Steering is an essential feature of non-locality in quantum theory

A physical theory is called non-local when observers can produce instantaneous effects over distant systems. Non-local theories rely on two fundamental effects: local uncertainty relations and steering of physical states at a distance. In quantum mechanics, the former one dominates the other in a well-known class of non-local games known as XOR games. In particular, optimal quantum strategies for XOR games are completely determined by the uncertainty principle alone. This breakthrough result has yielded the fundamental open question whether optimal quantum strategies are always restricted by local uncertainty principles, with entanglement-based steering playing no role. In this work, we provide a negative answer to the question, showing that both steering and uncertainty relations play a fundamental role in determining optimal quantum strategies for non-local games. Our theoretical findings are supported by an experimental implementation with entangled photons.

Proposition 1. The optimal quantum strategy for the game G (7) (achieving ω q (G (7) ) ≈ 0.782) violates the uncertainty principle -quantum game value correspondence, i.e., Alice is unable to steer Bob's system to the maximally certain states and vice versa.
In the proof, we will use the Lemmas 1 and 2 stated below.
Lemma 1 ( [1]). In the Bell scenario B(n, 2, 2) for any number of parties n, all extreme boxes P (a, b|x, y) of the quantum set Q(n, 2, 2) can be realized by measuring n qubit pure states with projective observables.
Lemma 2 (JORDAN'S LEMMA [2]). Any two binary observables A 1 and A 2 acting on a finite-dimensional Hilbert space C n with n ≥ 1 can be simultaneously block-diagonalized into 1 × 1 and 2 × 2 blocks.
(4) To find the optimum quantum value 1 4 λ max , we use the KKT conditions, i.e., we investigate the expression for λ in terms of α 1 and β 1 in four sectors.
where we have used π B (y = 0) = π B (y = 1) = 1 2 , and the uncertainty bounds are and with the optimal α 1 given in Supplementary Equation (11). Case I: (x,a) = (0,0). The maximally certain state for (x, a) = (0, 0) is Projecting onto the optimal state with the optimal projector Π x=0 a=0 = |+ +|, we see that Alice only manages to steer Bob's system to the state which achieves value 0.8446 < ξ for the corresponding uncertainty relation. Note that here we have stated the expression numerically simply to avoid the cumbersome notation associated with the exact analytical expression.
Case II: (x,a) = (0,1). The maximally certain state for (x, a) = (0, 1) is Projecting onto the optimal state with the optimal projector Π x=0 a=1 = |− −|, we see that Alice only manages to steer Bob's system to the state which achieves value 0.8446 < ξ (0,1) B for the corresponding uncertainty relation.
Case III: (x,a) = (1,0). The maximally certain state for (x, a) = (1, 0) is Projecting onto the optimal state with the projector Π x=1 a=0 , we see that Alice manages to steer Bob's system to the maximally certain state in Supplementary Equation (21) for this uncertainty relation.
We thus see that for the game G (7) , Alice is unable to steer Bob's system to the maximally certain states even for the non-trivial uncertainty relations corresponding to (x, a) = (0, 0) and (x, a) = (0, 1). In order to formally complete the argument for every optimal quantum strategy, we note that in the general case, one obtains a mixture of uncertainty relations over the outcomes i of the block index measurement by Alice and Bob. Since the uncertainty relation fails to be saturated in each block, this implies that the same holds true also in the convex mixture of uncertainty relations.
The quantum value of the game ω q (G (7) ) is thus lower than what could have been achieved if the non-locality of the theory were bounded by the uncertainty principle alone. The same holds true from the point of view of Bob steering Alice's system. While Bob is able to steer Alice's system to the maximally certain state for (y, b) = (1, 1), he is unable to do so for the non-trivial uncertainty relations corresponding to (y, b) = (0, 0) and (y, b) = (0, 1) as well as for the trivial uncertainty relation for (y, b) = (1, 0). Thus, this example proves that the non-locality of quantum theory is not determined by the uncertainty principle alone, and steering plays a definite role.

Supplementary Note 2 -Relation with the Schrodinger-Hughston-Jozsa-Wootters theorem
Let us remark upon a curious feature of the rewriting in Eq.(2) of the main text with regard to steerability of quantum systems. Consider a set of measurement operators M x a on Alice's side, i.e., positive operators M x a ≥ 0 satisfying a M x a = 1. Such a collection represents a positive-operator valued measure (POVM) for each x. For any fixed bipartite quantum stateσ AB , every measurement on Alice's side gives rise to an assemblage Here are the conditional (unnormalised) states of Bob's system prepared by Alice's measurement. We have that for every x, x ∈ X , in order to obey the no-signaling principle; i.e., without the knowledge of Alice's outcome a, Bob's state is independent of the measurement choice x. The well-known Schrödinger-Hughston-Jozsa-Wootters (SHJW) theorem [3,4] shows that every assemblage {P A|X (a|x),σ B a|x } a,x satisfying Supplementary Equation (24) has a quantum realization as in Supplementary Equation (23) for some quantum stateσ AB and for some set of measurement operators {M x a }. Now, the set of states {σ B a|x } achieving the maximum value of the uncertainty relations together with the optimal probabilities {P A|x (a|x)} forms an assemblage. One might then wonder whether the result of [5] is a direct consequence of the SHJW theorem, since Alice might steer to the assemblage corresponding to the maximally certain states. However, maximally certain states together with the optimal local probabilities P (a|x) do not guarantee that the no-signaling principle given by Supplementary Equation (24) holds. Thus, the UP-QGV correspondence found in [5] is a non-trivial property of the optimal states and measurements. It was posed as an open question in [5] whether the correspondence holds for all non-local games.

Supplementary Note 3 -Games vs. General Bell inequalities
As observed in the main text, it is crucial for the correspondence between uncertainty relations and optimal quantum strategy to be meaningful that the form of the Bell expressions is restricted, for instance to the form of non-local games as in the paper [5]. It is readily seen that if one allows an arbitrary freedom in rewriting the Bell expressions up to normalization and no-signaling equality constraints as suggested in [6], then one can always find a form of the Bell expression where the correspondence holds approximately, up to an arbitrarily small error. For instance, consider the following Bell expression, where the game has been supplemented with multiples λ x,y of the normalization constraints for input When λ x,y factorize as λ x,y = π A (x)β y , the resulting fine-grained uncertainty relation for fixed (x, a) is given as where we have used b P B|Y,X,A (b|y, x, a) = 1 for all y in the second term. Now clearly, the absolute value of the multipliers β y need to be bounded at least as |β y | ≤ 1 to be comparable with π B (y|x). Similar considerations hold also for the multipliers associated to the no-signaling constraints. Failing such restrictions, one might always choose appropriately large (in comparison with π B (y|x)) β y that lead to the saturation ofξ x,a in the uncertainty relation, up to an arbitrary small deviation. Even otherwise, the artificial addition of normalization and no-signaling constraints which are satisfied by all boxes in the set, leads to the question whether the resulting saturation of the uncertainty relations is intrinsic to the non-local correlations that maximally violate the inequality. To avoid such mathematical sleight of hand (which is also inherent in questions such as that of unbounded violation of Bell inequalities [7], the unique games conjecture [8], etc.) we follow [5] in restricting to non-local games, i.e., where P A,B|X,Y (a, b|x, y) appear in the Bell expression only with non-negative coefficients and all non-zero coefficients are equal (to π A,B (x, y)) for a fixed input pair (x, y). Note that this difference between non-local games and general Bell inequalities has also been noted previously in [9].

Supplementary Note 4 -Constructing counter-example games for all non-maximally entangled states
In the previous sections, we have seen an example of a non-local game with the optimal state being a nonmaximally entangled two-qubit state, for which the UP-QGV correspondence breaks down. In this section, we prove that this is not a one-off instance, indeed for every non-maximally entangled two-qudit state |ψ ∈ C d ⊗ C d (for an Hilbert space of arbitrary dimension d), one can construct a two-player game G ψ such that |ψ is optimal for G ψ and such that the UP-QGV correspondence does not hold for G ψ . We present the construction in this section as the proof of the following Proposition from the main text. The construction we use resembles that used by Coladangelo et al. in [11] to show that all bipartite pure entangled states can be self-tested.
Proposition 3. For any two-party entangled, but nonmaximally entangled, state |ψ ∈ C d ⊗ C d for arbitrary Hilbert space dimension d, there exists a game G ψ for which the optimal quantum strategy is given by suitable measurements on |ψ , and such that the correspondence between the uncertainty principle and the quantum game value does not hold for G ψ .
The corresponding uncertainty relations are given as where ξ (x,a) B are the bounds on the uncertainty expressions. We find that while the first two inequalities above are saturated by the optimal quantum strategy, the third and fourth inequalities fail to be saturated except when θ = π 4 , i.e., for the maximally entangled state |ψ π 4 . For the third expression, i.e., when (x, a) = (1, 0), the bound is with the maximally certain state being |+ = 1 √ 2 (|0 + |1 ). On the other hand, we see that for (x, a) = (1, 0) using the optimal strategy Alice steers Bob's state to |ψ (1,0) = cos θ|0 + sin θ|1 , so that only for the maximally entangled state (θ = π 4 , β = 0) does Alice manage to steer Bob's system to the least uncertain state. Similarly for the case (x, a) = (1, 1), the bound is with the maximally certain state being |− = 1 √ 2 (|0 − |1 ). On the other hand, for (x, a) = (1, 1) using the optimal strategy Alice steers Bob's state to |ψ (1,1) = cos θ|0 − sin θ|1 . So that it is again only for the maximally entangled state that Alice manages to steer Bob's system to the least uncertain state. Thus, the tilted CHSH inequality of [10] expressed as a game, shows that every non-maximally entangled two-qubit state serves as the optimal state for a game in which the uncertainty principle -quantum game value correspondence does not hold.
It now remains to generalize the result even further, to all two-qudit states that are non-maximally entangled. Consider a general two-qudit non-maximally entangled state, written as with the Schmidt coefficients λ i ∈ R obeying 0 < λ i < 1 for all i and i λ 2 i = 1 with not all λ i equal to 1 √ d . We first deal with the case when d is even. The idea is to design a game with d-outcome measurements on each side, such that the correlation tables for some measurement settings are block-diagonal with blocks of size 2×2 each. The j-th 2 × 2 block will correspond to a tilted CHSH game that is maximally violated by a two qubit state, which is a normalized version of λ 2j−1 |2j − 1, 2j − 1 + λ 2j |2j, 2j , with j = 1, . . . , d/2. Accordingly, we construct a game with two inputs x = 0, 1 for Alice, and 2d + 2 inputs y = 0, . . . , 2d + 1 for Bob, with d outputs each. Given |φ , the ratios λ2j λ2j−1 for j = 1, . . . , d/2 determine the game as follows. For inputs x = 0, 1 and y = 0, . . . , d/2 + 1, the players play a set of d/2 tilted CHSH games determined by the following procedure.
To complete the construction, we now consider another set of d/2 tilted CHSH games with parameters β 2j , this time played by Alice and Bob on the inputs x = 0, 1 and y = d/2 + 3j − 1, d/2 + 3j, d/2 + 3j + 1 for j = 1, . . . , d/2. It remains to specify the input distributions, these are given with τ : With the above input distributions, we can now directly calculate the value achieved by a quantum strategy given by the shared state |φ and observables z . In the j-th 2 × 2 sector, the strategy achieves a value of (λ 2 2j + λ 2 2j−1 ) ω * q τ , so that summing over all j = 1, . . . , d/2, we obtain the quantum value of our generalized tilted CHSH game to be by virtue of the fact that i λ 2 i = 1. Let us now verify that this is in fact the optimal quantum value of the game CHSH gen-tilt . This is seen by the fact that the game decomposes into 2 × 2 blocks, and the maximum quantum value within each block is ω * q τ , obtained from ω q (CHSH tilt ) presented earlier. Moreover, we see that the uncertainty relations fail to be saturated within each 2 × 2 block, except those which correspond to λ 2j = λ 2j−1 , from the results for the qubit case. Finally, the case for odd d works in a very similar manner to that for even d, we use the generalized tilted CHSH game corresponding to d − 1 which is even, and augment the game with a 1 × 1 block, i.e., the entries P A,B|X,Y (d, d|x, y) for x, y ∈ {0, 1} 2 and x = 0, 1, y = 2d, 2d + 1. Similarly, we augment the observables with the projector |d d|, i.e., z ⊕ |d d| . While the uncertainty relation corresponding to the 1 × 1 block is saturated, for all non-maximally entangled states, the uncertainty relations for the 2 × 2 blocks are not, as we have seen in the even d case. Thus, we have constructed for every d ≥ 2, a non-local game with the optimal strategy being given by the state |φ = i λ i |i, i and such that the correspondence between the uncertainty principle and the quantum game value is broken.

Supplementary Note 5 -Experimental implementation
In the experiment, we used single photon's polarization state as the physical qubit. To maximally violate the Bell inequality given in Eq. (5) of the main text, we prepare the following polarization entangled two-photons state, This state is produced as follows, firstly, we prepare maximally entangled photons pairs (see Fig. (3) of the main paper). For this an Ultraviolet light centered at wavelength of 390 nm was focused onto two 2 mm thick β barium borate (BBO) nonlinear crystals placed in cross-configuration to produce photon pairs emitted into two spatial modes a and b through the second order degenerate type-I spontaneous parametric downconversion (SPDC) process [12]. Any spatial or temporal distinguishability between the down-converted photons is carefully removed through quartz wedges placed in the pump beam (not shown in the figure) and a pair of Y V O 4 crystals located in each of the down-converted beams. The emitted photons were passed through the narrow-bandwidth interference filters (IF) (∆λ = 3 nm) and coupled into 2 m single mode optical fibers (SMF) to secure well defined spatial and spectral emission modes. Secondly, to prepare the desired state as outlined in [13], the pump polarization is altered to produce the state cos θ p |HH − sin θ p |V V with θ p = 31.50 • . Then, as the final step this state is rotated to |Ψ by the use of a half wave-plate (HWP) placed after the output fiber coupler in each of the mode (a) and (b) at an angle of 39.69 • and 84.69 • respectively.

Supplementary Note 6 -State tomography
To estimate the fidelity of the two-photon prepared state (ρ exp ) with respect to the ρ th = |Ψ Ψ|, we carried out quantum state tomography as described in Ref. [14], and where we have measured each of the two photons in three mutually unbiased basis (H/V, D/A, L/R). These polarization measurements were performed by using HWPs, quarter wave plates (QWP) and polarizing beam splitters (PBS) followed by single photon detectors (actively quenched Si-avalanche photodiodes (Si-APD)). An FPGA based timing system is used to record the number of coincidence events with a detection time window of 1.7 ns. For each setting, we have obtained approximately 1.6 Million events.
The obtained density matrix of the prepared state is The real and imaginary parts of this density matrix are shown in Supplementary Figure 1. From experimentally collected data, we reconstructed the density matrix ρ exp by considering maximum likelihood estimation, as described in Ref. [14]. Our figure of merit to quantify effectiveness of state reconstruction is the relative fidelity with respect to the theoretical state |Ψ . Errors for fidelity reconstruction have been estimated by taking into account the Poissonian statistical distribution of the photon number counting (see Section VI, Ref. [14]).
We obtained the following state fidelity

Supplementary Note 7 -Steering and Bell Violation
Alice and Bob can perform Bell test on experimentally prepared state with the following settings where γ = 4.794814.
To perform Bell test and the tomography of the steered state, Bob randomly chooses if he wants to realize tomographic measurement or the Bell test settings.

a b
Supplementary Figure 1. (Color online). Experimental density matrix. a Real and, b imaginary part, shown in computational basis (HH, HV,VH and VV), of the experimentally obtained elements of the two qubit density matrix ρExp, constructed using maximum likelihood quantum state tomography.
Supplementary For the Bell test we obtained ω q (G (7) ) = 0.7770 ± 0.0002, when the corresponding theoretical value is 0.78221. In the following, we report theoretical and the experimentally obtained density matrices of Bob states when Alice projected her photon.

Supplementary Note 8 -Error estimation
We have considered error originated from the measurement side only, as the error on the preparation side will just shift the experimentally prepared state away from the desire state and therefore will be apparent from the reported state fidelity or the value of Bell inequality violation.
To estimate the error in our experiment, we have considered errors due to cross-talks-originated from the PBS extinction and absorption-wave-plate setting errors, wave-plates offset error, wave-plates retardance tolerance and error due to Poissonian statistics of the incoming photons. Cross-talk is considered here as the used PBSs were not perfect. To calculate the PBS extinction, we have carefully estimated the extinction ratio of the PBSs used on both sides (Alice and Bob) with their transmission and absorption for each polarizations.
The wave-plate setting error is considered as one has to switch the settings during collecting data for the estimation of the state fidelity. In the experiment, we used motorized stages to rotate the wave-plates to switch among the different settings. These mounts have repeatability of less than 0.02 • . Therefore, for error estima-tion, we assumed that the wave-plates setting error has normal distribution with standard deviation of 0.02 • .
A normally distributed offset error of 0.1 • in setting the wave-plates is also assumed as the zero of a given wave-plate could not be adjusted better than 0.1 • .
The wave-plates retardance tolerance of λ 300 is also taken into account by assuming a normally distributed retardance error in each of the wave plate used. Note that, among all these errors, wave-plates retardance error is leading and it is over estimated as it is fixed with each wave-plate chosen for the experiment, moreover, we are carefully characterizing wave plates which we have not assumed here. Finally, we considered errors arising due to the photon counts following the Poissonian statistics.