Experimental detection of steerability in Bell local states with two measurement settings
Steering, a quantum property stronger than entanglement but weaker than non-locality in the quantum correlation hierarchy, is a key resource for one-sided device-independent quantum key distribution applications, in which only one of the communicating parties is trusted. A fine-grained steering inequality was introduced in [PRA 90 050305(R) (2014)], enabling for the first time the detection of steering in all steerable two-qubit Werner states using only two measurement settings. Here we numerically and experimentally investigate this inequality for generalized Werner states and successfully detect steerability in a wide range of two-photon polarization-entangled Bell local states generated by a parametric down-conversion source.
The notion of non-locality was first introduced in 1935 by Einstein, Podolsky and Rosen Einstein et al. (1935) by discussing a “spooky action at a distance”. This led Schrödinger to introduce the concept of steering, in his response Schrödinger (1935, 1936) to the EPR paper. The term steering describes a property of quantum mechanics that allows the subsystems of a bipartite entangled state to affect each other’s state upon measurement: indeed the measurement of one of the subsystems can project (or steer) the other subsystem in a particular state, which depends on the chosen measurement on the first subsystem. In 2007, this concept was reformulated in terms of a quantum information task by Wiseman, Jones and Doherty Wiseman et al. (2007), which allowed them to establish a strict hierarchy between quantum correlations of increasing strength: entanglement, steering and Bell non-locality. Seen in this light, steering is the resource that allows Alice to win a game in which she tries to convince Bob (who does not trust her) that she can prepare an entangled state and share it between them.
As was shown in 2012 by Branciard et al. Branciard et al. (2012), this resource can be exploited for quantum cryptography in a one-sided device-independent quantum-key-distribution (1SDI-QKD) scenario, intermediate between standard QKD in which both parties need to trust their measurement apparatus and device-independent QKD (DI-QKD) Ekert (1991); Acín et al. (2007) in which neither does: for 1SDI-QKD, only one of the parties needs a trusted measurement device. Compared to the DI-QKD scenario, this additional constraint of one trusted device must be fulfilled but, in return, the security can be based on the violation of a steering inequality rather than a Bell inequality, which lowers significantly the experimental requirements in terms of detection and transmission efficiencies Wittmann et al. (2012); Bennet et al. (2012); Smith et al. (2012) because, in particular, the detection loophole needs to be closed only on Alice’s side and the noise tolerance is higher.
In order to fully exploit this resource, practical and efficient ways of detecting steerability in experimentally-relevant bipartite states are needed. Over the last ten years, a lot of different steering inequalities have been proposed Cavalcanti et al. (2009); Walborn et al. (2011); Schneeloch et al. (2013); Skrzypczyk et al. (2014); Cavalcanti et al. (2015); Kogias et al. (2015); Zhu et al. (2016) and experimentally tested Saunders et al. (2010); Smith et al. (2012); Bennet et al. (2012); Wittmann et al. (2012); Kocsis et al. (2015); Wollmann et al. (2016); Sun et al. (2016) (for a recent review focused on semi-definite programming (SPD) see Ref. Cavalcanti and Skrzypczyk (2017)), however they are generally subject to some trade-off between the number of required measurement settings and their robustness to noise. In particular, they require at least three measurement settings to detect steering in Bell-local Werner states Saunders et al. (2010); Werner (1989).
In this article, we investigate numerically and experimentally a recently proposed steering inequality Pramanik et al. (2014) based on fine-grained uncertainty relations Oppenheim and Wehner (2010), which is more efficient in that it allows the detection of steerability in a large range of noisy two-qubit states, in particular in all steerable Werner states, with two measurement settings. We first recall the fine-grained inequality in Section I. In Section II we apply the inequality to generalized Werner states, give a procedure to correctly choose the measurement settings and compare its performance with coarse-grained inequalities. Finally in Section III we illustrate the procedure with a simple experiment.
I Fine-grained steering inequality
Let us first recall the fine-grained steering inequality introduced in Ref. Pramanik et al. (2014) and the underlying game scenario. Alice prepares a bipartite state , keeps the part labeled for herself and sends the part labeled to Bob. Bob then asks Alice to steer to any eigenstate of an observable , randomly chosen in the set (with and maximally incompatible). Alice measures with an observable (with if and if ) and sends her measurement outcome to Bob. Bob finally measures with the previously chosen observable and gets an outcome . After repeating these steps a large number of times, Bob is convinced that Alice can indeed steer his subsystem (i.e. that is steerable) if the following inequality is violated:
where is the conditional probability of Bob getting the outcome upon measurement of or when Alice announces the outcome , and range over all possible maximally incompatible measurements, and is the probability of obtaining an outcome upon a quantum measurement on a quantum system described by a hidden variable . The left-hand side of the inequality is the steering parameter which can be estimated from measurement results. The right-hand side is calculated by considering a local hidden state (LHS) model for : this model corresponds to the cheating strategy of a dishonest Alice who prepares a local state instead of an entangled state and announces values of that do not correspond to actual quantum measurements. As was shown in Ref. Pramanik et al. (2014), if Alice knows beforehand the set of Bob’s possible measurement settings (“scenario I”), then . If, however, Alice does not know the set before preparing (“scenario II”), then .
Ii Numerical analysis for generalized Werner states
In the following, we study the aforementioned inequality in both scenarios for states of the form:
where is a pure state, , is the identity matrix and subscripts refer to Alice’s and Bob’s qubit respectively. These states model well polarization-entangled two-photon pairs produced e.g. by bulk type-I spontaneous parametric down-conversion sources Kwiat et al. (1999) (see the dashed-line box in Fig. 1) where a laser beam, with a linear polarization adjusted by a half-wave plate (HWP) set at an angle , pumps two adjacent nonlinear crystals having optical axes orthogonal to each other. Then corresponds to the amount of pairs emitted in the first crystal (horizontally-polarized photons, denoted by ) with respect to those emitted in the second crystal (vertically-polarized photons, denoted by ); is the phase between the two possible pair emission processes; is linked to the amount of unpolarized noise present in the set-up (background light, fluorescence of the optical elements…); and is linked to a dephasing noise accounting for the partial distinguishability between the optical modes of photons emitted by each crystal.
These expressions, for given values of and , give lower bounds on the value of for which is entangled or Bell non-local (see black and red lines in Fig. 2). In particular, for Werner states Werner (1989) (i.e. and ), the state is entangled if and only if and it is Bell non-local if . Note that here we use the CHSH inequality to distinguish between Bell non-local states and states admitting a LHV model (which we call Bell-local here) since we restrict Alice and Bob to only two projective measurement settings Acín et al. (2006). This would not be valid for an arbitrary number of projective measurements since, in that case, it has been shown Acín et al. (2006) that, for Werner states, the separation occurs for , being Grothendiek’s constant of order 3 for which the best known lower Brierley et al. (2016) and upper Hirsch et al. (2017) bounds up to now are , i.e. . Hence, strictly speaking, we are sure that a Werner state is Bell-local only if .
To compute the steering parameter defined in Eq. (1), any combination can be chosen with but some choices may give larger values than others depending on the asymmetry of the state, thus we will define a more general steering parameter . Writing Bob’s and Alice’s measurement settings as (with and , and the Pauli matrices), and imposing and such that and are maximally incompatible measurements, can be expressed as:
where , (), and (), and denotes the sign of (see Appendix A for details). The angles , , and are optimized so as to maximize once and have been fixed. For scenario I, Bob’s measurements and are fixed to and respectively, thus with and . For scenario II, and must be chosen such that is minimized: . Note that for Werner states (i.e. and ) which are symmetric states, any choice of maximally incompatible settings and will give the same value of ; however for this is not the case and the minimization must be done in order to avoid overestimating (see Appendix B). This constraint comes from the requirements that and should be unknown by Alice when she prepares .
Equation (LABEL:Eq:RhoSteeringPr) gives a lower bound on for the state to be steerable, according to scenario I or II (see blue and green lines in Fig. 2). For the particular case of Werner states, using scenario I, we find a lower bound of for , the same as for Bell non-locality detected by the CHSH inequality and the same as for all coarse-grained steering inequalities with two measurement settings Cavalcanti and Skrzypczyk (2017). However, using scenario II, the theoretical bound Wiseman et al. (2007) is reached with only two measurement settings. Note that we can even conjecture that scenario II gives an optimal lower bound for generalized Werner states (i.e. for any ). Indeed Fig. 3 shows that this bound (dark green full line) lies between a lower bound for steerable states (light blue dotted line) and an upper bound for states with a LHS model (light green dashed line) that were both calculated numerically in Ref. Hirsch et al. (2016), respectively with a semi-definite program Skrzypczyk et al. (2014) and with a new iterative method for constructing LHS models Hirsch et al. (2016); Cavalcanti et al. (2016). We can also notice that, for both scenarios, the set of steerable states detected with two settings is strictly larger than the set of Bell non-local states seen by CHSH: for , the lower bound on for steering is strictly lower than the one for Bell non-locality. In particular, even for scenario I, Bell local steerable states can be detected for a large range of values of .
Iii Experimental implementation
We have experimentally tested this fine-grained steering inequality in both scenarios with a commercial source of polarization-entangled two-photon states (QuTools QuT ) based on the scheme of Ref. Kwiat et al. (1999) (see Fig. 1). The projective measurements corresponding to settings on Alice’s photon and on Bob’s photon, with (i.e. in the plane of Bloch’s sphere), are implemented with a polarizer whose axis is set at an angle (projection on the eigenstate of ) or (projection on the eigenstate of ) with respect to the vertical direction. Note that measurements are not needed when or (see Appendix A).
We first characterized the experimental state with visibility measurements () in the () basis, for different values of (Fig. 4(a)). Modeling the state with Eq. (2) and using the relation , one can show that and (see Appendix A). We thus deduced a noise parameter , a dephasing parameter and a phase .
In Fig. 4(b) we show the measured value of the Bell parameter , with , and its theoretical value given by Eq. (4), for the optimized measurement settings Gisin (1991) shown in Fig. 4(c). The state violates the CHSH inequality and is thus non-local for , with a maximal value of for . The corresponding measured values of the steering parameter in both scenarios and are shown in Fig. 4(d). For scenario II, Bob’s measurement angles have been set to and , which minimize Eq. (LABEL:Eq:RhoSteeringPr) for (for any value of , and , see Appendix B). For both scenarios, Alice’s settings have been optimized (with ) for each value of so as to maximize and are shown in Fig. 4(e). The state violates the fine-grained steering inequality (Eq. (1)) in scenario I for and in scenario II for . For scenario I, the maximum value of is obtained for and . For scenario II, the maximum is for .
In Fig. 5, we have plotted the measured values of and against , together with the simulations for and . We can identify five main zones in this plot, corresponding to quantum correlations of different strength. The shaded red area corresponds to Bell non-local states that violate the CHSH inequality (i.e. ) and also violate the fine-grained steering inequality. The blue (green) area corresponds to steerable Bell local states detected in scenario I or II: or , and . The white area corresponds to states that violate neither the CHSH inequality nor the fine-grained steering inequality. Finally the grey area corresponds to states that are Bell non-local but unsteerable which is in contradiction with the established hierarchy of quantum correlations Wiseman et al. (2007); none of our measurement results fall in this zone.
In conclusion, we have numerically and experimentally investigated the performance of the fine-grained steering inequality of Ref. Pramanik et al. (2014) for the detection of steerability in experimentally-relevant two-photon states which can be modeled as generalized Werner states with dephasing. We have shown that contrary to other steering inequalities which are coarse-grained Cavalcanti and Skrzypczyk (2017) and require strictly more than two measurement settings to detect steerability in Bell local states Saunders et al. (2010), this fine-grained inequality is able to detect a much larger set of steerable states with only two measurement settings, in particular all steerable Werner states and (most probably) all steerable generalized Werner states. We have also shown that even using the most conservative LHS bound of scenario I, the inequality allows the detection of steerability in states that do not violate the Bell-CHSH inequality. Finally, for scenario I, a key rate for 1SDI-QKD has been proven in Ref. Pramanik et al. (2014) against individual eavesdropping attacks; with our best value of , a rate secure bit per photon detected by Bob could thus be achieved. It will be interesting to extend this calculation to scenario II, which allows for a better noise tolerance.
We thank Ioannis Touloupas for his help with the data acquisition, and Nicolas Brunner and Flavien Hirsch for kindly providing numerical data for steering and LHS bounds in Ref. Hirsch et al. (2016). We acknowledge financial support from the BPI France project RISQ, the French National Research Agency projects QRYPTOS and COMB, and the Ile-de-France Region project QUIN. The work of MK is supported in part by EPSRC grant number EP1N003829/1 Verification of Quantum Technology.
Appendix A Detailed analytical calculations for generalized Werner states and optimal measurement angles
a.1 Projective measurement outcomes
When applying to the generalized Werner state defined in Eq. (2) a projective measurement described by an observable (i.e. a measurement setting) , with
where , and are the Pauli matrices and , the computational basis states are transformed as:
and the probabilities of the four possible combinations of outcomes for and for are:
The correlation function of this measurement is
a.2 Visibilities and
The visibility is in the basis and in the basis.
a.4 Bell-CHSH parameter
The Bell parameter for the CHSH inequality Clauser et al. (1969) is given by:
where are four measurement settings described by Eq. (6) with (although one could use different values of in case ).
The optimal measurement angles do not depend on the value of and can be calculated as in Ref. Gisin (1991): fixing () and (), the optimal measurement angles for Bob (that maximize ) are:
and the corresponding maximal value of the Bell parameter (Eq. (4)) is:
a.5 Fine-grained steering parameter
The fine-grained steering parameter given in Eq. (LABEL:Eq:RhoSteeringPr) of the main text is calculated as follows:
with and , with and .
Thus, if we impose that and (one possible choice for two maximally incompatible measurement operators and for Bob), we have, for any choice of :
is maximized for
which gives, for :
and is maximized for
which gives, for :
with , and .
We remark that
for and or for and ,
for and or for and ,
for and for ,
for and for ,
for and for .
Thus, with , we obtain the general expression of the fine-grained steering parameter (Eq. (LABEL:Eq:RhoSteeringPr)):
for the optimal angles (for )
or (for )
All the analytical expressions of optimal angles were calculated with Wolfram Alpha Wol .
Appendix B Bob’s choice of measurement settings and noise bounds for Scenario II
In Fig. 6, we plot the value of as a function of the measurement angle and the noise parameter for different values of , with . For these plots, and have been optimized as above and . We see that for a given value of , the minimum value of is obtained for , except for for which every choice of gives the same value of for a given value of . The steering bound on as a function of (for Scenario II) given in Fig. 2 and 3 corresponds to the value of for which with .
Appendix C Experimental probabilities and error bars
The joint probabilities are experimentally estimated from coincidence counting of single-photon detection events in both avalanche photodiodes of the source presented in Fig. 1. For each joint measurement setting , four such coincidence counts are recorded with Alice’s and Bob’s polarizers set at the angles , , and , corresponding to the measurement outcomes , , and respectively. Thus the experimental coincidence probabilities of obtaining the outcomes and for the measurement setting are:
Thus, the visibilities are given by:
The CHSH parameter is given by: