Abstract
Bell nonlocality between distant quantum systems—that is, joint correlations which violate a Bell inequality—can be verified without trusting the measurement devices used, nor those performing the measurements. This leads to unconditionally secure protocols for quantum information tasks such as cryptographic key distribution. However, complete verification of Bell nonlocality requires high detection efficiencies, and is not robust to typical transmission losses over long distances. In contrast, quantum or Einstein–Podolsky–Rosen steering, a weaker form of quantum correlation, can be verified for arbitrarily low detection efficiencies and high losses. The cost is that current steeringverification protocols require complete trust in one of the measurement devices and its operator, allowing only onesided secure key distribution. Here we present measurementdeviceindependent steering protocols that remove this need for trust, even when Bell nonlocality is not present. We experimentally demonstrate this principle for singlet states and states that do not violate a Bell inequality.
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Introduction
Entanglement provides a fundamental resource for a range of quantum technologies, from quantum information processing to enhanced precision measurement^{1,2,3,4}. In particular, the strong correlations inherent in shared entanglement—for example, between two parties—allows secure messaging and quantum information transfer, potentially over long distances^{1,5}. At the same time, the strong restrictions of quantum measurement theory prevents the extraction of useful information when an adversary has access to only one of the entangled systems^{6,7,8}. Furthermore, any adversary measuring one or more of the entangled systems reveals their presence to the communicating parties.
When correlations due to quantum entanglement are sufficiently strong, they allow the unconditionally secure sharing of a cryptographic key between two distant locations, without requiring any trust in the devices used or in the observers reporting the results^{9}. They also allow generation of unconditionally genuine randomness, again with no trust in the devices used or their operators^{10,11}. The corresponding verification protocols are thus deviceindependent, and can be put in the form of a ‘Bellnonlocal game’, played between a referee and two untrusted parties, which can be won by the latter only if they genuinely share a Bellnonlocal quantum state (Fig. 1a)^{12}, that is, an entangled state that violates a Bell inequality.
There are, however, practical difficulties in entanglement verification via Bellnonlocal games. Even if the entanglement is strong enough (compared with noise) to otherwise violate a Bell inequality, there may be too many null measurement results for unconditional verification—arising, for example, from detector inefficiencies or the typical transmission losses involved in implementations over long distances. Too many null results will make it impossible even for ‘honest’ devices to win a Bellnonlocal game. This is the wellknown ‘detection loophole’^{13}.
A promising alternative is based on a different test of nonlocality, called Einstein–Podolsky–Rosen (EPR) steering (or quantum steering). First identified by Erwin Schrödinger^{14}, and present in the EPR paradox^{15}, this corresponds to being able to use entanglement to steer the state of a distant quantum system by local measurements, and is strictly weaker than Bell nonlocality^{16,17}. Further, the detection loophole can be circumvented in the verification of steering, if the device and operator for one of the two entangled systems is completely trusted by the referee^{18,19,20} (Fig. 1b). This leads to the real possibility of onesided deviceindependent secure key distribution that is robust to both detector inefficiency and transmission loss^{21}. Unfortunately, however, an unconditionally secure protocol cannot rely on trust at all, even in one side.
Very recently, work on entanglement verification by Buscemi^{22} has been generalized to show that EPR steering can in fact be verified in the absence of trust in either side, via quantumrefereed steering (QRS) games^{23}. In comparison with Bellnonlocal games, the referee still sends classical signals to one party, but sends quantum signals to the other party (Fig. 1c). The quantum signals must be chosen such that they cannot be unambiguously distinguished, to prevent the possibility of cheating. Until now, only an existence proof for such games was known, with no explicit means of construction^{23}. For the case of entanglement witnesses, a recent measurementdeviceindependent protocol and demonstration has addressed a similar question^{24,25}, although EPR steering, Bell nonlocality and calibration of the quantum signals (see below) were not considered.
In this paper we give the first explicit construction of a QRS game, for the trustfree verification of steering entanglement. We also demonstrate a proofofprinciple implementation, for optical polarization qubits, in a scenario where no Bell nonlocality—as tested by the Clauser–Horne–Shimony–Holt (CHSH) inequality^{26}—is present. The results open the way to measurementdeviceindependent key distribution protocols that do not require Bell nonlocality, and which can circumvent the detection loophole.
Results
Quantumrefereed steering game
Consider the following scenario (Fig. 1c). On each run the referee, whom we shall call Charlie, chooses at random a pair of numbers labelled by k≡(j, s), with jε{1, 2, 3} and s=±1. Charlie sends Alice the value of j as a classical signal, and sends Bob a qubit in the seigenstate of the Pauli spin observable , that is, the state . The referee requires Alice and Bob to send back classical binary signals, a=±1 and b=0 or 1, respectively. The referee uses their reported results over many runs to calculate the payoff function
where ‹·›_{j, s} denotes the average over those runs with k=(j, s). Here r≥1 is a parameter that indicates how well the referee can prepare the desired qubit states , with r=1 for perfect preparation (see Methods section). Alice and Bob win the game if and only if P(r)>0.
In this QRS game Alice and Bob are allowed to plan a joint strategy beforehand, but are not allowed to communicate during the game. The latter could be enforced via spacelike separated measurement regions. The game is measurementdeviceindependent, because Charlie makes no assumptions about how Alice and Bob generate their values of a and b—they and their devices are untrusted. Remarkably, Alice and Bob cannot cheat—they are only able to win the game if Alice is genuinely able to steer Bob’s state (see Methods section).
For example, suppose that Alice and Bob share a twoqubit Werner state, , where 0≤W≤1 and Ψ^{−}›^{AB} denotes the singlet state^{27}, and adopt the following strategy: on receipt of signal j Alice measures , while Bob measures the projection operator Ψ^{−}›^{BC}‹Ψ^{−} onto the singlet state in the twoqubit Hilbert space spanned by his system and . It is straightforward to calculate that the corresponding theoretical value of the payoff function in equation (1) is
Hence Alice and Bob can, in principle, win the game whenever . This condition is in fact necessary for them to be able to win the game with a shared Werner state (see Methods section), and therefore the above strategy is optimal.
A QRS game can be constructed for every quantum state that is EPR steerable (see Methods). Quantum signals must be sent to Bob if it is to be verified that Alice can steer Bob’s state, and to Alice if the converse is to be verified. Note this directionality of EPR steering is nontrivial: for some quantum states only oneway EPR steering is possible^{28,29,30}.
Measurementdeviceindependent verification of EPR steering
We experimentally verified deviceindependent EPR steering using our quantumrefereed game. Alice and Bob’s shared state, and the states sent by Charlie to Bob, were encoded in photon polarization qubit states. The payoff function P(r) was calculated via singlequbit measurements and a partial Bellstate measurement (BSM), all using linear optics and photon counting (Fig. 2)^{31}.
The QRS game requires two components: the entanglement shared between Alice and Bob and the qubit encoding the state ω_{k} from Charlie. These were generated using a degenerate 820 nm polarization entanglement source, and a heralded singlephoton source at 820 nm, respectively. To play the game, different measurement apparatus is required for Alice and Bob: Alice only makes singlequbit measurements, while Bob implements a twoqubit measurement between his half of the entangled pair, and the incoming qubit from Charlie. This twoqubit measurement was implemented using a partial BSM device. Thus, Bob implemented projections onto the singlet subspace Ψ^{−}›^{BC}‹Ψ^{−} (corresponding to the outcome b=1) and the triplet subspace (−Ψ^{−}›^{BC}‹Ψ^{−}) (corresponding to b=0), of the twoqubit Hilbert space spanned by his and Charlie’s systems (see Methods). However, the inner workings of Alice and Bob’s apparatus need not be known, because the protocol is measurementdeviceindependent.
In particular, a key innovation of our protocol is that the payoff function P(r) in equation (1) cannot present ‘false positives’ of EPR steering. Alice and Bob do not have to be trusted and can try to cheat by any means, provided that they cannot communicate during the demonstration. This result requires r≥r* in the payoff function P(r) to assure measurementdevice independence, where r* characterizes the quality of Charlie’s preparation of the states he sends to Bob (r*≥1, r*=1 perfect; see Methods). In this work we choose r=r*.
To implement our protocol, Charlie sent Bob a qubit ω_{k} (derived from the heralded singlephoton source) encoded in the , , or basis, and announced to Alice a corresponding value of j=1, 2 or 3. Alice implemented a measurement on her half of the entangled state (projective, in the , or basis depending on Charlie’s announcement) and Bob implemented his partial BSM. Charlie received classical outputs from Alice (a=±1) and Bob (b=0 or 1) over many runs. Using this information Charlie calculated the payoff function P(r) in equation (1), and tested for positivity to verify EPR steerability.
We tested for deviceindependent steering both for highly entangled states and also in the regime where a Bell inequality cannot be violated, since there exist mixed entangled states such as Werner states that can satisfy this latter condition. In theory, violation of the bound P⩽0 for our steering test requires (see Methods section), while the best explicit Belltype inequality for Werner states is violated for W≳0.7056—the Vértesi bound^{32}. Note this is slightly below the wellknown CHSH bound of (ref. 26).
We carefully characterized Charlie’s state preparation to determine that r*=1.081±0.009. Using a Werner state with W=0.698±0.005 (below both the CHSH and Vértesi bounds) we observed P(r*)=0.05±0.04—a violation of our steering inequality (Fig. 3). This violation may be compared with the theoretical prediction P_{W}(r*)=0.22 from equation (2), for ideal states and measurements. Our experimental state has a fidelity of 97.6% with a Werner state having W=0.698. Although the fidelity is high, it is not unity: our experimental state has small imperfections including an undesired population imbalance and an undesired phase shift between logical states. From modelling, we find that this accounts for the imperfection to within experimental error, with the model predicting P(r*)=0.067.
With higher values of W (for example, W≈1) one would also expect a verification of steering, and indeed we observed P(r*)=1.09±0.03 for a state having a fidelity F≈0.98 with the ideal singlet Bell state (Fig. 3). This is close to the ideal value of for a singlet state, corresponding to W=1 in equation (2).
Even without ideal entangled states or measuring devices, our observations of measured payoffs P(r*)>0 meant that Charlie was able to verify that Alice could steer Bob’s state, without requiring any trust in them or their devices.
Discussion
EPR steering is a key quantum resource because, apart from its fundamental interest, it is known to be useful in secure quantum key distribution protocols^{21}. Compared with violation of a loopholefree Bell inequality—which provides fully deviceindependent QKD—EPR steering in its usual form provides a onesided deviceindependent protocol, requiring trust in one party (say, Bob) and their apparatus. Our demonstration of QRS removes the need to trust Bob and his apparatus, only requiring the assumption that quantum mechanics is a reliable description of reality. This lack of trust is possible essentially because Bob is unable to unambiguously distinguish between the states sent to him by Charlie^{23}.
Thus, as long as quantum mechanics is correct, the protocol has the advantages of Bell inequality violation, but can tolerate higher noise. It should also be noted that steering inequalities exist for an arbitrarily high degree of loss^{18}, and hence corresponding QRS games can be constructed using our methods for longdistance applications such as secure quantum networks^{33}.
We note that the r parameter that we have introduced is only required to characterize the degree of confidence in the preparation of the referee states. It is unnecessary to characterize the state that Bob eventually receives from Charlie; indeed, transmission through any quantum channel will not change the protocol nor increase r (see Methods section). Therefore, as long as Charlie can characterize his prepared states, the protocol can proceed. Our protocol imposes a more complex measurement procedure on Bob, a joint BSM, compared with onequbit Pauli projections required in a Bell test. As the protocol is robust against preparation and transmission imperfections of the referee states, this added complexity of Bob’s measurement is a reasonable overhead for removing all need for trust. We note that it is easier for Alice and Bob to demonstrate EPR steering to Charlie if he can prepare his states with a high degree of confidence, that is, with r≈1.
A future challenge is to demonstrate the closure of the detection loophole and spacelike separation loophole for our protocol. When this is achieved, it will be possible to perform fully deviceindependent entanglement sharing between two parties—with only the assumption that quantum physics holds—with application in quantum key distribution, random number generation and beyond.
Methods
Constructing QRS games
A quantum state ρ^{AB} on some Hilbert space H_{A}⊗H_{B}, shared between two parties Alice and Bob, is defined to be nonsteerable by Alice if and only if there is a local hidden state (LHS) model for Bob^{16}, that is, if and only if the joint probability of measurement outcomes a and b, for arbitrary measurements A and made by Alice and Bob, can be written in the form p(a, b)=∑λp(λ)p(aλ)p(bλ), with p(bλ) restricted to have the quantum form . Here is the positiveoperatorvalued measure (POVM) corresponding to . Such LHS models, and hence nonsteerable states, satisfy various EPR steering inequalities^{16}, of the form
where the a_{j} denote classical random variables generated by Alice, and the B_{j} denote quantum observables on Bob’s system. States nonsteerable by Bob are similarly defined in terms of LHS models for Alice, however, we may focus on EPR steering by Alice without any loss of generality. It is known that for any EPRsteerable state shared by Alice and Bob, there is a corresponding steering inequality of the above form^{34}. To construct a QRS game from any such steering inequality, we adapt a method recently used by Branciard et al.^{24} for constructing games for verifying entanglement per se .
In particular, for a given EPR steering inequality as in equation (3), we define a corresponding QRS game G (see Fig. 1c) in which on each run the referee, Charlie, sends Alice a classical label j and Bob a state defined on a Hilbert space H^{C} isomorphic to some subspace of H^{B}. These states must be such that the equivalent states on H^{B} form a linear basis for the observables B_{j}, that is, for some set of coefficients g_{jk}. Alice and Bob are not allowed to communicate during the game, but can have a prearranged strategy and perform arbitrary local operations. Alice returns a value a=a_{j}, and Bob returns a value b=0 or 1 corresponding to some POVM on H_{B}⊗H_{C}. The corresponding payoff function is defined by P_{G}:=∑_{j,k}g_{jk}‹ab›_{j,k}, where ‹·›_{j,k} denotes the average over runs with a given j and k. Alice and Bob win the game if P_{G}>0. The QRS game in the main text is equivalent to taking j=0, 1, 2, 3, k≡(j, s), a_{j}=±1 for j=1, 2, 3, , and g_{jk}=s(=1) for j≠0 (j=0). The factor of 2 in the payoff function equation (1) for this game is chosen to make P(r) equal to the left hand side of the steering inequality (ref. 34). This steering inequality can be violated for Werner states only if (ref. 34), and hence this condition is also necessary for Alice and Bob to be able to win the QRS game in the main text. For perfect state generation by the referee, that is, r=1 (see below), this reduces to . Note that this corresponds to the condition for a Werner state to allow EPR steering, with measurements limited to three Pauli directions, in the nonquantumrefereed scenario^{16,17}.
We now show that Alice and Bob can win game G only if Alice and Bob share a state that is EPR steerable by Alice. Restricting Alice and Bob to no communication during the game prevents them from generating a steerable state from a nonsteerable one^{23}, and hence we must show that if they share any nonsteerable state on any Hilbert space H_{A}⊗H_{B} then P_{G}≤0. Now, for such a state there is some LHS model (see above), and thus
where the normalization factor N, probability distribution q(λ) and density operator τ_{λ}^{C} are implicitly defined via on H_{C} is isomorphic to B_{j} on H_{B}, and the average is with respect to the LHS model {τ^{C}(λ); q(λ)}. Noting the average corresponds to the left hand side of steering inequality in equation (3) for this LHS model, one has P_{G}≤0 as required. Conversely, analogously to the entanglement verification games of Branciard et al.^{24}, it may be shown that Alice and Bob can in principle win the game if they share a state that violates the EPR steering inequality in equation (3), where Bob measures the projection B_{1} onto an appropriate Bell state on H_{B}⊗H_{C} (see, for example, equation (2)).
In practice, the referee cannot ensure perfect generation of the states . However, by performing tomography on these states, the referee can adjust the coefficients g_{jk} appropriately, to take this into account. We describe one method of doing so below, for the experiment carried out in this paper, which can be easily generalized to other QRS games. We observe that it does not matter if the generated states are acted on nontrivially by some completely positive channel, φ, before reaching Bob, as this is equivalent to simply replacing Bob’s measurement on H_{B}⊗H_{C} by (I_{B}⊗φ*)(), where φ* denotes the dual channel and I_{B} is the identity map on H_{B}.
In particular, for the QRS game corresponding to equation (1), suppose that the referee actually generates the states . The payoff function in equation (1) then evaluates to for a shared nonsteerable state, with N, q(λ) and defined as above and
where the inequality follows using a_{j}=±1 and v·σ≤v, and we define a=(a_{1}, a_{2}, a_{3}), . It is straightforward to show that the right hand side of the inequality is no more than zero for r≥r*, with
Hence, for r≥r*, the operator T_{λ}(r) is nonpositive, and hence P(r)≤0 for any nonsteerable state. It is straightforward to check that r*=1 for perfect state generation, . Determining r* experimentally involves tomographically characterizing (as below) Charlie’s state preparations {j, s} to find the Bloch vectors n^{(j, s)}. We experimentally found r*=1.081±0.009.
Experimental apparatus
The individual spontaneous parametric downconversion (SPDC) sources used in our demonstration consisted of a pair of sandwiched bismuth borate (BiBO) crystals, each 0.5 mm in length and cut for typeI degenerate downconversion from 410 nm (pump) to 820 nm (signal/idler), with their optic axes perpendicularly oriented. Charlie’s source was pumped with 200 mW of horizontally polarized light to generate polarizationunentangled photon pairs. One of Charlie’s photons (signal) was sent to a singlephoton counting module (PerkinElmer SPCMAQR14FC), to herald the arrival of a degenerate idler counterpart at the BSM device. The second SPDC source was pumped with 200 mW of diagonally polarized light, generating the polarizationentangled state ρ^{AB}≠ρ^{A}⊗ρ^{B} shared between Alice and Bob. The state from the SPDC source could be transformed into any of the four Bell states by implementing a local unitary with a fibre polarization controller (to generate anti/correlated statistics) combined with a halfwave plate tilted in the xy plane with its optic axis in the horizontal plane (to set the phase φ of the entangled Bell state). Alice’s photon (consisting of onehalf of the entangled state) was sent to her singlequbit measurement station, whereas Bob’s photon (consisting of the remaining half of the entangled state) was coupled into singlemode fibre and sent to Bob’s BSM device. Bob’s BSM device consisted of a central 50:50 beam splitter and polarization analysis at the output ports. The device combined Bob’s half of the entangled state ρ^{AB}, and the state that Charlie sent to him. Bob’s partial BSM device resolved the Ψ^{+}› and Ψ^{−}› Bell states through discrimination of orthogonally polarized photon pairs (the case of Ψ^{+}›) or through antibunching behaviour (the case of Ψ^{−}›). On the other hand, the Φ^{±}› states required number resolving detection (since these states saw pairs of photons degenerate in polarization bunched at the point of detection). Because our singlephoton counting modules were not number resolving, we instead opted for pseudonumber resolution by replacing the singlemode fibres at Bob’s BSM output with singlemode 50:50 fibre beam splitters. The initially bunched pairs of photons travelling down these fibre beam splitters were separated and numberresolved 50% of the time, a feature accounted for in the analysis of the payoff function.
The Bellstate analysis featured nonclassical HOM interference between the ρ^{B} and photons at the central 50:50 beam splitter. A HOM interference visibility of 89% was calculated, where a high interference visibility corresponded to effective resolution of the singlet state Ψ^{−}› and the other three triplet Bell states (for some local unitary). Bob performed a joint measurement on , where the fibre input coupler for the photon was kept on a linear z−translation stage to match temporal modes between the ρ^{B} and photons. A photon detection at Alice’s detector heralded the presence of the ρ^{B} photon at the 50:50 beam splitter, and a photon detection in Charlie’s heralding detector signified the presence of the photon. Our method to calculate the payoff function P(r) for an experimental Werner state ρ^{AB} was relatively straightforward, and used the fact that a Werner state can be expressed as a statistical mixture of all four Bell states. Data was taken with ρ^{AB} consecutively prepared in the four Bell states, and the data sets were aggregated to produce a value of the payoff function for the effective state ρ^{AB}. The Werner parameter was tuned by weighting the data collection time for the singlet state relative to the data collection time for the three triplet states (where the data collection interval for the three triplet states was identical). For example, to test the payoff function using a completely mixed state (W=0), data could be taken for an equal time with all four Bell states. Taking a relatively shorter collection time for the triplet states allowed us to obtain a mixture corresponding to a Werner state with W=0.0698±0.005, below the CHSH and Vértesi bounds for Bell inequality violation by a Werner state (see main text). It may be remarked that it remains an open question whether there exists a Bell inequality that can be violated for W below the Vértesi bound, although it is known to be impossible for W≲0.6595 (refs 32, 35).
Charlie’s ability to send the correct state to Bob was also experimentally characterized. An average fidelity of _{av}=98.7±0.6% was measured in the Bellstate analysis setup for the six Pauli operator eigenstates prepared by Charlie’s source.
All states were characterized using maximumlikelihood quantum state tomography as per ref. 36, and fidelities between (in general) mixed states ρ and σ given by the standard formula .
Experimental error analysis
Experimental uncertainties were derived from Poissionian counting statistics and standard error propagation techniques. Error bars quoted represent ±1 standard deviations. Where uncertainties are required in quantities derived from tomographic state reconstructions^{36}, the process was as follows. A large number of tomographic reconstructions on the state were performed, with each trial drawing from a Poissonian distribution of statistics for each measurement outcome. Each of the reconstructed density matrices were used to calculate the parameter of interest (for example, W), and the mean and s.d. of the distribution in that parameter produced the value and its uncertainty.
Additional information
How to cite this article: Kocsis, S. et al. Experimental measurementdeviceindependent verification of quantum steering. Nat. Commun. 6:5886 doi: 10.1038/ncomms6886 (2015).
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Acknowledgements
We thank Howard Wiseman for discussions and Raj Patel for helpful contributions. This work was supported by the Australian Research Council, Project No. DP 140100648. D.J.S. acknowledges MarieCurie Fellowship PIIFGA2013629229.
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M.J.W.H. and G.J.P. conceived the project. M.J.W.H. carried out the theoretical work. S.K., A.J.B. and D.J.S. performed the experiment, with assistance from G.J.P. All authors contributed to the analysis and writing the manuscript.
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Kocsis, S., Hall, M., Bennet, A. et al. Experimental measurementdeviceindependent verification of quantum steering. Nat Commun 6, 5886 (2015). https://doi.org/10.1038/ncomms6886
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