Simultaneous Certification of Entangled States and Measurements in Bounded Dimensional Semi-Quantum Games

Certification of quantum systems and operations is a central task in quantum information processing. Most current schemes rely on a tomography with fully characterised devices, while this may not be met in real experiments. Device characterisations can be removed with device-independent tests, it is technically challenging at the moment, though. In this letter, we investigate the problem of certifying entangled states and measurements via semi-quantum games, a type of non-local quantum games with well characterised quantum inputs, balancing practicality and device-independence. We first design a specific bounded-dimensional measurement-device-independent game, with which we simultaneously certify any pure entangled state and Bell state measurement operators. Afterwards via a duality treatment of state and measurement, we interpret the dual form of this game as a source-independent bounded-dimensional entanglement swapping protocol and show the whole process, including any entangled projector and Bell states, can be certified with this protocol. In particular, our results do not require a complete Bell state measurement, which is beneficial for experiments and practical use.

Introduction.-With the rapid development of quantum technologies in state preparation and dynamical evolution, quantum devices are expected to outperform their classical counterparts in the future. However, unexpected and uncharacteristic noise hampers the functioning of quantum devices. Thus certification of states and processes which have been actually implemented in the quantum devices is a central task in this field. Nonetheless, as a classical being, an observer can only take advantage of classical statistics together with a physical model he/she believes to characterise the system and its dynamical evolution. One way for certification is to employ a full tomography on the system (state tomography) [1][2][3] or on the process (process tomography) [4][5][6]. With an information-complete set of measurements (input states), one can reconstruct all the elements in the quantum state operator (quantum process). However, such approaches require a full knowledge on the devices. Unavoidable noise from the real environment can easily nullify the results. The states and measurements used in tomography need to be fully characterised and trusted, while sometimes they might be even more difficult to calibrate than the system/process investigated.
Luckily, quantum physics allows us to break away from the dilemma by Bell nonlocality [7,8]. Violation of a Bell inequality indicates entanglement in a nonlocal system and incompatibility of measurements. In fact, as first explicitly pointed out by Mayers and Yao [9], the state and measurements can be uniquely determined up to local isometries from certain classical correlations, or, the system "self-tests" itself [9] without any charactisation of the devices, but only relying on the validity of quantum physics. In this sense we say the certification is device-independent. So far, numerous remarkable results have been derived in the subject of state self-testing [10][11][12][13][14][15], and physical processes or measurements certification [16][17][18][19][20][21][22]. To make self-testing meet a real situation, robust self-testing, allowing the noises and imperfections to some extent, has been investigated, aiming to give a lower bound on the fidelity of the physical system from a reference system (in the sense of local isometries) based on the observed statistics. Much progress has been made in this field, too [14,[23][24][25][26] .
Realising loophole-free device-independent tests are highly challenging in practice, mainly suffering from detection and locality loopholes. Up till now, only a few device-independent experiments succeed in closing the detection loophole and the locality loophole simultaneously [27][28][29][30][31]. However, in these demonstrations, either only a rather small violation of Bell inequality is obtained, which is far from certifying a Bell state according to the current robust self-testing results [25], or the repetition rate is too low for any practical applications.
A compromising approach to balance deviceindependence and practicality is to apply a semidevice-independent scenario. One of the most prominent scenarios is the semi-quantum game proposed by Buscemi [32]. Semi-quantum games are similar to Bell tests, except for that general local quantum inputs are allowed. It has been proved that all entangled states exhibit non-locality in these games, bridging the gap between the concept of entanglement and non-locality. After Buscemi's seminal results, focusing on the prepared entangled states, there are following works presenting approaches to qualitatively witness entanglement [33][34][35][36][37] and quantitatively estimate the amount of entanglement in the system with semi-quantum games [38,39]. However, little attention has been received for channels or measurements in these games, with one exceptional work focusing on the quantum channel with quantum memories [40]. Moreover, it is left open whether we can "uniquely" certify certain systems and operations simultaneously in a manner similar to self-testing in device-independent tests.
In this letter, we consider the certification of entangled states and measurements in semi-quantum games with a given dimensional Hilbert space. We design a new type of semi-quantum game, showing the plausibility of simultaneous certification of any pure bipartite entangled state in H 2 ⊗H 2 and Bell state measurement operators. In particular, our design does not rely on a complete Bell state measurement but can be naturally generalised for such a case. Moreover, we transform the setting into the manner of a source-independent entanglement swapping protocol via a duality treatment of state and measurement operator. In its "dual" form of semi-quantum game, any entangled projector acting on the bounded dimensional space and Bell states can be certify simultaneously.
Preliminary.-We first briefly review the concept of semi-quantum games in [32] and results in [33]. Consider a nonlocal game with two players, Alice and Bob, as shown in Fig. 1. In each round of the game, a referee gives them quantum states ψ A0 x ∈ D(H A0 ), ψ B0 y ∈ D(H B0 ) individually. We denote the input systems as H A0 , H B0 for Alice and Bob, respectively. Then they are asked to give classical outputs a, b, and a score β x,y a,b is obtained based on the combination of {ψ A0 x , ψ B0 y , a, b}. In this game, generally, Alice and Bob can first share a bipartite state ρ AB ∈ D(H A ⊗H B ), and perform joint positive operatorvalued measure measurements (POVMs) on the party they each hold and the received quantum input state, individually. We denote the POVM elements as M Using the technique of partial POVM, we can treat inputting quantum states as a tomography process of the effective POVM elements (2) It has been proved in [33] that a separable ρ AB will result in separable effective POVM elementsM |ii . One can focus on one specific outcome, say (a, b) = (0, 0), by letting β x,y a,b = δ a,0 · δ b,0 · β x,y in a semi-quantum game. Consider any conventional entanglement witness W , Tr W ρ AB < 0 for a specific entangled state ρ AB while Tr[W σ] ≥ 0 for any separable operator σ. One can transform the witness W into a semi-quantum game with the parameters β x,y and quantum inputs {ψ A0 x , ψ B0 y } in a decomposition of W , and a negative score S < 0 implies the entanglement in systems A, B. Furthermore, these semi-quantum game can also be applied to detect detailed entanglement structure [36], estimate the robustness of entanglement, negativity and randomness [39].
Simultaneous certification of entangled state and measurements.-In previous works, only prepared states or channels are characterized via a semi-quantum game. To further elaborate the application of semi-quantum games, we now investigate the problem of certifying entangled states and joint measurements simultaneously. We first define the concept of entangled measurement operators and entangled measurements: , otherwise it is called a separable measurement operator. A POVM {M AB ab } is called entangled if at least one of its measurement operators is entangled.
Entangled measurements, especially Bell state measurement (BSM), play an important role in many quantum information processing tasks like entanglement swapping and teleportation. In the measurement-deviceindependent protocols, entangled measurement operators are essential for a nonlocal behaviour, too. we have the following result for semi-quantum games: Theorem 1.-In a semi-quantum game where the input states and appointed scores form an entanglement witness given by Eq. The proof is given in Appendix. This theorem together with previous results show that entangled measurements and state can be witnessed by investigating the effective POVM elements. Now we take one step further and try to certify them via a quantitative investigation on the effective POVM measurement operatorsM A0B0 ab . We consider the case with a bounded dimensional Hilbert space, specifically, the case where the unknown state is a pair of qubits, and we design the semi-quantum games such that all quantum inputs are qubits as well, i.e.
We follow the design of measurement-deviceindependent entanglement witness and also focus on one pair of measurement operators corresponding to a single outcome, say (a, b) = (0, 0). For brevity, we omit the subscripts and denote the measurement operators as M A0A , M BB0 and the effective POVM element as M A0B0 . In Eq. (3), we choose an operator W with the spectral decomposition W = 4 i=1 l i |ψ i ψ i | and design a following measurement-device-independent game: Design of the measurement-device-independent game: |ψ 1 is an entangled pure state. Now we give the main theorem in this letter: any two-qubit entangled pure state |ψ 1 and corresponding joint measurement operators M A0A , M BB0 can be certified, with the largest score l 1 /4 in the above game obtained. To be more specific, the observation of this score certifies that (ρ AB ) T ∼ |ψ 1 ψ 1 | , M A0A , M BB0 ∼ |Φ + Φ + |, where the equivalence refers to a freedom of local unitary operations.
A sketch of proof.-Our design of the semi-quantum game restricts thatM A0B0 must lie in the support of |ψ 1 ψ 1 | , supp{|ψ 1 ψ 1 |}, in order to maximize the average score S. This allows us to represent the problem in an optimization form: Subject to: Here arg refers to the arguments when the maximum of the objective function is taken. In order to maximize the objective function, M A0A , M BB0 need to be projectors. Yet we do not make any assumptions on their ranks. Furthermore, the state ρ AB is in general a mixed state. However, we have the following lemmas showing that only rank-one measurement operators and a pure state can guarantee thatM A0B0 is projected on the span of a pure entangled state: , a necessary condition forM A0B0 to be a rank-one operator is that both M A0A and M BB0 are rank-one operators.
We leave the proofs in Appendix. For a general state ρ AB and POVM elements M A0A , M BB0 , we can apply spectral decomposition to them, For fixed a, b, the operator is positive, thusM A0B0 , the sum of above operators, is rank-1 only if these operators all project on supp{|ψ 1 ψ 1 |}. Via Lemma 1, we relax ρ AB to be pure. Then with a pure ρ AB , we require M A0A , M BB0 both to be rank-1 operators so thatM A0B0 ∈ supp{|ψ 1 ψ 1 |}.
We now can focus on the case with rank-one projectors and a pure state. First we apply the Schmidt decomposition to |ψ 1 , and without loss of generality it can be written as |ψ 1 = cos χ |00 + sin χ |11 , where |0 , |1 represent orthogonal bases in systems A 0 , B 0 . Since the measurement operators can be treated as pure bipartite states now, we now denote them as In Eq. (4), the state vectors |0 , |1 acting on systems A 0 , B 0 are the same as the ones in representing |ψ 1 Requiring Ψ to be parallel to |ψ 1 indicates that Under this condition we have the following lemma: , and the subspaces of the two parties are spanned by While the lemma is obvious in the bounded dimension Hilbert space we now consider, it is worth noting that it does not rely on a restriction of dimensions of H A , H B . We leave the proof for this lemma in the general case in Appendix.
Certification in a source-independent entanglement swapping game.-It is interesting to notice that the entanglement swapping protocol shares a dual form of the semi-quantum game if we exchange the roles of state and measurements, shown in Fig. 2. We have two independent bipartite states, ρ A0A ∈ D(H A0 ⊗ H A ), ρ BB0 ∈ D(H B ⊗ H B0 ). Different from the standard entanglement swapping protocol, here we consider a scenario with uncharacteristic states ρ A0A , ρ BB0 and an uncharacteristic joint measurement {M AB i } acting on systems A, B. Compared to the semi-quantum game in Fig. 1 where trusted quantum inputs are utilised, suppose now we can carry out well-characterised local measurements {N A0 a }, {N B0 b } on systems A 0 , B 0 . Thus our entanglement swapping game is similar to a (non)bilocal scenario [41,42], except for that we can now apply trusted measurements on the systems A 0 , B 0 . Because the uncharacteristic underlying state ρ A0A ⊗ρ BB0 , we say the protocol shares a source-independent nature [43]. i . If we focus on one specific measurement result on systems A, B, with a duality treatment on states and measurements, the sourceindependent entanglement swapping protocol can be transformed into the measurement-device-independenttype semi-quantum game, except for a difference in the normalisation factors of states and measurement operators. Therefore, we may construct a dual semi-quantum game to certify the initial systems A 0 , A and B, B 0 and joint measurement operator simultaneously, with measurements on the final systems A 0 , B 0 only.
We now design a specific game as follows: Design of the source-independent entanglement swapping game: |ψ ψ| is an entangled projector, l 0.
In this game, the average score can be expressed as S = Tr Vρ A0B0 1 . In order to maximize the score, the operator ρ A0B0 1 needs to be embedded in the support of |ψ ψ|, which is much the same as the idea in the measurement-device-independent game. In our proofs for Lemma 1 and Lemma 2, we do not rely on the normalisation factors, but mainly the orthogonality between different eigenvectors in the spectral decomposition. Hence with a similar route, one can come to the following theorem: Theorem 3.-In the above source-independent entanglement swapping game with the assumption that all systems are two-dimensional, the largest score that can be obtained is 1/4, and it is obtained only if (M AB 1 ) T = |ψ ψ| , ρ A0A , ρ BB0 = |Φ + Φ + |, up to local unitary operations.
This theorem actually states that an entangled projector and Bell states can be certified in this game, if the stateρ A0B0 i / Tr ρ A0B0 i prepared on the systems A 0 , B 0 is a pure entangled state with a probability of 1/4. We can use the result to certify the case with multiple outputs of the joint measurement, by observing all subnormalised operators {ρ A0B0 i }. One special case is that we can certify an ideal entanglement swapping process corresponding to a complete Bell state measurement: Corollary.-Under the qubit system assumption, if all measurement outcomes occur with a probability of 1/4 and the corresponding final states on the systems A 0 , B 0 form a complete set of Bell states under a certain choice of local bases, one can certify that the joint measurement {M AB i } is a set of complete Bell state measurements, and ρ A0A , ρ BB0 are Bell states.
Conclusions.-Via focusing on the effective POVM operator seen from the input ports in a semi-quantum game, we show that any pure entangled state and Bell state measurement operators can be certified up to local unitary operations. Moreover, we present the certification of Bell states and an entangled joint measurement operator in a source-independent entanglement swapping protocol, following a similar technique. This technique can also be expected to to certify other type of semiquantum game, for instance, certification of a quantum memory channel [40].
Due to a dimension restriction, we cannot treat our certification as a "self-testing" result. However, we conjecture that the same certification result can hold even if we do not restrict on the unknown system's dimension in the manner of self-testing. Lemma 3 in our approach relies only on the ranks of the measurement operators and quantum system rather than the system's dimension, and it may be a starting point for self-testing. Furthermore, robust semi-device-independent self-testing results can also be expected. It is also worth noting that our methods do not require a complete Bell state measurement result, which is often not easy to achieve in practice (in particular, it has been proved impossible to carry out a complete Bell state measurement in linear optics). A robust semi-device-independent self-testing may be beneficial for practical blind quantum computing tasks [44,45]. For future directions, it is also interesting to extend semi-quantum games into the non-i.i.d. region and parallel setting. We hope our work can shed light on the further exploration of applications of semi-quantum non-local games.
For convenience, we first restate the theorem in Main text:  is a rank-1 operator, ρ AB needs to be a pure state.

Proof of Lemma 3
Lemma 3. For rank-one projectors where H A0 , H B0 ∼ = C 2 , fix the measurement bases of H A0 , H B0 to be {|0 , |1 } A0,B0 and express the projectors as In the condition that Ψ ϕ AφB = Ψ φ A ϕ B = 0, we have Ψ ϕ A ϕ B 2 + Ψ φ AφB 2 ≤ 1. The equality is taken iff ϕ A φ A = ϕ B φ B = 0, and the subspaces of the two parties are spanned by ϕ A , φ A , ϕ B , φ B , respectively. In particular, no restriction on the dimensions of H A , H B is made.
For the right side of this inequality, it can be regarded as a function of the form f (x) = ax 2 + bx √ 1 − cx 2 + d, with the variant In our problem, c ≥ 1, hence it is required that Let f (x) = 0 and we have where the term a 2c 1 √ a 2 +b 2 c ≤ a 2c 1 a = 1 2c , so we are able to take the value x = ± 1 2c ± a 2c 1 √ a 2 +b 2 c . By some calculation with respect to f (x), for the maximum value of f (x), we only need to consider its value at the points x = 1 c and x = and the equality is taken iff cos θ = cos γ = 0. This indicates that ϕ A φ A = ϕ B φ B = 0.