Self-testing quantum states and measurements in the prepare-and-measure scenario

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Self-testing quantum states and measurements in the prepare-and-measure scenario

TAVAKOLI, Armin, et al.

Abstract

The goal of self-testing is to characterize an a priori unknown quantum system based solely on measurement statistics, i.e. using an uncharacterized measurement device. Here we develop self-testing methods for quantum prepare-and-measure experiments, thus not necessarily relying on entanglement and/or violation of a Bell inequality. We present noise-robust techniques for self-testing sets of quantum states and measurements, assuming an upper bound on the Hilbert space dimension. We discuss in detail the case of a 2→1 random access code with qubits, for which we provide analytically optimal self-tests. The simplicity and noise robustness of our methods should make them directly applicable to experiments.

TAVAKOLI, Armin, et al . Self-testing quantum states and measurements in the prepare-and-measure scenario. Physical Review. A , 2018, vol. 98, no. 062307

DOI : 10.1103/PhysRevA.98.062307

Available at:

http://archive-ouverte.unige.ch/unige:127273

Disclaimer: layout of this document may differ from the published version.

1 / 1

Armin Tavakoli,¹J˛edrzej Kaniewski,²Tamás Vértesi,³ Denis Rosset,^{4, 5}and Nicolas Brunner¹

1Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

2QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

3Institute for Nuclear Research, Hungarian Academy of Sciences, P.O. Box 51, 4001 Debrecen, Hungary

4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5

5Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria

The goal of self-testing is to characterise an a priori unknown quantum system based solely on measurement statistics, i.e. using an uncharacterised measurement device. Here we develop self-testing methods for quantum prepare-and-measure experiments, thus not necessarily relying on entanglement and/or violation of a Bell in- equality. We present noise-robust techniques for self-testing sets of quantum states and measurements, assuming an upper bound on the Hilbert space dimension. We discuss in detail the case of a2→1random access code with qubits, for which we provide analytically optimal self-tests. The simplicity and noise robustness of our methods should make them directly applicable to experiments.

Introduction.—Predicting the results of measurements per- formed on a given physical system has traditionally been the main concern of physics. However, with the advent of device- independent quantum information processing [1–3], the op- posite question has become relevant. More specifically, given an initially unknown system and an uncharacterised measure- ment device, what can be inferred about the physics of the experiment based solely on the observed measurement statis- tics? Despite the apparent generality of this question, certain cases do allow for a precise characterization of the system.

This is referred to as self-testing [4,5].

The possibility to self-test quantum states and measure- ments usually relies on quantum nonlocality. Consider two distant observers performing local measurements on a shared quantum state. When the resulting statistics leads to viola- tion of a Bell inequality [6], it is necessarily the case that the shared quantum state is entangled, and moreover, that the local quantum measurements are incompatible; see e.g. [7].

Furthermore, for specific Bell inequalities, maximal violation (i.e. the largest possible value in quantum theory) implies that the quantum state and the measurements can be uniquely iden- tified (up to local isometries). For instance, a maximal vio- lation of the Clauser-Horne-Shimony-Holt (CHSH) Bell in- equality [8] implies maximally incompatible measurements (two anti-commuting Pauli observables) and a shared maxi- mally entangled two-qubit state [9–12]. More recently, it has been demonstrated that all bipartite pure entangled states can be self-tested [13], as well as certain multipartite entangled states [14–16]. Another important progress is the develop- ment of self-testing methods robust to noise [17–23]. For in- stance, given a certain level of violation of a Bell inequality (but not necessarily maximal), the fidelity between the ini- tially unknown state and a given target state can be lower- bounded.

Self-testing thus offers promising perspectives for the cer- tification of quantum systems in experiments (see e.g. [24]), as well as for device-independent quantum information pro- tocols [25]. It is therefore natural to ask whether the concept of self-testing can be applied to more general quantum exper- iments, beyond those based on entanglement and nonlocality.

In the present work, we develop self-testing methods tai- lored to the prepare-and-measure scenario. This covers a

broad class of experiments, where quantum communication schemes (e.g. the BB84 quantum key distribution (QKD) pro- tocol) are prominent examples. In this setting, a preparation device initially prepares a quantum system in different possi- ble states. The system is then transmitted to a measurement device, which performs different possible measurements on it. While it is still possible in this case to characterise certain physical properties of the system based only on statistics, this requires in general an assumption on the devices. One pos- sibility, which we will follow here, is to assume that the set of quantum states and measurements admit a full description in a Hilbert space of given dimension [26–28]. Intuitively this means that the amount of information communicated from the preparation device to the measurement device is assumed to be upper bounded. Such a scenario considering quantum systems of fixed dimension, but otherwise uncharacterised, is referred to as semi-device-independent, and opens interesting possibil- ities for quantum information processing [29–33].

Here we demonstrate techniques for robustly self-testing sets of prepared quantum states, as well as sets of quantum measurements. These methods allow one to (i) assess the compatibility of given sets of preparations and measurements with the observed statistics, and (ii) lower-bound the average fidelity between the unknown preparations (measurements) and a set of ideal quantum states (measurements). We discuss in detail a simple prepare-and-measure scenario, namely the 2→1random access code (RAC). This allows us to provide analytically optimal self-tests for a pair of anti-commuting Pauli observables, and for a set of four qubit states corre- sponding to the eigenstates of two anti-commuting Pauli ob- servables. We then generalise these results to other prepare- and-measure scenarios. The simplicity and robustness of our methods should make them directly applicable to experi- ments. We conclude with a number of open questions.

Scenario.— We consider a quantum prepare-and-measure experiment. Upon receiving input x, a preparation device emits a physical system in a quantum stateρ_x. The system is then transmitted to a measurement device, which, upon receiv- ing an inputy, performs a quantum measurement returning an outcomeb. Formally, the measurement is described by a set of positive operatorsM_y^b, that equal identity when summed over b. Importantly both the specific statesρ_xand measurements

arXiv:1801.08520v2 [quant-ph] 11 Sep 2018

M_y^bare a priori unknown to the observer. The statistics of the experiment is then given byP(b|x, y) = tr(ρxM_y^b). In this setting, any possible probability distribution can be obtained, given that the prepared statesρxcan be taken in a sufficiently large Hilbert space. This is however no longer the case when we limit the Hilbert space dimension; specifically we impose thatρx ∈ L(C^d)for some givend <|x|(where|x|denotes the number of possible inputsx). In this case, limits on the set of possible distributions can be captured via inequalities of the form

A=X

x,y,b

αxybP(b|x, y)≤Qd, (1) where α_xyb are real coefficients. These “dimension wit- nesses” allow one to place device-independent lower bounds on the dimension of the quantum system [26].

Subsequently, one can ask what the limitations are on the set of distributionsP(b|x, y)given that the preparations ad- mit a classicald-dimensional representation, i.e. there exists ad-dimensional basis such that all states ρxare diagonal in this basis. We denote byCdthe maximal value of the quan- tityA in this case. Interestingly, for well-chosen quantities A, one finds thatCd < Qd. Thus, for a given system dimen- sion d, quantum systems outperform classical ones, in the sense that certain quantum distributions cannot be reproduced classically [26]. This quantum advantage can be viewed as the origin for the possibility of developing self-testing meth- ods for the prepare-and-measure scenario; in analogy to Bell inequality violation being the root for self-testing entangled states.

In the following we present robust self-testing techniques based on specific dimension witnesses A. Based only on the value ofA, which is directly accessible from the exper- iment statistics, we characterise the (initially unknown) pre- pared states and measurements. In particular, when the max- imal value of the witness is obtained, i.e. A = Q_d, then a specific set of pure statesρx = |ψxihψx|and a specific set of projective measurementsM_y^b must have been used (up to a unitary). Moreover, when a non-maximal valueA< Qdis obtained, the compatibility of given sets of preparations and measurements can be assessed. Finally, one can efficiently lower bound the fidelity between the prepared states and mea- surements and the ideal (or target) states and measurement leading toA=Qd.

Note that a recent series of works followed a related though conceptually different approach, based on hypothesis testing [34–36]. This method does however not allow for self-testing.

The2 → 1random access code.—We discuss in detail a simple prepare-and-measure experiment. This involves four possible preparations, denoted byx= (x0, x1)(wherexj ∈ {0,1}), and two possible binary measurements, y ∈ {0,1}

andb∈ {0,1}. The score is given by A2= 1

8 X

x₀,x₁,y

P(b=xy|x0, x1, y). (2) This means that, upon receiving inputy, the measurement de- vice should return the outputb =xy, i.e. they-th bit of the

input bit-stringxreceived by the preparation device. Hence the name of a2 → 1RAC [37–39]. Note that all inputs are assumed to be chosen uniformly at random. Indeed, this task is nontrivial only whend <4; here we will consider the case d = 2, i.e. qubits. In this case, one finds the tight bounds C2 = 3/4andQ2 = (1 + 1/√

2)/2 ≈0.85[37]. The clas- sical boundC2can be obtained by simply always sending the bitx0. The quantum boundQ2is obtained via the following

“ideal” strategy. The four qubit preparations correspond to the pure states

ρ^ideal₀₀ =¹1+σ_x

2 ρ^ideal₀₁ = ¹1+σ_z

2 ρ^ideal₁₁ =¹1−σx

2 ρ^ideal₁₀ = ¹1−σz

2 . (3)

These are simply the eigenstates of the Pauli observablesσx

andσz. Next, the measurements are projective and given by two anti-commuting Pauli observables

M_y^ideal= (M_y⁰)^ideal−(M_y¹)^ideal= σx+ (−1)^yσz

√2 . (4) These qubit preparations and measurements represent the ideal situation, where the maximal value A2 = Q₂ is achieved. In the following we will determine what restric- tions apply to the possible preparations and measurements, given that a particular value ofA2is observed. In particular, when the maximal valueA2 =Q2is attained, both the states and the measurements must be the ideal ones as given above (up to a unitary).

Self-testing preparations.—Here we find restrictions on the set of prepared states given an observed value of A2. For convenience, we write the qubit preparations as ρx₀x₁ = (11+m~x₀x₁·~σ)/2, where m~x₀x₁ denotes the Bloch vector (satisfying|m~x₀x₁| ≤ 1) and~σ = (σx, σy, σz)denotes the vector of Pauli matrices.

The first step consists in re-expressing A2= 1

2+1 8

X

y

tr M_y⁰V_y

≤ 1 2+1

8 X

y

q

tr M_y⁰V_y²

tr M_y⁰

(5) whereVy = P

x₀,x₁(−1)^xyρx₀x₁. In the second step we used that for a positive semi-definiteO and a Hermitian op- eratorR, it holds that |tr (OR)|² ≤ tr OR²

tr (O) [23].

Without loss of generality, we can restrict to extremal qubit measurements, which are here projective rank-one operators.

Consequently, we have thattr M_y⁰

= 1. Next, we obtain V_y²= ¹₂(β+ (−1)^yα)11, where

β= 1 2

X

x₀,x₁

|m~x₀x₁|²−m~00·m~11−m~01·m~10 (6) α= (~m₀₀−m~₁₁)·(~m₀₁−m~₁₀). (7) Finally, we find that Eq. (5) reduces to

A2≤1 2 + 1

8√ 2

hpβ+α+p β−αi

. (8)

This provides a tight self-test of the prepared states (in terms of their Bloch vectors), for any given value ofA2. Let us start with the caseA2 =Q2. Since√

β+α+√

β−α= q

2β+ 2p

β²−α², we see that Eq. (8) is maximal iffα= 0 andβ is maximal. This turns out to be achievable. In order to maximiseβ, we need (i)∀x₀x₁ : |~m_x0x1| = 1, i.e. that all preparations are pure states, and (ii) that m~₀₀ ·m~₁₁ =

~

m01·m~10 =−1, i.e. the states correspond to (pairwise) an- tipodal Bloch vectors. We define~r0 = m~00 = −~m11 and

~

r1 = m~01 = −~m10. Consequently, we find α = 4~r0·~r1. Therefore, in order to haveα= 0, we must choose~r0·~r1= 0.

This implies that the right-hand-side of Eq. (8) is upper- bounded byQ₂. Therefore, we conclude that when observing maximal valueA2=Q2, the set of four prepared states must be equivalent (up to a unitary rotation) to the set of four ideal states; we note that this was also shown in Ref. [40] in the context of QKD.

More generally, for any value A2, one can find a set of preparations (and corresponding measurements) such that the inequality (8) is saturated, see Supplementary Material (SM) part A. For the case of classical preparations (i.e. diagonal in a given basis), the Bloch vectors can simply be replaced by numbersm_x0x1∈[−1,1], and we getA₂≤C₂.

Self-testing measurements.— Let us now consider self- testing of measurements. Using thatMy = M_y⁰−M_y¹, we write

A2= 1 2+ 1

16 X

x0,x1

tr ρ_x0x1[(−1)^x0M₀+ (−1)^x1M₁]

≤ 1 2+ 1

16 X

x₀,x₁

λmax[(−1)^x0M0+ (−1)^x1M1], (9) whereλmax[X]is the largest eigenvalue of the (Hermitian) op- eratorX. Since the upper bound corresponds to choosing the optimal preparations for a fixed pair of observables, it sim- ply quantifies the optimal performance achievable using these observables. IfM₀andM₁are qubit observables the upper bound can be evaluated exactly (see SM part A) to give

A2≤ 1 2+ 1

16 q

2µ+ 2ν−η²₊+ q

2µ−2ν−η₋²

, (10) whereµ = tr M₀²+M₁²

, ν = tr{M0, M₁} andη_± = tr(M0±M1). The right-hand side reaches the optimal value Q2 iff µ = 4, η_± = 0 and ν = 0, which implies anti- commuting projective observables (i.e. projective measure- ment operators). In other words, observingA2=Q2implies that the measurements are unitarily equivalent to the ideal ones. Moreover, note that inequality (10) is tight; for any value ofA2 one can find measurements (and corresponding states) such that inequality is saturated (see SM part A). It fol- lows that any pair of projective, rank-one observables that is incompatible (|ν|<4) can lead toA₂> C₂.

Robust self-testing of the preparations.—We now discuss the problem of characterizing the fidelity between the realised preparations and the ideal ones. This will allow us to quantify the distance of the prepared states with respect to the ideal

ones. Again, we want to develop self-testing methods which are based only on the value ofA2.

More formally, given an arbitrary set of preparations, we define the average fidelity with the ideal preparations to be S({ρx0x1}) = maxΛP

x₀,x₁F(ρ^ideal_x

0x₁,Λ[ρ_x0x1])/4, where Λ is a quantum channel i.e. a completely positive trace- preserving map. Here the fidelitiesF(ρ, σ) = tr p√

ρσ√ ρ simplify toF(ρ^ideal_x0x1,Λ[ρx0x1]) = tr Λ[ρx0x1]ρ^ideal_x0x1

, as the ρ^ideal_x0x1are pure states. We derive lower bounds on the smallest possible value ofSgiven a value ofA2, i.e.,

F(A₂) = min

{ρx0x1}∈R(A2)

S[{ρ_x0x1}]. (11) Note that this involves a minimisation over all sets of four preparations R(A2) that are compatible with an observed valueA2.

In order to lower bound F, we use an approach in- spired by Ref. [22]. From Eq. (9), we have A₂ = ¹₂ + P

x₀,x₁tr (Wx₀x₁ρx₀x₁), whereWx₀x₁ = ₁₆¹ P

y(−1)^xyMy. We define operators corresponding to some suitably chosen channel acting on the ideal preparations:

K_x0x1(M₀, M₁) = Λ^†(M₀, M₁)[ρ^ideal_x

0x₁], (12) whereΛ^† is the channel dual toΛ. We aim to construct oper- ator inequalities of the form

K_x0x1(M₀, M₁)≥sW_x0x1+t_x0x1(M₀, M₁)11, (13) for all inputs(x₀, x₁), for any given measurements, where sandtx₀x₁(M0, M1)are real coefficients. Finding such in- equalities, as well as a suitable channelΛallows us to lower boundSas follows:

S ≥1 4

X

x₀,x₁

tr (Kx₀x₁ρx₀x₁)≥ s 4

X

x₀,x₁

tr (Wx₀x₁ρx₀x₁)

+1 4

X

x0,x1

t_x0x1 = s

4(A2−1/2) + 1 4

X

x0,x1

t_x0x1, (14) Applying a minimisation overM0andM1to the right-hand- side, the above inequality becomes valid for all preparations.

Consequently, F(A2)≥ s

4(A2−1/2) +t≡L(A2), (15) wheret≡1/4 minM₀,M₁P

x₀,x₁tx₀x₁(M0, M1). In SM part B, we construct explicitly the channel and derive an operator inequality leading to a lower bound, given bys= 4 1 +√

2 andt= 2−√

2 /4.

This provides a robust self-testing for the preparations. A maximal valueA2 = Q2 implies F = 1, i.e. the prepara- tions must be the ideal ones (up to a unitary). ForA2 =C2, i.e. a maximal value given a set of classical states, we get that F ≥ 3/4. This bound can be attained via the set of pure statesρx₀x₁ = (11+ (−1)^x0x1σz)/2 (diagonal in the same basis, hence classical), combined with the measure- mentsM0 = M1 = σz. Therefore, we see that our bound

FIG. 1. Average fidelityF(F⁰)for prepared states (measurements), as a function of the observed value ofA2. The black line is our analytical lower bound of Eq. (15). The blue region is accessible via single qubit strategies without shared randomness, as confirmed by strong numerical evidence (see SM). When allowing for shared randomness between the devices, the accessible region (obtained by taking the convex hull of the blue region) now also includes the grey area, and our analytic lower bound is tight in general.

F(A₂) ≥L(A₂)is optimal, as far as linear inequalities are concerned (see Fig. 1). It is then interesting to consider the intermediate regionC2<A2< Q2. First, focusing on strate- gies involving a single set of states and measurements, we observe numerically that the linear boundF(A2)≥ L(A2) cannot be saturated anymore, and conjecture the form of op- timal states and measurements; see red curve in Fig. 1 and SM part C for details. Second, allowing for shared random- ness between the preparation and measurement device (such that convex combinations of qubit strategies are now possi- ble), the linear bound becomes tight, a direct consequence of the linearity ofFandA₂in terms of the states and measure- ments.

Robust self-testing of the measurements.— Similarly, we can quantify the average fidelity of the measure- ments with respect to the ideal ones: S⁰ {M_y^b}

= max_ΛP

y,bF (M_y^b)^ideal,Λ[M_y^b]

/4, whereΛmust be a uni- tal channel (i.e. mapping the identity to itself), in order to ensure that measurements are mapped to measurements. In analogy with the case of preparations, our goal is to lower bound the following quantity:

F⁰(A2) = min

{M_y^b}∈R⁰(A2)

S⁰ {M_y^b}

, (16)

whereR⁰(A2)represents all sets of measurements compatible with a certain value ofA₂.

We first rewrite A2 = P

y,btr(M_y^bZyb), where Zyb =

1 8

P

x0,x1ρ_x0x1δ_b,xy. Next, we construct operator inequalities Kyb({ρx₀x₁})≥sZyb+ty({ρx₀x₁})11, (17) given the unital channelK_yb = Λ^†[(M_y^b)^ideal]. Similarly to the case of preparations, strong operator inequalities can be derived by choosing carefully the channel; all details are given in SM part D. Finally, this leads to a lower bound on the aver-

age fidelity

F⁰(A2)≥ min

{ρx0x1}

1 4

X

y,b

tr KybM_y^b

≥L(A2). (18)

That is, we find thatF⁰can be lower-bounded by a linear ex- pression in terms ofA2, which turns out to be the same as for the case of preparations.

This provides a robust self-test for the measurements. Ob- servingA2 = Q2implies that F⁰ = 1, hence the measure- ments are equivalent to the ideal ones (up to a unitary). For A₂ =C₂, we have thatF ≥3/4. This lower bound can be attained by choosingM₀ =σ_z andM₁ = 11, with the states ρ00 =ρ01 = (11+σz)/2andρ10 =ρ11= (11−σz)/2. For C2 <A2 < Q2, we find numerically that the inequality (18) cannot be saturated using a single set of measurements and states (see Fig. 1). Details, in particular a conjecture for the form of the optimal measurements, are given in SM part C.

Similarly as for the case of states, when allowing for convex combinations of qubit strategies, our linear bound is tight.

Generalisations.—The above results can be generalised in several directions. Firstly, a generalisation of the2→1RAC enables self-testing of any pair of incompatible Pauli observ- ables (see SM part E). Secondly, we derive compatibility rela- tions for theN →1RAC with qubits (see SM part F). Thirdly, we self-test qutrit preparations and projective measurements in the2→1RAC (see SM part G).

Finally, we present a numerical method for robust self- testing of preparations applicable in scenarios beyond RACs.

The method is based on semi-definite programing and com- bines (i) the swap-method [21] used for self-testing in Bell scenarios with (ii) the hierarchy of finite-dimensional quan- tum correlations [41–43]. The idea is to first construct a swap operator, based on the measurement operators, which maps the state of the preparation onto an ancilla. The average fi- delity between the ancilla and the ideal states can then be ex- pressed in terms of strings of products of measurements oper- ators and the extracted states. The last step is to miminize this average fidelity over all quantum realisations that are compat- ible with a given witness value, using the hierarchy of Refs [41–43]. Although typically returning sub-optimal bounds on F, this method is widely applicable. In SM part H, we de- scribe in detail the methodology and apply to two examples, including the2→1RAC.

Outlook.— We presented methods for self-testing quan- tum states and measurements in the prepare-and-measure sce- nario. These techniques demonstrate strong robustness to noise, and should therefore be directly amenable to exper- iments, providing useful certification techniques in a semi- device-independent setting. Moreover, these ideas should find applications in quantum communications. Our methods apply to the states and measurements used in QKD (e.g. in BB84), as well as in semi-device-independent QKD and randomness generation protocols [29–33].

It would be interesting to develop robust self-testing tech- niques for more general scenarios, e.g. for higher dimensional quantum systems. Another direction would be to consider sce- narios beyond prepare-and-measure, for instance adding be-

tween the preparation and measurement devices a transforma- tion device [44,45], and self-test the latter.

Finally, while we have focused here on self-testing based on an assumption on the dimension, one could develop methods based on different assumptions, such as a bound on the mean energy [46], the overlap [47], or the entropy [48].

Acknowledgements.— This work was supported by the Swiss national science foundation (Starting grant DIAQ, QSIT, and Early Postdoc Mobility fellowship

P2GEP2_162060), the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Action ROSETTA (grant no. 749316), the European Research Council (grant no. 337603), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (grant no. 10059), and the National Research, Development and Innovation Office NKFIH (Grants No. K111734, and No.

KH125096).

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Appendix A: Self-testing relations for preparations and measurements

In this section we provide a simple example of preparations that saturate the compatibility bound forA2given in the main text. Moreover, we derive the upper bound for compatibility of measurements given in the main text.

First, we consider the case of preparations. Consider preparations such thatρ00 andρ11, and ρ01 andρ10 correspond to antipodal Bloch vectors with a relative angleθ, the maximal quantum value ofA2, is obtained from

A2=1 2 +1

8 X

y

λ_max[Vy], (A1)

whereV_y =P

x0,x1(−1)^xyρ_x0x1. We represent the preparations on the Bloch sphere asρ_x0x1 = 1/2 (11+m~_x0x1·~σ), where

~

m₀₀ = [cos(θ/2),0,sin(θ/2)]andm~₀₁ = [cos(θ/2),0,−sin(θ/2)], withm~₁₁ =−m~₀₀andm~₁₀ =−m~₀₁. This givesV₀ = 2 cos(θ/2)σxandV1= 2 sin(θ/2)σz. The respective largest eigenvalues areλmax[V0] = 2 cos(θ/2)andλmax[V1] = 2 sin(θ/2), leading to

A₂=1 2 + 1

4√ 2

h√

1 + cosθ+√

1−cosθi

. (A2)

It is straightforward to see that this achieves the upper bound in the main text; indeed the above choice of preparations leads to β= 4andα= 4 cosθ.

In order to derive the upper bound onA2for compatibility of measurements in the main text we evaluate X

x₀,x₁

λ_max

(−1)^x0M₀+ (−1)^x1M₁

(A3) for arbitrary qubit observablesM₀, M₁. We take advantage of the fact that

λ_max[T] +λ_max[−T] =λ_max[T]−λ_min[T], (A4) which for a2×2matrix can be evaluated analytically. More specifically, ifT is a 2×2Hermitian matrix with eigenvalues λ0≥λ1, let

χ:= trT =λ0+λ1, ζ:= trT²=λ²₀+λ²₁

and then

λ0−λ1=p

2ζ−χ². (A5)

Evaluating this expression forT =M0±M1gives the desired upper bound given.

Appendix B: Operator inequalities for robust self-testing of preparations In this section we provide a detailed derivation of the lower bound on the average fidelityF(A2).

For a real constants >0, to be chosen later, consider for each pair(x0, x1)the operatorKx₀x₁ −sWx₀x₁, whereWx₀x₁ =

1 16

P

y(−1)^xyM_yandK_x0x1 = Λ^†[ρ^ideal_x

0x₁], for some channelΛ. Suppose now thatt_x0x1 ∈Ris a lower bound on its eigenvalues, or, equivalently, that the operator inequality

K_x0x1 ≥s W_x0x1+t_x0x111 (B1) holds. Then, computing the trace of this inequality withρ_x0x1and averaging over inputs leads to

S≥ 1 4

X

x₀,x₁

F(ρ^ideal_x

0x₁,Λ[ρ_x0x1])≥ s 4

A₂−1

2

+t, t≡1 4

X

x₀,x₁

t_x0x1 , (B2) where the first inequality holds becauseS is defined as maximisation over all possible channels, and the Λused there is one possible choice. In turn, if (B1) holds as an operator inequality, it is valid for any set of preparations {ρx₀x₁}, and thus F(A2)≥ ^s₄ A2−¹₂

+t. Note that (B1) has a dependence onM0,M1throughWx₀x₁. If (B1) holds for a particular choice of measurement operatorsM0,M1, then the bound onF(A2)holds for all preparations, for that particular choice ofM0,M1.

However, if (B1) holds forallpossibleM0,M1, then the bound onF(A2)is valid for all quantum setups and is thus a robust self-testing inequality. To derive the appropriate constantssandtx₀x₁, we first allowtx₀x₁andΛto have a dependence onM0

andM1. We then minimise overM0andM1the constantstx₀x₁, for a suitable choice ofs, such that, at the end, (B1) holds regardless of the choice of measurement operators.

We choose a dephasing channel of the form

Λθ(ρ) = 1 +c(θ)

2 ρ+1−c(θ)

2 Γ(θ)ρΓ(θ), (B3)

where for0≤θ≤π/4we useΓ =σ_x, while forπ/4< θ≤π/2we useΓ =σ_z. The functionc(θ)∈[−1,1]will be specified later.

In the interval0≤θ≤π/4, the action of the channel leads to K00= ¹1+σx

2 K01= ¹1+c(θ)σz

2 K10= ¹1−c(θ)σz

2 K11= ¹1−σx

2 , (B4)

whereas in the intervalπ/4< θ≤π/2, we have K₀₀= ¹1+c(θ)σ_x

2 K₀₁=¹1+σ_z

2 K₁₀= ¹1−σ_z

2 K₁₁=¹1−c(θ)σ_x

2 . (B5)

As discussed in the main text, for any given set of preparations, the optimal measurements are projective and rank-one.

Furthermore, any two such measurements can be represented on an equator of the Bloch sphere. Due to the freedom of setting the reference frame, we can without loss of generality represent the two measurements in thexz-plane, i.e.

M0= cosθ σx+ sinθ σz M1= cosθ σx−sinθ σz.

We can therefore writeW_x0x1 as W00=1

8cosθσx W01=1

8sinθσz W10=−1

8sinθσz W11=−1

8cosθσx. (B6) We can reduce the number of operator inequalities (B1) by exploiting the apparent symmetries in the expressions forW_x0x1 and Kx0x1: we restrict ourselves so thatto≡t01=t10andte≡t00=t11. Thus, we have to consider two operator inequalities in each intervalθ∈[0, π/4]andθ∈(π/4, π/2]. In the first interval, the two operator inequalities are

1 +σx

2 −s

8cosθσx−te11≥0 1 +c(θ)σz

2 −s

8sinθσz−to11≥0. (B7) In the second interval, the two operator inequalities are

1 +c(θ)σx

2 −s

8cosθσx−te11≥0 1 +σz

2 −s

8sinθσz−to11≥0. (B8) We now focus on the former interval. Solving the two inequalities fortoandtewe obtain

te≤1−s

8cosθ, to≤ 1

8(4 + 4c(θ)−ssinθ), (B9) te≤ s

8cosθ, to≤ 1

8(4−4c(θ) +ssinθ). (B10) Any choice oft_oandt_e satisfying these constraints gives rise to valid operator inequalities. In order to obtain the strongest bound, we choose the largest values oftoandteconsistent with their respective constraints, i.e.,

t_e= minn 1−s

8cosθ,s 8cosθo

t_o= min 1

8(4 + 4c(θ)−ssinθ),1

8(4−4c(θ) +ssinθ)

. (B11)

A similar procedure for the intervalθ∈(π/4, π/2]leads to t_e= min

1

8(4 + 4c(θ)−scosθ),1

8(4−4c(θ) +scosθ)

t_o= minn 1−s

8sinθ,s 8sinθo

. (B12)

It is worth pointing out that the two intervals only differ by exchangingte↔toandsinθ↔cosθ. Hence, for any givenθ, we have constructed operator inequalities of the form (B1).

As shown in the main text, we obtain our lower bound on the average fidelity from F(A2)≥ s

4(A2−1/2) + min

M₀,M₁t(M₀, M₁)≡L(A2), (B13)

wheret(M0, M1) = (te+to)/2. To compute this quantity we fix the value ofsto be s= 4 1 +√

2

(B14) and choose the dephasing function asc(θ) = min{1,₄^ssinθ}whenever θ ∈ [0, π/4], and c(θ) = min{1,^s₄cosθ} whenever θ∈(π/4, π/2]. It is easy to see thatc(θ)∈[0,1], which ensures thatΛθis a valid quantum channel, and thatc(θ)is continuous atθ=π/4. A simple calculation shows that in this case

t= 2−√ 2

4 , (B15)

which gives the lower bound

F(A2)≥ 1 +√

2

A2− 3 2√

2 ≡L(A2). (B16)

One can check that choosing distinct values ofswill not lead to improved lower bounds.

Appendix C: Tightness of fidelity bounds

In the main text, we have derived fidelity bounds for both the preparations and the measurements, based on operator inequali- ties. Specifically, we obtain a lower bound on the average fidelityFof the prepared states (with respect to the ideal ones) given by the linear expression

F(A2)≥ 1 +√

2

A2− 3 2√

2 ≡L(A2). (C1)

For measurements, a similar bound is obtained on the average fidelityF⁰with respect to the ideal ones. In the present appendix, we discuss the tightness of these bounds.

We start with our bound on the fidelity of the states. As discussed in the main text, obtainingA2=Q2impliesF = 1, i.e. the states are the ideal ones (up to a unitary). Let us refer to the optimal strategy (with the ideal states) as strategyS₁. Then, forA₂= C₂, our bound givesF ≥3/4. This bound is tight and can be obtained via the set of pure statesρ_x0x1 = (11+ (−1)^x0x1σ_z)/2 (diagonal in the same basis, hence classical), combined with the measurementsM0 =M1=σz. Let us refer to this strategy as S2.

The above shows that our bound (C1) is tight as far as linear inequalities are concerned. More generally, the bound is in fact tight in general, when shared randomness between the preparation and measurement devices is taken into account. In this case, taking a convex combination between strategiesS₁andS₂allows us to get any point on the line (i.e. pair of valuesFandA₂) betweenS₁andS₂.

It is also interesting to understand what happens when shared randomness between the devices is not taken into account. In this case, the end points (A2 =Q2,F = 1) and (A2 =C2,F = 3/4) can still be obtained. To understand what happens in the intermediate regionC2 <A2 < Q2, we first performed a numerical analysis. Specifically, we chose randomly four qubit states, and compute (i) the maximal value ofA₂(optimizing over the measurements), and (ii) the average fidelityF (where the optimization over channels is restricted here to unitaries). The resulting points are shown on Fig. 2 (blue circles). This indicates that forC2<A2 < Q2, the bound (C1) cannot be saturated anymore. Moreover, we conjecture that an optimal class of strategies is given by the pure states

|ψ00i=|0i |ψ11i=|1i |ψ01i= cosθ|0i+ sinθ|1i |ψ10i= cosθ|0i −sinθ|1i (C2) and the measurementsMy= cos(ϕ)σz+ (−1)^ysin(ϕ)σx. Straightforward calculations show that, takingtanϕ= sin 2θ, leads to

A₂= 1 2+1

4 q

1 + tan²(ϕ) and F= 1

4(3 + tanϕ). (C3)

This gives a parametric curve, as a function ofϕ∈[0, π/4], given by the red curve in Fig. 2. This curve is in excellent agreement with the numerical results obtained before. Note that this class of strategies interpolates between the strategiesS1(settingϕ= 0) andS2(settingϕ=π/4).

FIG. 2. The black line is the analytic lower bound on the average fidelityF(F⁰)for prepared states (measurements), as a function of the observed value ofA2. To characterise the region accessible via pure qubit strategies (i.e. without shared randomness), we perform numerics generating randomly sets of qubit preparations (blue circles and crosses); here we show the numerical results for the case of states, but similar results are obtained for the case of measurements. In the regionC2 <A2 < Q2, we conjecture that the class of strategies given in the text (corresponding to the red curve) are optimal; both forFandF⁰. Finally, the green dashed line is our conjectured upper bound on the average fidelity.

Next we discuss the bound on the average fidelity of measurements. As discussed in the main text, the linear boundF⁰(A₂)≥ L(A2)is optimal as far as linear inequalities are concerned. Moreover, when allowing for shared randomness the bound is tight in general forC2 ≤ A2 ≤ Q2. This is obtained by considering convex combinations of strategyS⁰₁(defined as the optimal strategyS1, up to a rotation ofπ/8around theyaxis; see below), and the following strategy (referred to asS3): takeM0 =σz

andM1=11, with the statesρ00=ρ01= (11+σz)/2andρ10=ρ11= (11−σz)/2.

Similarly to the case of states, we now consider the situation where shared randomness between the devices is not allowed.

Performing a numerical analysis similar to the one described above (except that measurements are no generated randomly), we observe that the accessible region (in terms ofF⁰vsA2) appears to be exactly the same as for the case of states (i.e. the blue region in Fig. 2). We conjecture that the lower bound is given by the following class of optimal strategies: take the measurements:

M0=σz and M1=ησx+ (1−η)11 (C4) with the states|ψ00i= cosθ|0i+ sinθ|1i,|ψ01i= cosθ|0i −sinθ|1i,|ψ10i = cosθ|1i+ sinθ|0i, and|ψ11i= cosθ|1i − sinθ|0i. Settingη= tan 2θ, we get

A2= cos²(θ) 2 +1

4 +sin²(2θ)

cos(2θ) and F =1

4(3 + tan(2θ)). (C5)

This gives a parametric curve, as a function ofθ∈[0, π/8], given by the red curve in Fig. 2. This curve is in excellent agreement with the numerical results obtained before. Also, this curve turns out to be exactly the same as the curve we obtained above for the case of states. Note that this class of strategies interpolates between the strategiesS₁⁰ (settingθ =π/8) andS₃(setting θ= 0).

Finally, note that the numerics also suggests that there is a linear upper bound on the average fidelitiesF(F⁰) as a function of A2(see Fig. 2); specificallyF ≤ _Q^1−Q2

2−3/4A2+^Q_Q^22−3/4

2−3/4and similarly forF⁰. It would be interesting to provide a proof of these upper bounds.

Appendix D: Operator inequalities for robust self-testing of measurements

In this section, we account for the detailed derivation of the lower bound on the average fidelity of the measurementsF⁰(A2).

The approach bears significant resemblance to the case of robustly self-testing preparations, as outlined in AppendixB.

We aim to derive operator inequalities of the form

Kyb({ρx₀x₁})≥sZyb+ty({ρx₀x₁})11, (D1)

where Zyb = ¹₈P

x₀,x₁ρx₀x₁δb,x_y and Kyb({ρx₀x₁}) = Λ^†[(M_y^b)^ideal]. For the sake of simplicity, we first apply a uni- tary channel to(M_y^b)^ideal align these operators with the eigenstates of σ_x and σ_z. Then, we adopt the same (unital, trace- preserving) channelΛas specified in the main text, with the same coefficients as used to robustly self-test the preparations:

c(θ) = min{1,^s₄sinθ}whenθ∈[0, π/4]andc(θ) = min{1,₄^scosθ}whenθ∈(π/4, π/2].

It is straightforward to see that, for any given pair of measurements, the optimal choice of preparations are four pure qubit states, such thatρ00andρ11, andρ01andρ10respectively, correspond to antipodal vectors on the Bloch sphere. Therefore, we can without loss of generality restrict to such preparations since these impose the weakest constraints on the measurements of our interest. We can therefore parametrise the preparationsρ_x0x1= 1/2 (11+m~_x0x1·~σ)by Bloch vectors

~

m00= [cosθ,0,sinθ] m~11=−[cosθ,0,sinθ] m~01= [cosθ,0,−sinθ] m~10= [−cosθ,0,sinθ]. (D2) ExpressingZybin terms of these preparations gives

Z00=1

8(11+ cosθσx) Z01=1

8(11−cosθσx) Z10=1

8(11+ sinθσz)σz Z11=1

8(11−sinθσz). (D3) Due to symmetries, we restrict ourselves so thatt_o ≡t₀₁ =t₁₀andt_e ≡t₀₀ =t₁₁. Thus, we have to consider two operator inequalities in each intervalθ∈[0, π/4]andθ∈(π/4, π/2]. In the first interval, the two operator inequalities are

1 +σx

2 −s

8(11+ cosθσ_x)−t_e11≥0 1 +c(θ)σz

2 −s

8(11+ sinθσ_z)−t_o11≥0. (D4) In the second interval, the two operator inequalities are

1 +c(θ)σ_x

2 −s

8(11+ cosθσ_x)−t_e11≥0 1 +σ_z 2 −s

8(11+ sinθσ_z)−t_o11≥0. (D5) Just as in AppendixB, we solve these inequalities fort_eandt_o, and choose the largest value compatible with the solutions. In the first interval, this gives

te= min 1

8(8−s−scosθ),s

8(cosθ−1)

to= min 1

8(4c(θ)−ssinθ−s+ 4),1

8(−4c(θ) +ssinθ−s+ 4)

. (D6) A similar procedure for the intervalθ∈(π/4, π/2]leads to

t_e= min 1

8(4c(θ)−scosθ−s+ 4),1

8(−4c(θ) +scosθ−s+ 4)

t_o= min s

8(sinθ−1),1

8(8−s−ssinθ)

. (D7) For any choice ofθ, we have constructed operator inequalities of the form (D1).

In order to obtain our lower bound onF⁰, we must minimise the quantityt(θ) = (t_e+t_o)/2for a specific choice ofs. In analogy with the procedure in AppendixD, we chooses= 4 1 +√

2

, which returnsmin_θt(θ) =−3/(2√

2). Hence, we have obtained the lower bound

F⁰(A2)≥ 1 +√

2

A2− 3 2√

2 =L(A2). (D8)

Appendix E: Self-testing all pairs of incompatible Pauli observables

Consider a generalisation of the2→1RAC, in which we introduce a bias on the score associated to certain inputs. Specifi- cally, whenever the game is successful, i.e.b=xy, the awarded score isq/2ifx0⊕x1= 0, and(1−q)/2ifx0⊕x1= 1, for someq∈[0,1]. The average score reads

A^q₂=1 2

X

x₀,x₁,y

r(x0, x1)P(b=xy|x0, x1, y), (E1) wherer(x0, x1) =q/2ifx0⊕x1= 0andr(x0, x1) = (1−q)/2ifx0⊕x1= 1. Note that forq= 1/2, we recover the standard 2→1RAC. Based on the quantityA^q₂, we will now see how to derive a self-testing condition for any pair of incompatible Pauli observables, i.e. any pair of non-commuting projective rank-one qubit measurements.

We start by expressingA^q₂for a quantum strategy:

A^q₂=1 2+1

4 X

x₀,x₁

r(x0, x1) tr (ρx₀x₁((−1)^x0M0+ (−1)^x1M1))≤ 1 2+1

4 X

x₀,x₁

r(x0, x1)λmax[(−1)^x0M0+ (−1)^x1M1].

(E2) Denotingµk =λmin

M0+ (−1)^kM1

andνk =λmax

M0+ (−1)^kM1

, fork= 0,1, we obtain A^q₂≤ 1

2 +1

8[q(µ0−ν0) + (1−q) (µ1−ν1)]. (E3) Following a derivation analogous to that appearing in AppendixAto obtain, we obtain

A^q₂≤1 2 +1

8 h

qp

β+α+ (1−q)p β−αi

, (E4)

whereβ = 2 tr M₀²+M₁²

−tr (M₀)²−tr (M₁)² andα = 2 tr ({M₀, M₁})−2 tr (M₀) tr (M₁). Treatingαandβ as independent variables, we obtain the largest value of the right-hand-side of Eq. (E4) by demanding that the derivative with respect toαequals zero, and checking that the second derivative is negative at this point. We obtain the optimality constraint

α= 2q−1

1−2q+ 2q²β. (E5)

Inserting this value back into Eq. (E4), we find an upper bound onA^q₂as obtained by independent variablesαandβ. It turns out that this bound can be saturated by the de facto coupled variablesαandβ. From Eq. (E4), it is clear that a necessary condition for optimality is to maximiseβ. This amounts to the observablesM0andM1being traceless and such thatM₀²=M₁²=11, leading toβ = 8. This implies that the observables represent projective rank-one measurements. Hence, we can writeMy =~ny·~σ where the Bloch vector satisfies|~ny|= 1. Hence, we haveα= 8~n0·~n1. Thus, Eq.(E5) becomes

~

n0·~n1= 2q−1

1−2q+ 2q², (E6)

which has a solution for any choice ofq. Note that settingq= 1/2reduces the above to~n0·~n1= 0which we recognise as the optimality constraint for the standard2→1random access code. In conclusion, for any pair of incompatible Pauli observables (characterised by the scalar product~n₀·~n₁), we have a gameA^q₂(whereqis chosen in order to satisfy the above equation), such that the maximal score can only be attained by using that specific pair of Pauli observables. We thus obtain a general class of self-tests for any pair of Pauli observables, corresponding to saturating the maximal quantum value ofA^q₂for a given value ofq:

A^q₂≤ 1 2

1 +p

1−2q+ 2q²

. (E7)

Appendix F: Self testing for theN→1random access code

In this appendix, we extend the results presented in the main text to self-test the preparations and measurements in anN →1 RAC. The latter is a straightforward generalisation of the2→1RAC considered in the main text. The input of the preparation device is a randomN-bit stringx≡(x1, . . . , xN), while the input of the measurement device isy∈ {1, . . . , N}. The average score is

AN = 1 N2^N

X

x,y

P(b=xy|x, y). (F1)

Considering qubit statesρ_x, and measurement observablesM_y, we get A_N = 1

2+ 1 N2^N+1

X

x,y

(−1)^xytr (ρ_xM_y). (F2)