are Pauli matrices. The concurrences of the ~ ~ { where y+i y+ y+ y+ y+ are the components of jy6æ states jY6(h)æ and jW6(h)æ are all equal to C~jjsin 2h . The states ~ z orthogonal to y+ , and V jiUz hj0 a k jiU{ hj1 a k , with jiU+ jY6(h)æ are orthogonal to the states jW6(h)æ, but {jY1(h)æ, jY2(h)æ} (or {jW1(h)æ, jW2(h)æ}) are not mutually orthogonal. To teleport the unknown state jQæAlice from Alice to Bob with perfect fidelity (equals to 1), state discrimination16–18 is generally required. It should also be noted that only the maximally entangled states (with h 5 p/4) can realize the perfect teleportation with unit success probability.

Remark 2. Through quantum teleportation, we see that our model recover the scheme in13, in which the principal qubit is randomly prepared in one of the two pure states jy1æ or jy2æ. Let us conisder replacing the entangled resource jY1(h)æ by maximally entangledÀÁ p Figure 1 | The General AOSD Protocol Illustration. Alice and Bob share a states randomlyÀÁ prepared ÀÁ with a probabilitiesÀÁ as {p1 : Yz 4 , p p p { z { pure entangled state | fæ of qubits S and C. To discriminate the two p2 : Y 4 , p3 : W 4 , p4 : W 4 }. Although they are all states | y1æ or | y2æ of S, Alice performs a joint unitary transformation U maximally entangled states and each of them is a resource for perfect between qubits S and A, followed by two independent von Neumann teleportation, perfectly faithful teleportation cannot be realized in measurements on the two qubits. Her state discrimination is successful if this case. It can be shown that the fidelity of teleportation is the the outcome of A is 0, but unsuccessful if outcome 1. one corresponding to the average state16

SCIENTIFIC REPORTS | 3 : 2134 | DOI: 10.1038/srep02134 2 www.nature.com/scientificreports

    p p p p ~ z rres p1jYz ihYz j p2jY{ ihY{ j 4 4 4 4 p p p p zp jWz ihWz jzp jW{ ihW{ j: 3 4 4 4 4 4 Consequently, the amount of entanglement contributing to tele- portationÀÁ ÀÁ is not just theÀÁ averageÀÁ valueÀÁ of theÀÁ entanglement ÀÁ whichÀÁ is p z p z p z p p1C Yz 4 p2C Y{ 4 p3C Wz 4 p4C W{ 4 , but the entanglement of the average state as CðÞrres . Therefore the amount of entanglement available depends crucially on the know- ledge of the entangled state. The amount of quantum entanglement that is needed for the AOSD scheme considered here, as well as the one in Ref. 13, refers to the entanglement of the average state, CðÞrSAÀÁ, and not to theÀÁ average value of the entanglement as Figure 2 | Quantum dissonance in the AOSD. We plot the dissonance z y z { y p CU z jik a p CUji{ jik a . versus | a | , for p1 5 1/2 (solid line), 1/4 (dashed line), and 1/8 (dot-dashed We are now ready to investigate the roles of correlations in the line). Dissonance is greater than zero for case (i), and is zero for case (ii). ~ AOSD. To this end, let us first calculate the concurrence of rSA: The critical point for D(rSA) 5 0 occurs at jja a. ~ 2 2 z 2 2 { CðÞrSA 2 Yz sin b Y{ cos b ð5Þ ÈÀÁ ÀÁ 1=2 ~ 2z 2 { 2 2 z 2YzY{ sin b cos b cosðÞhzd , tS 4 p{jja{ pzjjaz p{ 1 jja{ cos b ÀÁà ÀÁÀÁÉð12Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 pz 1{jjaz sin b zpzp{ 1{jjaz 1{jja{ , 2 with Y+~ 1{jja+ ja+jp+. When b 5 p/4 and d 5 0, Eq. (5) 13 and the three-tangle is reverts to the result in . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let us impose the constraint CðÞr ~0 for any a, a and p .Itis 2 SA 1 1 ~ { 2 z { 2 id then easy to see that tðÞjiY 4pzp{ 1 jja{ az cos b 1 jjaz a{ sin be :ð13Þ d~{h, b~arctanðÞY{=Yz : ð6Þ

Based on Eq. (6), state (2) is a separable state as Dissonancepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for casesp ffiffiffiffiffi (i) and (ii). For case (i), upon the substitution az ~ 4 {= z a 5 2 r ~jig hjg 6ji0 hj0 zjiW hjW 6jig hjg , ð7Þ jj p p jj, p2 1 p1, and Eqs. (6)(11)(12)(13) into SA 1 1 a 2 a 2 Eq. (10), one has the analytical expression for the dissonance, which depends only on jaj and p . In Fig. 2, we plot the curves of the where jg1æ andffiffiffiffiffiffiffiffiffiffiffiffijg2æa qareffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two unnormalizedqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi states as 1 p dissonance versus jaj for p1 5 1/2, 1/4, 1/8, respectively (see the pzp{ 2 2 jig ~ 1{jjaz a{ji0 { 1{jja{ azji1 , curves with D(r ) . 0). For case (ii), because ja j 5 1, one has b 5 1 Z SA 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p/2 and the state rSA is ð8Þ ~ 6 2 z 2 rSA ji1 hj1 ra, ð14Þ Yz Y{ az ~ z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffi jig2 a ji0 a ji1 a: Z jjaz 2 id r 5 1 m m m ~ { a za{ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a p1j1æaÆ1j p2j æaÆ j, jia 1 jjji0 a e ji1 a. 2 2 The state (14) is clearly a direct-product state hence its dissonance is where Z~ pzjjaz zp{jja{ . zero. In Fig. 2, for case (ii), we also plot the curves of dissonance Note that the state (2) has rank two, and it is really the reduced versus jaj for the same p1’s (see the curves with D(rSA) 5 0). Fig. 2 state of the followingffiffiffiffiffiffiÀÁ tripartite pure stateffiffiffiffiffiffiÀÁ shows that dissonance is a key ingredient for AOSD other than ~p zp jiY pz U yz jik a ji0 c p{ Ujiy{ jik a ji1 c: ð9Þ

Its discord can be derived analytically as D(rSA) 5 S(rA) 2 S(rSA) 1 E(rSC) 5 S(rA) 2 S(rC) 1 E (rSC) using the Koashi-Winter iden- 20 tity , where S(r) is the von Neumann entropy, E(rSC) is the entan- glement of formation19 between the principal system and the qubit C. The explicit expression for the discord is ~ { z DðÞrSA HðÞtA HðÞtC HðÞtSC , ð10Þ where pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1z 1{x 1z 1{x 1{ 1{x 1{ 1{x HðÞx ~{ ln { ln , t 2 2 2 2 A is the tangle between A and SC, tC is the tangle between C and SA, and ~ 2 tSC C ðÞrSC is the concurrence between S and C in the state rSC. One can obtains ÀÁ 2 2 2 tA~tSAz4pzp{ jjaz zjja{ {2jja , ð11aÞ ÀÁ 2 tC~4pzp{ 1{jja , ð11bÞ Figure 3 | Geometric picture for optimal success probability based on pffiffiffiffiffiffi pffiffiffiffiffiffi POVM strategy. The sides jj~ z, jj~ {, the angle c 5 arccos tSC~tS{tSA{tðÞjiY , ð11cÞ OA p OB p | a | , and AC H OB, BD H OA, EB H OB, EA H OA. For jja va, the point E ~ 2 % a §a with tS the tangle between S and AC, tSA C ðÞrSA , and t(jYæ) the locates inside of the angle AOB; for jj , the point E coincides with the 21 three-tangle . The tangle between S and AC is given by point B for p1 , p2 (or A for p1 . p2).

SCIENTIFIC REPORTS | 3 : 2134 | DOI: 10.1038/srep02134 3 www.nature.com/scientificreports

ƒ! y~ BD correspondpffiffiffiffiffiffi to the unnormalized states +i with the coefficients { p+. The third POVM element giving the inconclusive result is P0~ s{Pz{P{. The elements P6,0 are required to be positive - this is a constraint on the POVM strategy. Finally, the probability of successful discrimination is PPOVM 5 2 (r1p1 1 r2p2)(1 2 jaj ), which is ƒ ƒ ƒ ƒ ! ! : ! ! PPOVM~ OA{OB r{ BD{rz AC : ð15Þ

Figure 4 | Realization of the General Unitary Transformation. For the When jja va, the optimal P is attained at r ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi POVM ƒ ƒ +ƒ 2 2 ! ! ! initial states | y1æ 5 | 0æ, jiy{ ~ 1{jja ji1 zaji0 and | kæa 5 | 0æa, the 1{cos c p+=p+ sin c. The vectors r{ BD ~ EA and rz AC ƒ! unitary transformation U in Eq. (1) can be realized in four steps: (i) ~ EB , where E is the intersection point of AE and BE (see Fig. 3). controlled-UA with the system S being the control qubit; (ii) controlled-US The maximum value offfiffiffiffiffiffiffiffiffiffiffiffiPPOVM is the square of jABj, nanmely where the system S is controlled by the ancilla A; (iii) local unitary VA on 2 p PPOVM~jjAB ~1{2 pzp{jja , which recovers Eq. (4a). When the auxiliary qubit; (iv) controlled-UW with the same control qubit and ~ ~ jja a, the point E coincides with B for p1 , p2 (or A for p1 . target as the second step. The single qubit operations U A jiwA a z w ~ w z w ~ w z w p2), for the optimal PPOVM one has r2 5 1 (or r1 5 1) and PPOVM 5 hj0 A ahj1 , US jiS hj0 S hj1,qVffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA jiV ahj0 V ahj1 , and 2 ÀÁÀÁ p2(1 2 jaj ). For jja ~a and p1 , p2, E lies outside of the angle ~ z ~ { 2 { 2 z ƒ ƒ UW jiW hj0 W hj1 , with jiwA a 1 jjaz 1 jja{ ji0 a ! ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi %AOB and EB is opposite to AC . Consequently, we do not get a 2z 2{ 2 { 2 ~ { 2 z physically realizable value of r1. The optimal PPOVM strategy then jjaz jja{ 2jja ji1 aÞ 1 jja , jiwS 1 jja{ azji0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi occurs at r2 5 1 and r1 5 0 (i.e., E coincides with B), one has ƒ! ƒ! ÀÁ 2 2 2 2 ~{ : ~ { 2 1{jjaz a{jiÞ1 jjaz zjja{ {2jja , and jiw ~ PPOVM OB BD p{ 1 jja , which is Eq. (4b). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V a 2 1{jjaz ji0 zazji1 . Here, the states with a bar, w , w , and a a A a S a Discussion w , denote (is | w æ )*,(2is | w æ)*, and (is | w æ )*. V a y,a A a y S y,a V a In summary, based on a sufficiently general AOSD protocol, we found that the entanglement between the principal qubit and the entanglement for case (i), and that the classical state can accomplish ancilla is completely unnecessary. Moreover, this quantum entangle- the task of AOSD for case (ii). ment can be arbitrarily zero by adjusting the parameters in the joint unitary transformation without affecting the success probability. Geometric picture. It can be observed that the optimal success Theoretically, this fact clearly indicates that dissonance plays a key probability Psuc,max in Eq. (4) can be analyzed in two different role in assisted optimal state discrimination other than entangle- regions: jja va and jja §a. Here based on the positive-operator- 15 ment. Experimentally, the absence of entanglement can be more valued measure (POVM) strategy , we provide a geometric picture easily observed because there is no restriction on the a priori pro- of Psuc,max. Since the success probability, the concurrence and the babilities. In Fig. 4, we present a realization of the unitary trans- discord of state rSA under the constraints in Eq. (6) are all formationqffiffiffiffiffiffiffiffiffiffiffiffiffiffiU in Eq. (1) for the initial states jy1æ 5 j0æ, independent of the phase h of a, one can simply set h 5 0, and 2 2 jiy{ ~ 1{jja ji1 zaji0 and jkæ 5 j0æ by using single-qubit regard the states jy6æ as two unit vectors in R with the angle c 5 a a arccos jaj between them. The square roots of the a priori gates and two-qubit controlled-unitary gates. These gates can be pffiffiffiffiffiffi pffiffiffiffiffiffi 23,24 probabilities, i.e., pz and p{, behave like wave amplitudes, demonstrated experimentally in many systems in recent years. jfæ The success probability of state discrimination is determined by steps and the effects of the coherence canƒ! be seenƒ from! the states and jYæ. In Fig. 3, we plot two vectors OA and OB with %BOA~c to (i) to (iii), which transformqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the system-ancilla state into pffiffiffiffiffiffi pffiffiffiffiffiffi denote pz yz and p{jiy{ ffiffiffiffiffiffi, respectively. The two POVM { 2 z p yz jik a? 1 jjaz ji0 ji0 a azji0 ji1 a, ð16aÞ elements that identify the states p+ y+ can be implementedƒ as ~ ~ ~ ~ ~ ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P+ r+ y+ihy+ =ðhy+ y+iÞ, with r6 $ 0. The vectors AC and { 2 z jiy{ jik a? 1 jja{ ji1 ji0 a a{ji0 ji1 a: ð16bÞ

It is not affected by the controlled-UW in step (iv), which can adjust the correlations in state (2). Let us also reiterate a necessary criterion for the requirement of dissonance in AOSD based on linear entropy. Under the general protocol, Alice and Bob share the entangled state jfæ, encoded in the basis of the polarization of the qubit, Bob can acquire knowledge of the linear entropy SLðÞr of Alice’s qubit. If SLðÞr w1=2, he can be sure that Alice needs dissonance for her AOSD (see Fig. 5). Finally, we would like to mention that local distinguishability of multipar- tite orthogonal quantum states was studied in Ref. 22 where again the local discrimination of entangled states does not require any entanglement.

Figure 5 | Necessary criterion for requiring dissonance in AOSD based 1. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum on linear entropy or purity. ÀÁThe linear entropy reads entanglement. Rev. Mod. Phys. 81, 865–942 (2009). ~ { 2~ { 2 ~ { SLðÞr 2 2Trr 4pzp{ 1 jja , andÀÁ the purity P.ðÞrÀÁ1 SLðÞr . 2. Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics (Long Island City, 2 2 2 2 1 For a given amount of | a | , when SLðÞr ƒ41{jja jja 1zjja , the N.Y.) , 195–200 (1964). 3. Ollivier, H. & Zurek, W. H. Quantum discord: a measure of the quantumness of dissonance for the AOSD is zero (seeffiffiffi thep region below the dashed line). we max ~ ~ w correlations. Phys. Rev. Lett. 88, 017901 (2001). note that SL ðÞr 1=2 when jja 1 3. This means that if SLðÞr 1=2, 4. Henderson, L. & Vedral, V. Classical, quantum and total correlations. J. Phys. A then the dissonance is necessarily needed for the AOSD. 34, 6899–6905 (2001).

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