Semi-device-independent certification of entanglement in superdense coding
George Moreno, Ranieri Nery, Carlos de Gois, Rafael Rabelo, Rafael Chaves
SSemi-device-independent certification of entanglement in superdense coding
George Moreno, Ranieri Nery, Carlos de Gois, Rafael Rabelo, and Rafael Chaves , International Institute of Physics, Federal University of Rio Grande do Norte, - Natal, Brazil Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, CEP - , Campinas, Brazil School of Science and Technology, Federal University of Rio Grande do Norte, - Natal, Brazil (Dated: February , )Superdense coding is a paradigmatic protocol in quantum information science, employing a quan-tum communication channel to send classical information more efficiently. As we show here, it canbe understood as a particular case of a prepare and measure experiment, a scenario that has at-tracted growing attention for its fundamental and practical applications. Formulating superdensecoding as a prepare and measure scenario allows us to provide a semi-device-independent witnessof entanglement that significantly improves over previous tests. Furthermore, we also show howto adapt our results into self-testing of maximally entangled states and also provide a semidefiniteprogram formulation allowing to efficiently optimize, for any shared quantum state, the probabilityof success in the superdense coding protocol. I. INTRODUCTION
Quantum communication [ ] is arguably among thefirst offspring of quantum technologies to break out ofthe lab. Recent milestones, such as quantum teleporta-tion using metropolitan networks [ ] and satellites shar-ing entanglement across continental and intercontinen-tal distances [ , ], are paving the way for the realisticimplementation of many of the quantum communica-tion protocols discovered over the last years. Of par-ticular relevance is the possibility of large scale quan-tum networks, the so-called quantum internet [ , ], notonly allowing for more efficient communication [ , ]but also for fundamental information security [ ].In such applications it is of utmost importance tobe able to certify the non-classicality of the quantumresources, typically the presence of quantum entan-glement [ ] between the communicating parties. Forinstance, entangled states allow for better teleportedstates [ ], improved communication efficiency in thesuperdense coding protocol [ ] and quantum cryptog-raphy [ ]. However, in order to detect any quantumenhancement in these examples, one needs to have fullcontrol over the preparation as well as of the measure-ment apparatuses. In practice, noise is unavoidable,potentially leading to erroneous conclusions [ ] andopening the way to hacker attacks [ ]. To cope withthat, the device-independent (DI) framework has beenestablished [ ]. Based on mild general assumptions, itallows to certify quantumness simply from the obser-vational data, not requiring any detailed knowledge ofthe underlying physical mechanisms at play.The DI framework emerged in the context of Bell’stheorem [ ], finding use in practical applications rang-ing from quantum key distribution [ – ] to commu-nication complexity [ ] and self-testing [ , ]. Inspite of its clear importance, however, the Bell scenarioturns out to be rather restrictive in the context of quan- tum communication, since only pre-established corre-lations but no communication are allowed. More re-cently, device-independent scenarios allowing for com-munication have started to attract growing attention.Of particular relevance is the so-called prepare andmeasure (PAM) scenario, a fairly general structurethat, apart from its foundational relevance [ – ], hasfound applications in quantum networks [ , ], self-testing [ , ], quantum key distribution [ ], random-ness certification [ ], random access codes [ ] andas non-classicality witnesses [ – ]. Apart from ex-ploratory attempts in [ ], in all these works the com-municating parties share classical correlations; the non-classicality can only arise due to the communicating(non-entangled) quantum states. As a consequence, thePAM scenario and the kind of device-independence itentails have not yet found any use in the most relevantentanglement-enhanced quantum communication pro-tocols. That is precisely the problem we solve here.As we show, the paradigmatic superdense coding [ ]can be cast as a particular instance of the prepare andmeasure scenario. As a consequence, a dimension wit-ness quantifying the probability of success of the su-perdense coding [ ] can also be used to certify, in asemi-DI manner, the non-classicality of the shared cor-relations between the communicating parties. As op-posed to the typical Bell scenario that is fully DI, quan-tum communication scenarios have to impose a limiton the amount of communication exchanged, otherwisethe communication problem becomes trivial. In linewith the superdense coding protocol, we achieve thatby imposing a limit on the Hilbert space dimensionof the quantum system being communicated. Strik-ingly, no other information about the preparation andmeasurement devices are required. Nicely, any purebipartite entangled state as well as a large family ofentangled mixed states violate our witness. Our re-sults largely improve over other semi-DI witnesses of a r X i v : . [ qu a n t - ph ] F e b ρ x x y b Λ/ ρ FIG. . Directed acyclic graph (black box representation) ofthe prepare and measure scenario where two parties sharesome correlation, which in principle could be either classical,represented above by the set of variables Λ , or quantum, rep-resented by a shared state ρ . According to some input x Aliceprepares a state ρ x and sends it to Bob—this being the onlycommunication between them—, who performs a measure-ment labeled by some input y obtaining an output b . entanglement: not only they reduce the experimentalrequirements and increase the tolerance to noise, butalso do not require partial state tomography to work,such as in quantum steering [ ]. We also provide asemi-definite program formulation allowing to obtainlower bounds for the optimal probability of superdensecoding success for arbitrary shared states. Followingthat we show how the non-classicality in the super-dense coding naturally leads to a self-testing protocol,also discussing its limitations in cryptographic scenar-ios. Finally, we also go beyond the superdense coding,analyzing a more general prepare and measure scenarioallowing for quantum correlations and a measurementdevice with several inputs. II. SUPERDENSE CODING AS A PREPARE ANDMEASURE SCENARIO
The prepare and measure scenario consists of an ex-periment performed between two parties, which wewill label Alice and Bob. Alice prepares a system ina state represented by x ∈ {
0, . . . , N − } and sendsit to Bob, who chooses a measurement setting y ∈{
0, . . . , m − } and obtains an output b ∈ {
0, . . . , k − } (see Fig. ). The whole experiment is described by theconditional probability distribution p ( b | x , y ) .In a classical description, depending on her input x ,Alice prepares a message a ∈ {
0, . . . , l − } , where l is the size of the alphabet of the message a , or thedimension of the system, that is a probabilistic func-tion not only of x but also of λ , the source of possi-ble pre-shared correlations between Alice’s preparationand Bob’s measurement apparatus. Similarly, Bob’smeasurement outcome will depend on the message a being received, the choice of measurement y and thepre-shared correlations. Thus, if the observed distribu- | | AliceBob | i ⟩ | j ⟩ H HZ X
FIG. . Quantum circuit (device-dependent) representation ofthe super dense coding. Alice wants to send a two bit mes-sage to Bob, represented here by the states | i (cid:105) , | j (cid:105) ∈ {| (cid:105) , | (cid:105)} by sharing an entangled state with Bob (the two bottom qubitsin the circuit, the first of which is held by Alice). The goal isachieved by applying σ z conditioned to | i (cid:105) and σ x conditionedto | j (cid:105) on the qubit in possession of Alice, which is, then, sentto Bob, who, in turn, retrieves the values of i and j by per-forming a Bell-state measurement on both qubits. tion has a classical explanation, it can be written as p ( b | x , y ) = ∑ a ∈ A ∑ λ ∈ Λ p ( λ ) p ( a | x , λ ) p ( b | a , y , λ ) . ( )In turn, a quantum description will explicitly de-pend on which resources are made non-classical. Forinstance, Alice might be allowed to prepare and sendquantum states to Bob, but only share classical correla-tions with him. In this case, the prepared states are de-scribed by the set { ρ x } x = N − ⊂ D ( H ) , where D ( H ) represents the set of density operators acting on someHilbert space H . A set of positive semidefinite oper-ators { M ( y ) b } b = k − ⊂ Pos ( H ) for y =
0, . . . , m − ∑ k − b = M ( y ) b = ∀ y , describes the possiblemeasurements performed by Bob. By Born’s rule, theobserved distribution is then given by p ( b | x , y ) = tr (cid:16) ρ x M ( y ) b (cid:17) . ( )In particular, notice that the quantum and classical de-scriptions become equivalent if the prepared states ρ x form a mutually commuting set.In the most general case, not only Alice preparesand sends quantum states to Bob but might also shareentangled states with him. That is precisely the caseof the paradigmatic superdense coding protocol [ ],where, by sharing entanglement with Bob, Alice cansend him dits of information by actually transmit-ting only one qudit. To illustrate it, suppose Al-ice wants to send two bits of information to Bob, ( x , x ) ∈ {
00, 01, 10, 11 } . If they share a maximallyentangled state | Φ + (cid:105) = ( | (cid:105) + | (cid:105) ) / √
2, Alice can en-code the information to be sent in different local uni-taries applied to the qubit in her possession, for in-stance {
00, 01, 10, 11 } → { , σ z , σ x , σ z σ x } , where σ i arethe Pauli matrices. Thus, after Alice’s local operation,the entangled state shared between them correspondsto the orthonormal Bell basis {| Φ + (cid:105) , | Φ − (cid:105) , | Ψ + (cid:105) , | Ψ − (cid:105)} (depending on which bits Alice wants to send) and thatcan be discriminated if Alice sends her qubit to Bob andBob measures both qubits in his possession in the Bellbasis. Notice that in this formulation, however, for thesuperdense conding protocol to work, not only Alicehas to know her state preparations but also Bob has tobe sure he is measuring in the Bell basis (see Fig. ).That is, both the preparation and measurement deviceshave to be under full control of the parties and be wellcharacterized. In this standard form, the superdensecoding protocol is device dependent.The first hint for the possibility of a semi-DI for-mulation of the superdense coding is given by thefact that it can be understood as a particular instanceof a PAM scenario: one where both the states be-ing communicated as well as the correlations sharedbetween the preparation and measurement devicesare quantum. The scenario can be described as fol-lows. Consider a set of states { ρ x } x = N − ⊂D ( H A ⊗ H B ) and a set of positive semidefinite op-erators { M ( y ) b } b = k − ⊂ Pos ( H A ⊗ H B ) for y =
0, . . . , m −
1, for which ∑ k − b = M ( y ) b = ∀ y . Noticethat H A and H B represent the Hilbert spaces of thesystems held by Alice and Bob respectively, such thatdim ( H A ) = d A and dim ( H B ) = d B . Thus, the observedprobability distribution obtained in the PAM scenariodescribing superdense coding is given by p ( b | x , y ) = tr (cid:16) ρ x M ( y ) b (cid:17) , ( )where, necessarily,tr A ( ρ x ) = tr A ( ρ x (cid:48) ) ∀ x , x (cid:48) . ( )The condition above is crucial, since it subsumes theidea that Alice’s operations (encoding the message shewishes to send) are local and thus cannot affect themarginal quantum state of Bob. Notice that if Aliceaims to send two dits of information to Bob, this willcorrespond to | x | = N = d preparations of Alice.Also notice that in the standard superdense coding, | y | = m =
1, that is, Bob always measures the sameobservable. In this case, assuming that all preparations(the possible values of the dits Alice wishes to send)are equiprobable, we can define a measure of the su-perdense coding success as p suc = N N − ∑ x = P ( b = x | x ) . ( ) We highlight that this is a device-independent measureof success, since it only depends on observational dataand does not assume anything about Alice’s prepara-tions neither on Bob’s measurements. A. The Schmidt number
A concept that will play a fundamental role in ourresults is that of entanglement and its detection via theSchmidt number [ ]. Any pure bipartite state | Ψ (cid:105) ∈H A ⊗ H B can be represented as | Ψ (cid:105) = r − ∑ j = η j | ψ j (cid:105) ⊗ | φ j (cid:105) , ( )in which η j are real positive numbers which are or-dered in a way that η ≥ η ≥ · · · ≥ η r − . TheSchmidt rank r of | Ψ (cid:105) is such that 1 ≤ r ≤ min ( d A , d B ) .Importantly, the notion of Schmidt rank can be gen-eralized for mixed states via the concept of Schmidtnumber [ ]. The Schmidt number s of a mixed state ρ = ∑ j p j | Ψ j (cid:105)(cid:104) Ψ j | is defined via an optimization over allpossible pure decompositions (cid:8) | Ψ j (cid:105) (cid:9) of ρ . The Schmidtnumber s is the smallest possible highest Schmidt rankof the pure states | Ψ j (cid:105) , that is, s = min { | Ψ j (cid:105) } max r ( | Ψ j (cid:105) ) .Clearly, for pure states, the Schmidt number coincideswith the Schmidt rank. Furthermore, this concept al-lows a natural classification of the set of bipartite quan-tum states given by the set S k ⊆ D ( H A ⊗ H B ) com-posed by all the states with Schmidt number less thanor equal to k . Those sets are trivially convex and theirextremal points are given by pure states, being alsoclear that S ⊂ S ⊂ · · · ⊂ S min ( d a , d b ) . III. SEMI-DI ENTANGLEMENT CERTIFICATION INTHE SUPERDENSE CODING
Interestingly, if the preparation and measurement de-vices are allowed to share only classical correlations, theprobability of success ( ) is the same irrespectively ofwhether Alice sends classical or quantum states to Bob[ ]. In both cases the probability of success is boundedas p suc ≤ d A N , where d A is the dimension of the classi-cal/quantum system Alice sends to Bob. This can beseen as a consequence of Holevo’s bound [ , ] thatlimits the amount of information that may be retrievedin such a scenario, implying that quantum messagescannot transmit more information than their classicalcounterparts.However, as shown by the superdense coding proto-col, that is no longer the case if an entangled state isshared between the parties. Our first result, for whicha detailed proof is given in the Appendix, is a formal and quantitative proof of that claim. It shows that theoptimal probability of success depends not only on thedimension of the quantum system communicated fromAlice to Bob, but also on the amount of entanglement ofthe quantum state shared between them, as quantifiedby the Schmidt number. Result . In a prepare and measure scenario with N prepa-rations and a single measurement with N outcomes, the su-perdense coding probability of success ( ) is limited asp suc ≤ min (cid:18) d A sN , 1 (cid:19) , ( ) where d A is the Hilbert space dimension of the quantum sys-tem sent from Alice to Bob and s is the Schmidt number ofquantum state shared between Alice and Bob. For N = d A K,with K ≥ s, the bound is tight. In particular, we notice that for s =
1, that is,only classical correlations are shared between the Al-ice and Bob, we recover the usual Holevo’s bound p suc ≤ d A / N .A direct application of the result above is in the con-text of semi-DI certification of entanglement. The semi-DI comes from the fact that we have to assume theHilbert space dimension H A to be at most d A . As dis-cussed before, unless one limits the ammount of com-munication sent by Alice, the problem becomes trivial.Since for any separable state p suc ≤ d A N , any probabil-ity of success violating this bound is then an unam-biguous proof that the shared state must be entangled.We will consider a range of examples of shared states ρ ∈ D ( H A ⊗ H B ) , dim ( H A ) = d A and dim ( H B ) = d B ,for which a set of N = d A preparations is enoughto violate the classical bound that can be rewritten as p suc ≤ d A .If ρ is the state under test, we can always define theset of states being prepared by Alice { ρ x } x = N − as ρ x = ( Λ x ⊗ )[ ρ ] , ( )in which Λ x is a local channel, Λ x : D ( H A ) (cid:55)→ D ( H A ) ,for all x . Since Λ x is a local channel, all the statesin { ρ x } x = N − have a Schmidt number less than orequal to that of ρ , so witnessing that { ρ x } x = N − isnot contained in S s is sufficient to witness that ρ (cid:54)∈ S s .Our next result, proven in the Appendix, states thatevery pure bipartite entangled state allows for a quan-tum enhancement in the superdense coding protocol. Result . For N = d A , the probability of success in thesuperdense coding that can be achieved with a bipartite pureentangled state | Ψ (cid:105) = ∑ s − j = η j | j (cid:105) ⊗ | j (cid:105) is lower bounded asp suc ≥ + Γ d A with Γ ≡ ∑ j (cid:54) = k η j η k > and which violates the classicalbound for any non-separable state (s > ). In particular, notice that for maximally entangledstates of dimension d A we have coefficients η i = √ d A and then Γ = ( d A − ) implying that p suc = ] of a general bipartite quantum state ρ , given by ζ ( ρ ) = max Φ (cid:104) Φ | ρ | Φ (cid:105) , ( )where, for some unitary operators U and U , | Φ (cid:105) =( U ⊗ U ) | Φ + d A (cid:105) with | Φ + d A (cid:105) = ( √ d A ) ∑ d A − i = | ii (cid:105) beingthe maximally entangled state of dimension d A . Dimension χ c r i t Bell NonlocalitySteeringSuperdense coding
FIG. . Upper bounds for critical visibilities for entanglementdetection in the isotropic state [Eq. ( )] with three differentmethods: Bell nonlocality (red, dot-dashed curve); quantumsteering (blue, dashed curve); superdense coding (black, solidcurve). Solely with assumptions on the dimensionality of thedistributed system, superdense coding enables entanglementdetection for all entangled isotropic states, thus, with lowervisibilities as compared with quantum steering [ ] and Bellnonlocality. Values for Bell nonlocality were obtained fromthe violation of the Collins-Gisin-Linden-Massar-Popescu in-equality [ ]. Better estimates are known for d = d → ∞ [ ], which reduce χ Bcrit to 0.67 and 0.5, respectively, but arestill greater than the value provided by the superdense codingmethod.
Result . The best probability of success in the superdensecoding method provided by a shared bipartite state ρ is lowerbounded as p suc ≥ ζ ( ρ ) , ( ) in which ζ ( ρ ) is the maximal singlet fraction of ρ , and d A isthe dimension of the quantum state sent from Alice to Bob. As a particular case we can consider the family ofisotropic states, an usual benchmark for the utility ofa non-classicality witness [ ]. These states are given by ρ χ ∈ D ( H A ⊗ H B ) , for dim ( H A ) = dim ( H B ) = d A with ρ χ = ( − χ ) d A + χ | Φ + d A (cid:105)(cid:104) Φ + d A | . ( )As can be seen, the singlet fraction is given by ζ ( ρ χ ) = χ + ( − χ ) / d A . In particular, the critical visibilitybelow which the isotropic state becomes separable isgiven by χ crit = d A + )which coincides with the critical χ below which theprobability of success ( ) becomes classical. Strikingly,our semi-DI witness detects the non-classicality of anyentangled isotropic state.As a comparison we can consider the paradigmaticBell [ ] and steering tests [ ], both involving sharedentangled states and measurement devices for both Al-ice and Bob. A Bell test is fully DI and for this reasonleads to higher constraints over χ . In turn, the steeringscenario is semi-DI, because tomography on Bob’s stateis required, thus implying that not only the dimensionof the state has to be known but one also has to trust themeasurement device. In this sense, the semi-DI require-ments in the superdense coding protocol are milder ascompared to the steering, since the former only requirean assumption on the state dimension. For d =
2, thebest known Bell test [ ] requires χ Bcrit = ] χ Scrit = χ SDcrit = d → ∞ the best known Bell test implies χ Bcrit = χ Scrit = ( H d A − ) / ( d − ) , where H n = ∑ ni = i is the n -th harmonic number, imply-ing that for any dimension there will be a gap betweenthe steering and the superdense coding/entanglementtests. See Figure for more details. IV. SELF-TESTING MAXIMALLY ENTANGLED STATES
An important application of the DI framework is thepossibility to infer properties of the shared quantumstate without the need of knowing precisely the mea-surement apparatus, a feature known as self-testing[ , ]. As we show next, under the assumption ofthe dimension d A of the shared bipartite state, the PAMscenario can be employed to self-test maximally entan-gled states. Result . For N = d A , the saturation of inequality ( ) fors = d A self-tests, up to a local unitary, the presence of abipartite maximally entangled state. It is worth highlighting that self-testing in the super-dense coding is likewise the one in a Bell scenario and differs from usual self-testing results in PAM scenario[ , ]. Typically, the PAM scenario without shared en-tanglement can self-test a set of prepared states. Here,in contrast, we are self-testing the shared quantum stateand not the preparations.Another curious feature of this self-testing process re-lies on a strong dependence on the hypothesised causalstructure. Self-testing in superdense coding only cer-tifies that Alice and Bob share a maximally entangledstate with someone else, but not necessarily with eachother. For instance, Alice and Bob might share a max-imally entangled state with an eavesdropper and stillsaturate the superdense coding witness ( ). This is anunusual feature, for instance, when compared with self-testing in Bell scenarios that are robust to the insertionof an extra part, a crucial property in applications suchas quantum key distribution. In the case of the prepareand measure scenario, entanglement might be used tobreak existing semi-DI quantum key distribution proto-cols [ ].This shows that even though our witness is semi-DI(as it only assumes the dimension of the state but noother information from the preparation and measure-ment devices), in principle it is not robust against anexternal malicious part. As pointed out above, in thesuperdense coding an eavesdropper can retrieve the in-formation being sent from Alice to Bob without beingdetected. The source of cryptographic insecurity comesfrom the fact that the measurement device of Bob hasa single input, thus allowing the eavesdropper (shar-ing entanglement with Bob) to retrieve the informationwithout being detected.To avoid that, at least in a device dependent frame-work, one possibility is to adapt the BB protocol[ ]. Say that Alice randomly decided whether ornot to apply a Hadamard gate H—such that H | (cid:105) =( √ )( | (cid:105) + | (cid:105) ) and H | (cid:105) = ( √ )( | (cid:105) − | (cid:105) ) —tothe qubit she sends to Bob. Without knowing if Aliceapplied or not the Hadamard gate, the eavesdropperwill unavoidably make detectable mistakes.For instance, if Alice wanted to send the classicalmessage 00 and did not apply the Hadamard to herqubit, in the absence of an eavesdropper the state Bobwould receive is ( √ )( | (cid:105) ) + | (cid:105) . The eavesdrop-per, however, does not know whether the Hadamardwas applied or not. If he randomly decides to applythe Hadamard gate to the qubit he intercepts (eventhough Alice has not done it) he will then with prob-ability half wrongly conclude that the message beingsent by Alice was 10. The eavesdropper will not only re-send the wrong information to Bob but also with prob-ability one-half he will choose the wrong encoding. If,similarly to what happens in the BB protocol, Aliceand Bob use some rounds to publicly compare theirencoded and decoded messages, they will unavoidably detect the presence of the eavesdropper.In summary, combining the BB the superdense cod-ing protocol makes the latter robust in a cryptographicsense [ ]. Notice, however, that in this case the proto-col becomes device-dependent (we have explicitly usedthe quantum description of a Hadamard gate). A pos-sibility to achieve a semi-DI cryptographic formulationwould be to use a witness such that the measurementdevice takes more than just one possible input. We de-rive an example of such witness below but leave openthe possibility of whether it allows to secure the flow ofquantum information from an eavesdropper. V. OPTIMIZING THE PROBABILITY OF SUCCESS
Although useful lower bounds for the best proba-bility of success can be found analytically for specificstates by taking advantage of their special structures,in a more general case, for a generic shared state, find-ing good guesses for preparations and measurementsmight be a cumbersome task. An interesting alternativeis to find such lower bounds numerically. Given somestate shared between Alice and Bob, in order to achievethe optimal probability of success one has to optimizeover all possible preparations of Alice and the possiblemeasurements of Bob. As we show next, this optimiza-tion can be performed via semidefinite programming(SDP) [ ]. In particular, if the preparations of Alice arefixed, optimization over Bob’s measurement is given bythe following program:Given ρ x = ( Λ x ⊗ )[ ρ ] ,Maximize M x tr [ ρ x M x ] ,subject to M x ≥ ∑ x M x = . ( )In turn, fixing the measurements of Bob allows foroptimization over possible preparations in terms of anSDP by using the Choi-Jamiolkowski representation ofthe different channels [ , ] asMaximize L x tr [( L x ⊗ B ) · ( ρ T A ⊗ A (cid:48) ) · ( A ⊗ M x )] ,subject to L x ≥ A (cid:48) [ L x ] = A , ( )where L x are the operators that act on H A ⊗ H B andcorrespond to each preparation Λ x , ρ T A is the partialtransposition of the state ρ shared between Alice andBob, and the constraints ensure that the resulting mapsare completely positive and trace preserving.By alternating between preparation optimization andmeasurement optimization, starting with a random α d × p s u c d p s u c FIG. . Lower bounds on success probabilities for super densecoding, computed via the alternated optimization method forWerner states. The curves correspond to different local di-mensions d , ranging from to , from top to bottom (purplerhombuses correspond to d =
2, yellow circles to d =
3, redcrosses to d =
4, and blue triangles to d = d , so that the bound for separable states for allcases coincide in value 1. In every case computed, detectionof entanglement occurs after α ≈ ( d − ) / d . Inset: Compar-ison between p suc for α = d for dimensions 2, ..., 7 (dashed curve). measurement, we can obtain a lower bound on the op-timal p suc for any given state ρ .As an application of this method, we consider theWerner states [ ], described by ρ W ( α ) = d − α Sd − α d , ( )where d is the local dimension of each subsystem, S isthe swap operator ∑ di , j = | ij (cid:105)(cid:104) ji | , and α is a parameter inthe range [ −
1, 1 ] . Using N = d preparations and out-comes for Bob’s measurement and applying the methodto ρ W ( α ) for d =
2, . . . , 5, we find that p suc saturates thebound 1/ d [Eq. ( )] for all values of α below approxi-mately ( d − ) / d , and violates the bound for all valuesabove. This is shown in Fig. , where p suc is plottedfor α ≥ ( d − ) / d isstrictly lower than the threshold for establishing quan-tum steering [ ], given by d / ( d + ) . That is, the semi-DI test provided by the superdense coding once morebeats steering tests. In the inset of Fig. we showthe values computed for p suc when α = d =
2, . . . , 7. Ascan be seen, the gap p suc − d decreases quickly withthe dimension, indicating that Werner states of largerdimension provide smaller quantum enhancements inthe superdense coding. VI. PREPARE AND MEASURE SCENARIO WITHMORE THAN ONE MEASUREMENT SETTING
As discussed above, the fact that in the superdensecoding the measurement device of Bob only has one in-put opens the way to attacks of an external maliciouspart. An adaptation of the BB protocol is enough toguarantee the security of the superdense coding, how-ever, in a device dependent manner. Motivated by thatwe provide below another prepare and measure testthat witnesses the entanglement of the shared quan-tum state but, in this case, relying on several differentmeasurements of Bob. Whether this witness or a varia-tion of it can be combined with the superdense codingto guarantee its cryptographic security is an interestingproblem that we leave open for future research.Consider the prepare and measure framework featur-ing N preparations and N ( N − ) /2 dichotomic mea-surement settings. This has been analysed in Ref. [ ]under the hypothesis that the shared correlations areclassical. Here we drop this hypothesis and show thatnew bounds must be considered when the participantsshare a quantum state. Remarkably, in this case, asin the superdense coding scenario, we observe that theSchmidt number of the shared state plays a central rolein defining the bound. Result . In a prepare and measure scenario with N prepara-tions x and N ( N − ) /2 dichotomic measurements ( y , y ) ,for y > y , where y , y ∈ {
0, . . . , N − } , the set of prob-ability distributions is bounded by the inequality:V N ≤ N (cid:18) − ( d A s , N ) (cid:19) , ( ) where d A is the Hilbert space dimension of the quantum sys-tem sent from Alice to Bob, s is the Schmidt number of quan-tum state shared between Alice and Bob, andV N = ∑ x > x (cid:48) (cid:12)(cid:12) P ( | x , ( x , x (cid:48) )) − P ( | x (cid:48) , ( x , x (cid:48) ) (cid:12)(cid:12) . ( ) Furthermore, if s = d A and N < d A or N = cd A , for integerc, expression ( ) is tight. A detailed proof of the results in provided in the Ap-pendix.
VII. DISCUSSION
The ability to certify entanglement is a crucial bench-mark in quantum information processing. A stan-dard tool for that are entanglement witnesses [ ], ex-perimentally observable quantities allowing to distin-guish between separable and entangled states. How-ever, and in spite of its wide applicability, entanglementwitnesses are fully device-dependent. Unless one has perfect characterization of measurement devices, onemight incur in false positive results [ ].The best one can hope for is a fully device-independent certification of entanglement, such as thatprovided by the violation of Bell inequalities [ ]. Theproblem, however, is the fact such violations are still anexperimental challenge and furthermore are unable towitness the non-classicality of a wide range of entan-gled states [ , ]. A promising approach to achieve acompromise between our ability to witness entangle-ment with fewer assumptions as possible and at thesame time to achieve experimental feasibility is that of-fered by semi-device-independent protocols. In quan-tum steering [ ], for instance, one can detect a largerset of entangled states at the cost, however, of being ableto perform quantum tomography on some parts of theentangled system.Here we show that a paradigmatic protocol in quan-tum information science, the superdense coding proto-col [ ], offers a new platform for entanglement certifi-cation. As we show, superdense coding can be seen as aparticular case of a prepare and measure scenario [ ],one where both the communication and the shared cor-relations are allowed to have a quantum nature. Withinthis context, we provide a semi-device independentwitness—requiring only an assumption on the Hilbertspace dimension of the quantum state—that is upperbounded by the Schmidt number of the shared quan-tum state. Not only this witness has a clear operationalmeaning—the probability of success of the superdensecoding protocol—but also can be connected to an im-portant entanglement quantifier, the so-called singletfraction [ ], implying in particular that any pure bi-partite entangled state offers a semi-DI advantage inthe superdense coding protocol. Furthermore, our ap-proach provides a significant advantage in comparisonwith steering [ ], the standard semi-DI test in the liter-ature. As opposed to a steering test, not only our wit-ness does not require quantum state tomography butalso can witness the non-classicality of any entangledisotropic state, an important family of mixed entangledstates used as a benchmark in DI and semi-DI certifica-tion of entanglement. Nicely, our witness can also beused to self-test maximally entangled states of any di-mension. Finally, we provide a semi-definite programformulation allowing to obtain, for any shared quan-tum state, lower bounds for the best probability of suc-cess that can be obtained in the execution of superdensecoding.In the scenario where no shared quantum correla-tions are allowed, the prepare and measure scenariohas been employed in a variety of quantum informationtasks [ – ]. Thus, an interesting question is whetherthe fully quantum version of the PAM scenario we con-sider here can also lead to relevant practical applica- tions. For instance, we have shown that in its standardform, the dense coding is not cryptographically secure,as a malicious part could retrieve the information beingsent without being detected. As discussed, the sourceof insecurity comes from the fact that the measurementdevice has a single input. This has motivated us to alsoderive a witness with several measurement inputs. Per-haps a combination of both witnesses could provide thedesired cryptographic security. Another possibility isto investigate whether the PAM scenario with quantumcorrelations can also be employed to detect the dimen-sion of physical systems. So far, dimension witnesses[ – ] make the strong assumption that only classicalcorrelations are allowed between the preparation andmeasurement devices, an assumption that if not ful-filled ruins the current applications of such witnesses [ – ]. We believe our results might trigger furtherdevelopments in this direction. ACKNOWLEDGMENTS
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In this appendix we provide a detailed proof of theresults introduced in the main paper, each of which isrestated below for convenience.
Result . In a prepare and measure scenario with N prepa-rations and a single measurement with N outcomes, the su-perdense coding probability of success ( ) is limited asp suc ≤ min (cid:18) d A sN , 1 (cid:19) , (A ) where d A is the Hilbert space dimension of the quantum sys-tem sent from Alice to Bob and s is the Schmidt number of quantum state shared between Alice and Bob. For N = d A K,with K ≥ s, the bound is tight.Proof. First, let us notice that p suc is a linear functiondefined in D ( H A ⊗ H B ) . Since S s ⊆ D ( H A ⊗ H B ) isconvex for all possible values of s , it must hold that p suc is a convex function defined in S s for all s .We are interested in setting an upper bound on thevalue of p suc for a set { ρ x } x = N − ∈ S s for somefixed s . Since p suc is a convex function in S s , its max-imum value must happen for extremal points in S s ,which are pure states. Thus, we focus in the case {| Ψ x (cid:105)(cid:104) Ψ x |} x = N − ∈ S s .Given that Alice’s preparations cannot affect Bob’sside [Eq. ( )] and using the Schmidt decomposition ofeach | Ψ x (cid:105) [Eq. ( )], we get to | Ψ x (cid:105)(cid:104) Ψ x | = s − ∑ j , k = η ( x ) j η ( x ) k | ψ ( x ) j (cid:105)(cid:104) ψ ( x ) k | ⊗ | φ j (cid:105)(cid:104) φ k | ,(A )for | ψ ( x ) j (cid:105) ∈ H A and | φ j (cid:105) ∈ H B .Let us consider the orthonormal basis of H B given by {| φ j (cid:105)} j = d B − , in which for 0 ≤ j ≤ s − | φ j (cid:105) are exactly the same that appear in the Schmidt de-composition of | Ψ x (cid:105) . Plus, let H aux be the space gen-erated by the set of orthogonal vectors {| φ j (cid:105)} j = s − .Then, we have that | Ψ x (cid:105) ∈ H e f f ective = H A ⊗ H aux for x ∈ {
0, . . . , N − } , and dim ( H e f f ective ) = d A s .Thus, in this case, p suc = N N − ∑ x = tr ( | Ψ x (cid:105)(cid:104) Ψ x | M x )= N N − ∑ x = tr e f f ective (cid:0) | Ψ x (cid:105)(cid:104) Ψ x | M (cid:48) x (cid:1) ≤ N N − ∑ x = tr e f f ective (cid:0) M (cid:48) x (cid:1) = d A sN , (A )in which M (cid:48) x a positive semidefinite operator acting on H e f f ective and ∑ N − x = M (cid:48) x = e f f ective , where e f f ective isthe identity acting in H e f f ective .To verify that the bound is tight for N = d A K , with K ≥ s , consider the unitary operators W ( K ) x , x = d A − ∑ j = e π ijx / K | j ⊕ x (cid:105)(cid:104) j | , (A )defined for x ∈ {
0, . . . , d A − } and x ∈ {
0, . . . , K − } , which coincide with the Weyl operators for K = d A [ ]. Assume that Alice’s preparations are given byapplication of the W ( K ) x , x on her side of the shared state and that the shared state is maximally entangled withSchmidt rank s , i.e. | ψ (cid:105) = s − ∑ j = √ s | j (cid:105)| j (cid:105) . (A )Let us define then the resulting states as | ˜ Ψ ( K ) x , x (cid:105) : = s − ∑ j = √ s (cid:16) W ( K ) x , x | j (cid:105) (cid:17) | j (cid:105) . (A )It is straightforward to show that, for K ≥ s , ∑ x , x | ˜ Ψ ( K ) x , x (cid:105)(cid:104) ˜ Ψ ( K ) x , x | = ( K / s ) ∑ s − j = A ⊗ | j (cid:105)(cid:104) j | . As-sume then that Bob’s measurement operators are givenby M x , x = sK | ˜ Ψ ( K ) x , x (cid:105)(cid:104) ˜ Ψ ( K ) x , x | + N d B − ∑ j = s A ⊗ | j (cid:105)(cid:104) j | , (A )so that M x , x | ˜ Ψ ( K ) x , x (cid:105) = ( s / K ) | ˜ Ψ ( K ) x , x (cid:105) for all x , x .With this prescription, we obtain that p suc = sK . (A )Since N = d A K , we obtain precisely that p suc = d A s / N . Result . For N = d A , the probability of success in thesuperdense coding that can be achieved with a bipartite pureentangled state | Ψ (cid:105) = ∑ s − j = η j | j (cid:105) ⊗ | j (cid:105) is lower bounded asp suc ≥ + Γ d A with Γ ≡ ∑ j (cid:54) = k η j η k > and which violates the classicalbound for any non-separable state (s > ).Proof. Let | Ψ (cid:105) be the state under test and assume itsSchmidt rank as s , for some 1 ≤ s ≤ d A . Define the setof states ρ ( x , x ) = s − ∑ j , k = η j η k ( W x , x | j (cid:105)(cid:104) k | W † x , x ) ⊗ | j (cid:105)(cid:104) k | , (A )where W x , x are the Weyl operators [ K = d A on Eq.(A )] and the detection operators as˜ M b , b = | ˜ Ψ b , b (cid:105)(cid:104) ˜ Ψ b , b | , (A )where | ˜ Ψ b , b (cid:105) = | ˜ Ψ ( d A ) b , b (cid:105) [Eq. (A )]. Then, we obtain ρ ( x , x ) M x , x = d A d A − ∑ m = s − ∑ j , k = η j η k ( W x , x | j (cid:105)(cid:104) m | W † x , x ) ⊗ | j (cid:105)(cid:104) m | ,which proves the result. Result . The best probability of success in the superdensecoding provided by a shared bipartite state ρ is lower boundedas p suc ≥ ζ ( ρ ) , (A ) in which ζ ( ρ ) is the maximal singlet fraction of ρ and d A isthe dimension of the quantum state sent from Alice to Bob.Proof. We start by recalling the fact that any state ρ pre-senting a maximal singlet fraction ζ ( ρ ) can be convertedvia shared randomness and local unitary operationsinto an isotropic state ρ χ with same maximal singletfraction, ρ χ = ( − χ ) d A + χ | Φ + d A (cid:105)(cid:104) Φ + d A | , (A )in which χ = ζ ( ρ ) d A − d A − , via a twirling operation [ , ].This implies that, without any loss of generality, we canrestrict our demonstration to isotropic states only.Define the states given by ρ x , x χ = W x , x ρ χ W † x , x = − χ d A + χ W x , x | Φ + d A (cid:105)(cid:104) Φ + d A | W † x , x ,and the measurement basis defined in equation (A ).Then, using the fact that tr [ ρ x , x χ M x , x ] = (cid:104) Φ + d A | ρ χ | Φ + d A (cid:105) , we obtaintr (cid:2) ρ x , x χ M x , x (cid:3) = ζ ( ρ χ ) , (A )given that ζ ( ρ χ ) = ζ ( ρ ) , we get p suc = ζ ( ρ ) .Hence, using local operations and shared random-ness, it is possible to certify any state with maximalsinglet fraction satisfying ζ ( ρ ) > d A .Remarkably, for isotropic states, this is precisely thecondition for nonseparability [ ], implying such statesare entangled if and only if they can violate our witness. Result . The saturation of inequality ( ) for s = d A , withN = d A preparations, self-tests the presence of a shared bi-partite maximally entangled state up to local unitary opera-tions.Proof. First of all, we recall the fact that for a fixed di-mension d A of the Hilbert space of Alice’s system, H A ,any set of preparations contained in an effective Hilbertspace of dimension d A , H e f f ective .Given that for reaching such bound we must have p ( b = x | x ) =
1, we can assure that the preparations { ρ x } x = N − do not overlap, i.e., tr ( ρ x ρ x (cid:48) ) = x (cid:54) = x (cid:48) , otherwise no measurement would perfectly dis-tinguish them.This, associated with the condition N = d A , imposesthat the states must also be pure, { ρ x } x = N − = {| Ψ x (cid:105)(cid:104) Ψ x |} x = N − . To get this result, we use thespectral decomposition of ρ x : ρ x = d A − ∑ j = λ ( x ) j | Ψ ( x ) j (cid:105)(cid:104) Ψ ( x ) j | , (A )in which (cid:104) Ψ ( x ) j | Ψ ( x ) k (cid:105) = δ j , k . Then, we obtaintr ( ρ x ρ x (cid:48) ) = d A − ∑ j , k = λ ( x ) j λ ( x (cid:48) ) k tr ( | Ψ ( x ) j (cid:105)(cid:104) Ψ ( x ) j | Ψ ( x (cid:48) ) k (cid:105)(cid:104) Ψ ( x (cid:48) ) k | )= d A − ∑ j , k = λ ( x ) j λ ( x (cid:48) ) k |(cid:104) Ψ ( x ) j | Ψ ( x (cid:48) ) k (cid:105)| .For x (cid:54) = x (cid:48) we get that d A − ∑ j , k = λ ( x ) j λ ( x (cid:48) ) k |(cid:104) Ψ ( x ) j | Ψ ( x (cid:48) ) k (cid:105)| = ( ρ x ) = d A , given that {| Ψ ( x (cid:48) ) k (cid:105)} x = N − form a basis of H e f f ective , the abovesum cannot be null. At most, for a fixed value of x and x (cid:48) , rank ( ρ x ) = d A − ( ρ x (cid:48) ) =
1. By extend-ing this analysis to other values of x (cid:48) , we conclude thatrank ( ρ x ) = x .Because those states are pure, they can be written as | Ψ x (cid:105) = d A − ∑ j = η ( x ) j | ψ ( x ) j (cid:105) ⊗ | φ ( x ) j (cid:105) . (A )Considering now the condition ( ) plus the unitaryequivalence of the purifications [ ], we have that: | Ψ x (cid:105) = d A − ∑ j = η j (cid:0) U x | ψ j (cid:105) (cid:1) ⊗ | φ j (cid:105) , (A )in which U † x U x = and there is no loss of generality insetting U = .It holds that d A − ∑ x = | Ψ x (cid:105)(cid:104) Ψ x | = ,which implies thattr A d A − ∑ x = d A − ∑ j , k = η j η k ( U x | ψ j (cid:105)(cid:104) ψ k | U † x ) ⊗ | φ j (cid:105)(cid:104) φ k | = d A B . On the other hand we have thattr A d A − ∑ x = d A − ∑ j , k = η j η k ( U x | ψ j (cid:105)(cid:104) ψ k | U † x ) ⊗ | φ j (cid:105)(cid:104) φ k | = d A d A − ∑ j = η j | φ j (cid:105)(cid:104) φ j | which can only happen if η j = √ d A , which then con-cludes the proof. Appendix B: N preparations and N(N- )/ dicotomicmeasurements Result . In a prepare and measure scenario with N prepara-tions x and N ( N − ) /2 dicotomic measurements ( y , y ) ,for y > y , where y , y ∈ {
0, . . . , N − } , the set of prob-ability distributions is bounded by the inequality:V N ≤ N (cid:18) − ( d A s , N ) (cid:19) , (B ) where d A is the Hilbert space dimension of the quantum sys-tem sent from Alice to Bob, s is the Schmidt number of quan-tum state shared between Alice and Bob, andV N = ∑ x > x (cid:48) (cid:12)(cid:12) P ( | x , ( x , x (cid:48) )) − P ( | x (cid:48) , ( x , x (cid:48) ) (cid:12)(cid:12) . (B ) Furthermore, if s = d A and N < d A or N = cd A , for integerc, expression (B ) is tight.Proof. First notice that if s = ], and the result holds. We herebyanalyse the remaining quantum cases, for s > { ρ x } x = N − ⊂ S s be the set of preparations forwhich relation holds, and { M y , y (cid:48) b } b ∈{ } the measure-ments settings, so: V N = ∑ x > x (cid:48) (cid:12)(cid:12)(cid:12) tr (cid:16) ( ρ x − ρ x (cid:48) ) M ( x , x (cid:48) ) b = (cid:17)(cid:12)(cid:12)(cid:12) . (B )Defining D ( ρ x , ρ x (cid:48) ) : = || ρ x − ρ x (cid:48) || , it is known that[ ]: D ( ρ x , ρ x (cid:48) ) = max P ∈P ( H A ⊗H B ) tr (( ρ x − ρ x (cid:48) ) P ) ,where P ( H A ⊗ H B ) is the set of positive operators thatact in H A ⊗ H B . Clearly the following relation is alwayssatisfied: V N ≤ ∑ x > x (cid:48) | D ( ρ x , ρ x (cid:48) ) | . (B )Because of the triangle inequality, the right-hand sideof the equation is a convex function of D ( ρ x , ρ x (cid:48) ) , whichbeing a norm is also a convex function of ρ x − ρ x (cid:48) , whichcan be seen as a map F , whose domain is dom ( F ) = { χ = ρ ⊗ σ | χ ∈ D ( H A ⊗ H B ⊗ H A ⊗ H B ) ρ , σ ∈D ( H A ⊗ H B ) } , given by: F ( χ ) = tr A B ( χ ) − tr A B ( χ ) . (B )It follows that F is a linear map, and that dom ( F ) is aconvex set whose extremal points are given by elements χ = ρ ⊗ σ for which ρ and σ are pure states.With this we can say that the right-hand side of equa-tion (B ) is a convex function defined in dom ( F ) andthus has its maximal value at some extremal point indom ( F ) . This implies that: V N ≤ ∑ x > x (cid:48) | D ( | Ψ x (cid:105)(cid:104) Ψ x | , | Ψ x (cid:48) (cid:105)(cid:104) Ψ x (cid:48) | ) | = ∑ x > x (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − |(cid:104) Ψ x | Ψ x (cid:48) (cid:105)| (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) = ∑ x > x (cid:48) (cid:16) − |(cid:104) Ψ x | Ψ x (cid:48) (cid:105)| (cid:17) = N ( N − ) − ∑ x > x (cid:48) |(cid:104) Ψ x | Ψ x (cid:48) (cid:105)| = N ( N − ) − (cid:32) ∑ x , x (cid:48) |(cid:104) Ψ x | Ψ x (cid:48) (cid:105)| − N (cid:33) . (B )Now, define Ω as follows: Ω = N N − ∑ x = | Ψ x (cid:105)(cid:104) Ψ x | , (B )so, the equation (B ) can be expressed as: V N ≤ N − N ( Ω ) . (B )At this point, we recall that: | Ψ x (cid:105) = s ∑ j = η ( x ) j | ψ ( x ) j (cid:105) ⊗ | φ j (cid:105) , (B )and condition ( ) plus the unitary equivalence of thepurifications lead to: | Ψ x (cid:105) = s ∑ j = η j (cid:0) U x | ψ j (cid:105) (cid:1) ⊗ | φ j (cid:105) , (B )This implies that Ω ∈ D ( H e f f ective ) , i.e., Ω is adensity operator acting on H e f f ective , where H e f f ective was defined in appendix A and has dimensiondim ( H e f f ective ) = d A s . So we must have that:tr ( Ω ) ≥ d A s , (B )which leads to: V N ≤ N (cid:18) − d A s (cid:19) . (B )Whenever N ≤ d A s , V N we are working under thecommunication capacity of the channel, and V N always reaches its maximum algebraic value, so we can rewrite(B ): V N ≤ N (cid:18) − ( d A s , N ) (cid:19) . (B )Now we show that if N < d A or N = cd A , for an inte-ger c , the above expression is saturated using the mea-surements that optimally discriminate | Ψ x (cid:105) from | Ψ x (cid:48) (cid:105) given that: | Ψ x (cid:105) = √ d A d A − ∑ j = U x | j (cid:105) ⊗ | j (cid:105) . (B )in which { U x } is a to be defined set of unitary operatorsacting on H A . We prove that by showing that if thestate defined as Ω is such that tr ( Ω ) = ( d A , N ) , forthe above preparations, then there exist measurements { M y , y (cid:48) b } b ∈{ } , for y > y , with y , y ∈ {
0, . . . , N − } leading to a saturation of our witness, even thoughthese are never specified.From equation (B ): Ω = Nd A N − ∑ x = d A − ∑ j , k = (cid:16) U x | j (cid:105)(cid:104) k | U † x (cid:17) ⊗ | j (cid:105)(cid:104) k | ,and: Ω = N d A N − ∑ x , x (cid:48) = d A − ∑ j , k , m = (cid:16) U x | j (cid:105)(cid:104) k | U † x U x (cid:48) | k (cid:105)(cid:104) m | U † x (cid:48) (cid:17) ⊗ | j (cid:105)(cid:104) m | . Straight forward calculations lead to:tr ( Ω ) = N d A N − ∑ x , x (cid:48) = tr (cid:16) U † x (cid:48) U x (cid:17) tr (cid:16) U † x U x (cid:48) (cid:17) Fixing the set { U x } as the set of Weyl operators { W x } x = d A − acting on H A , and letting c = (cid:22) Nd A (cid:23) , i.e. c is the integer part of Nd A , we have: N = cd A + N mod d A . (B )Define N = N mod d A . Now, we are going to dividethe set of preparations {
0, . . . , N − } into d A groups.If x and x (cid:48) are in the same group, then | Ψ x (cid:105) = | Ψ x (cid:48) (cid:105) .There will be N groups with c + N − N groups with c members. Each group is defined by aWeyl operator W X so we have that:tr ( Ω ) = N d A N − ∑ x , x (cid:48) = d A δ X , X (cid:48) = N N − ∑ x = ( c + ) + N − ∑ x = N mod d A c = N (( c + ) N + c ( N − N ))= N ( N + cN ) .Clearly the above expression is N if N < d A (in thiscase c = N = N ), and d A if N = cd A , for integer c (here we have that N =13