Abstract
Einstein–Podolsky–Rosen (EPR) steering describes a quantum nonlocal phenomenon in which one party can nonlocally affect the other’s state through local measurements. It reveals an additional concept of quantum nonlocality, which stands between quantum entanglement and Bell nonlocality. Recently, a quantum information task named as subchannel discrimination (SD) provides a necessary and sufficient characterization of EPR steering. The success probability of SD using steerable states is higher than using any unsteerable states, even when they are entangled. However, the detailed construction of such subchannels and the experimental realization of the corresponding task are still technologically challenging. In this work, we designed a feasible collection of subchannels for a quantum channel and experimentally demonstrated the corresponding SD task where the probabilities of correct discrimination are clearly enhanced by exploiting steerable states. Our results provide a concrete example to operationally demonstrate EPR steering and shine a new light on the potential application of EPR steering.
Introduction
In the original discussion of Einstein–Podolsky–Rosen (EPR) paradox,^{1} Schrödinger^{2,3} described a quantum nonlocal phenomenon that Alice can steer Bob’s state through her local measurements. Since then, great efforts have been made to understand quantum nonlocality. It was not until 2007, Wiseman, Jones, and Doherty revisited Schrödinger’s discussion and formulated the concepts of quantum nonlocality as quantum entanglement, EPR steering, and Bell nonlocality in terms of quantum information tasks.^{4,5} It is now clear that all steerable states are entangled, but not all steerable states exhibit the Bell nonlocality,^{4,5} which implies that EPR steering sits between quantum entanglement and Bell nonlocality. This hierarchy also holds for all possible positive operator valued measures.^{6} EPR steering has recently drawn plenty of attention.^{7} For example, several theoretical studies including the verification of EPR steering based on steering inequalities^{8} and allversusnothing proof,^{9} nocloning of quantum steering,^{10} temporal steering,^{11,12,13,14,15} quantification of steerability,^{16,17,18} and oneway EPR steering^{19} have been reported as well as the corresponding experiments.^{20,21,22,23,24,25,26} There are also other interesting steering experiments, such as the highorder steering^{27} and loopholefree steering.^{28,29,30} Moreover, the parallel works based on the continuous variable systems^{31,32,33,34,35,36,37,38,39} have been reported.
Similar to the necessary and sufficient verification of quantum entanglement with a quantum information task named quantum channel discrimination,^{40} which refers to the task of distinguishing among different quantum operations,^{41,42,43} EPR steering can be characterized necessarily and sufficiently based on a quantum task named subchannel discrimination (SD).^{17} As an extension of the quantum channel generally representing the physical transformation of information from an initial state to a final state in which the quantum operation is tracepreserving for all input states,^{44} a subchannel is a completely positive operator that does not increase the trace in the density matrix space.^{17} A series of subchannels {Λ_{ h }}_{ h }, that constitute a channel Λ satisfying \({\mathrm{\Lambda }} = \mathop {\sum}\nolimits_h {\kern 1pt} {\mathrm{\Lambda }}_h\), can be treated as a decomposition of the channel into its different evolutionary branches with the corresponding probability Tr(Λ_{ h }[ρ]) for any state ρ, as shown in Fig. 1a. Here, \({\mathrm{\Lambda }}_h[\rho ] = K_h\rho K_h^\dagger\), where the Kraus operators K_{ h } are the explicit matrix descriptions of Λ_{ h } and satisfy \(\mathop {\sum}\nolimits_h K_h^\dagger K_h = {\mathbb{I}}\). The SD task allows one to distinguish in which subchannel the quantum evolution occurs, whereas this information is lost if the process is described simply in the framework of the quantum channel. Moreover, SD tasks might lead to the emergence of new quantum phenomena and applications in quantum information processing, such as the SDbased quantum key distribution.^{45}
Recently, it has been proven that for any bipartite state, we can verify it is steerable if there exists an SD task in which the successful discrimination probability is enhanced by this state compared with the case employing singlequbit states; otherwise, if no such SD tasks exist, it is unsteerable.^{17} Also, such an SD task presents an operational method to characterize EPR steering. However, the detailed construction of such subchannels has not been investigated up to now. In this article, we design a feasible collection of concrete subchannels and experimentally demonstrate EPR steering with the corresponding SD task.
Results
SD task for the twosetting case
First, we would like to introduce the detailed SD task in the simplest case with two measurement settings. In this work, we consider a channel consisting of four subchannels Λ_{ ij } (i, j = 0 or 1), where the corresponding Kraus operators are denoted by K_{ ij }. We exploit an entanglementbreaking channel (EBC)^{46} to limit the bound established in the singlequbit protocol. The Kraus operators K_{ ij } are implemented with the EBC, as illustrated in Fig. 1b, where A_{ j } (j = 0 or 1) is regarded as the intermediate subchannel and satisfies \(K_{ij} = \left i \right\rangle \left\langle i \right \cdot A_j\) (i, j = 0 or 1) (see Methods). Since the information of i is included in the output ρ_{ out }, the SD task is transformed into the task of distinguishing A_{ j } based on i. To realize {A_{ j }}_{ j }, a unitary operation U is performed on a quantum system consisting of a target qubit in the state ρ and an auxiliary qubit initially in the state \(\left 0 \right\rangle\),^{47} as shown in Fig. 1c. In this work, the operation is represented as follows,
A_{ j } is determined according to the output j measured along the z direction on the auxiliary qubit. The SD task in singlequbit protocol is completed by guessing j according to the output b that is measured along a direction \(\vec n\) on the target qubit. Since the target qubit only carries the classical information after the EBC, \(\vec n\) is optimized to be z to maximize the success probability \(P_\rho ^{\mathrm{s}}\). With the input state ρ, the results of different strategies for guessing j are denoted by \(p_\rho ^{c0},p_\rho ^{c1}\) (guessing j is the constant 0 or 1 regardless of b, respectively), \(p_\rho ^{00},p_\rho ^{01}\) (guessing j = b or j = b ⊕ 1 where ⊕ represents addition modulo 2, respectively). The success probability is denoted as \(P_\rho ^{\mathrm{s}} = {\mathrm{max}}\left\{ {p_\rho ^{c0},p_\rho ^{c1},p_\rho ^{00},p_\rho ^{01}} \right\}\), and the upperbound probability P^{s} in the singlequbit case is obtained by optimizing the input state, which implies \(P^{\mathrm{s}} = {\mathrm{max}}_\rho \left\{ {P_\rho ^{\mathrm{s}}} \right\}\).
We now consider the twoqubit Werner states ρ_{ AB }^{48} with the form of,
where η ∈ [0,1], \(\left {\mathrm{\Phi }} \right\rangle\) is the maximally entangled state, and \({\mathbb{I}}{\mathrm{/}}4\) is the maximally mixed state. As illustrated in Fig. 1d in the twosetting case (m = 1, and g_{1} is identical), the task is that Alice guesses j and announces to Bob based on a which is obtained by measuring along \(\vec n_i\) (chosen according to b). Since b ∈ {0, 1}, there are two directions \(\vec n_i\) along which Alice can choose to measure. In this work, we follow two rules to design the SD tasks, i.e., (i) the success probability of maximally entangled state is 100%; (ii) the success probability of maximally mixed state is 50%. Thus, the success probability of SD task \(P_{\rho _{AB}}\) equals to 1/2 + η/2. In the linear EPR steering inequalities, \(C_n^{{\mathrm{LHS}}}\) denotes the bound established by the local hidden state model where n is the number of measurement settings.^{20} In the case of n = 2, \(C_2^{{\mathrm{LHS}}} = \eta _2^ \ast\) where \(\eta _2^ \ast = 1{\mathrm{/}}\sqrt 2\) is the visibility bound of the Werner states. When \(\eta > \eta _2^ \ast\), \(\rho _{\rho _{AB}}\) is steerable. For the singlequbit protocol, by directly calculating, we find \(P^{\mathrm{s}} = 1{\mathrm{/}}2 + C_2^{{\mathrm{LHS}}}{\mathrm{/}}2\) (see Section I of the Supplemental Material for details (See the Supplemental Material.)). Thus, if Bob finds \(P_{\rho _{AB}} > P^{\mathrm{s}}\), the steerability from Alice to Bob is observed.
SD task for the multisetting cases
EPR steering from Alice to Bob relates to the number of settings measured by Alice.^{4,49} For some predictably steerable states, steering fails because of the very limited number of measurement settings.^{25} To capture as much information about the states as possible to demonstrate EPR steering, it is necessary for Alice to apply multiple measurement settings to approach the predictions of infinite measurement settings. In this work, we consider the regularly spaced directions which are given by the Platonic solids with the number of measurement settings n corresponding to 2, 3, 4, 6, and 10.^{20} Compared with the optimal measurement settings introduced in,^{49} here, the twosetting and threesetting measurements are optimal and EPR steering can be affirmed necessarily and sufficiently. For other multisetting cases, the optimal measurements don’t correspond to the regularly spaced directions and are difficult to realize in experiment. Moreover, the diffence between the results in this work and the predictions of the optimal measurements is very small (see Section I of the Supplemental Material (See the Supplemental Material.)). To experimentally realize such a task, on Bob’s side, the qubit equiprobably evolves through several unitary gates g_{ m } before the Kraus operators K_{ ij }, as illustrated in Fig. 1d, and the details can be found in Section I of the Supplemental Material (See the Supplemental Material.). The corresponding measurement setting based on Bob’s result bg_{ m }, which denotes that b is obtained under the gate operation g_{ m }, is then implemented on Alice’s side. In fact, for each g_{ m }, the SD process can still be regarded within the framework of two measurement settings. In the singlequbit protocol with n measurement settings, denoting \(P_{m,\rho }^{\mathrm{s}}\) for each g_{ m }, the total success probability is obtained as \(P_n^s = {\mathrm{max}}_\rho \left\{ {\mathop {\sum}\nolimits_m {\kern 1pt} 1{\mathrm{/}}mP_{\rho ,m}^{\mathrm{s}}} \right\}\). Similarly, for the twoqubit state ρ_{ AB }, \(P_{\rho _{{{AB}}},{\kern 1pt} n} = \mathop {\sum}\nolimits_m {\kern 1pt} 1{\mathrm{/}}m{\kern 1pt} P_{\rho _{{{AB}}},m}\). Furthermore, similar with the twosetting case, we have \(C_n^{{\mathrm{LHS}}} = \eta _n^ \ast\) and \(P_n^{\mathrm{s}} = 1{\mathrm{/}}2 + C_n^{{\mathrm{LHS}}}{\mathrm{/}}2\) (see Section I of the Supplemental Material for details (See the Supplemental Material.)) when the multiple measurement settings are selected based on the Platonic solids.^{20} As a result, the constructed SD task provides an operational method to characterize the steerability of Werner states. If the success probability of SD is enhanced by using the twoqubit state ρ_{ AB }, i.e., \(P_{\rho _{AB}} > P^{\mathrm{s}}\), then ρ_{ AB } is steerable from Alice to Bob regardless of the number of measurement settings; otherwise, under n measurement settings performed by Alice, i.e., \(P_{\rho _{{{AB}}},{\kern 1pt} n} \le P_n^{\mathrm{s}}\), Alice fails to steer Bob’s states.
Compared with the twosetting case in which there are four subchannels, the multisetting cases can be regarded as multisubchannel discrimination tasks where more subchannels consisting of gates g_{ m } and the corresponding Kraus operators K_{ ij } are required, and the reconstructed subchannels could be expressed as \(K_{ijm}^\prime = K_{ij} \cdot g_m\). Following the similar method designing subchannels for Werner states, we can also create the corresponding subchannels for other types of twoqubit states, like the Bell diagonal states (see Section I of the Supplemental Material (See the Supplemental Material.)).
Experimental setup
The unitary operation U shown in Fig. 1c can be decomposed into several parts, including two controlnot (CNOT) gates (C_{NOT1} and C_{NOT2}) and the other unitary evolutions E, V_{1}, V_{2}, and V_{3}, and implemented in an optical Sagnaclike interferometer (SLI), as illustrated in Fig. 2 (see Methods). And U could be expressed as
where \({\mathbb {I}}_C\) is the identical operation on the control qubit. To obtain the bound P^{s} and verify the setup, the singlequbit protocol is performed with the input state denoted as \(\rho (\theta ) = {\mathrm{cos}}{\kern 1pt} \theta \left {\mathrm{H}} \right\rangle + {\mathrm{sin}}{\kern 1pt} \theta \left {\mathrm{V}} \right\rangle\) where \(\left {\mathrm{H}} \right\rangle\) and \(\left {\mathrm{V}} \right\rangle\) represent the horizonal and vertical polarizations of the photons, respectively. For the twoqubit protocol, Werner states are prepared via the spontaneous parametric down conversion process by pumping the nonlinear crystal of periodically poled KTiOPO_{4} (PPKTP) which is placed in a polarization Sagnac interferometer.^{50} Here, \(\left {\mathrm{\Phi }} \right\rangle\) is prepared to be \(\left( {\left {\mathrm{HH}} \right\rangle + \left {\mathrm{VV}} \right\rangle } \right){\mathrm{/}}\sqrt 2\). The experimental Werner states ρ_{ AB } are prepared with an average fidelity of 98.3 ± 0.2%. The detailed experimental preparation can be found in Methods.
Experimental results
In the case of two measurement settings, the results \(p_\rho ^{c0},p_\rho ^{c1},p_\rho ^{00}\) and \(p_\rho ^{01}\) are presented in Fig. 3a which show that the input states ρ(θ) should be optimized to obtain the upperbound value P^{s}. More results in the singlequbit protocol with multiple measurement settings are presented in Section III of the Supplemental Material (See the Supplemental Material.). The Werner state ρ_{ AB } is identified to be steerable when the SD performance is enhanced with \(P_{\rho _{AB}} > P^s\), see Fig. 3b, c. By contrast, when \(P_{\rho _{AB},{\kern 1pt} n} \le P_n^s\) (n = 2, 3, 4, 6, 10), Alice fails to steer Bob’s state via the corresponding SD task. As the number of measurement settings increases, the bound established for the singlequbit approach decreases, whereas the success probability achieved by employing steerable resources remains constant, as illustrated in Fig. 3d. All error bars in this work are estimated as the standard deviation from the statistical variation of the photon counts, which is assumed to follow a Poisson distribution. As the error bars on the experimental probabilities are very small, roughly 0.002, they are not shown in the figures.
Moreover, by means of the SD task, the difference between EPR steering and entanglement can be characterized in an operational way. It is found that the success probability achieved using unsteerable Werner states ρ_{ AB } cannot surpass the singlequbit bound when \(\eta \le \eta _{10}^ \ast \approx 0.524\) in the case of ten measurement settings.^{20} However, ρ_{ AB } is still entangled when η > 1/3. This implies that the success probability of SD cannot be enhanced by using unsteerable entangled states, which is experimentally verified by the two pink dots in Fig. 4a. The concurrences of these two pink dots are measured to be 0.154 ± 0.008 and 0.223 ± 0.009, which verify that the states are entangled.^{51} We further investigate EPR steering with Belllocal states. Theoretically, the Bell inequality will be violated when \(\eta > 1{\mathrm{/}}\sqrt 2\),^{48} and according to ref. ^{52}, ρ_{ AB } is a Belllocal state when η ≤ 0.683. We measure the BellCHSH parameter S^{53} which is shown as the function of \(P_{\rho _{{AB}}}\) in Fig. 4b. The success probabilities of SD using three Belllocal states which are represented by the dark red dots in Fig. 4 are enhanced, and therefore, these states are steerable.
Discussion
Based on the proof of the necessary and sufficient characterization of EPR steering, we designed and experimentally implemented an SD task to demonstrate EPR steering using twoqubit Werner states. The methods for decomposing a quantum evolution into subchannels can be helpful for gaining a thorough understanding of complex opensystem dynamics. The enhanced probabilities of successful discrimination achieved using EPR steering provides a concrete example of the application. Moreover, this practical task offers an intuitive means of operationally distinguishing the different concepts of quantum nonlocality.
Compared with the previous experiments using steering inequalities to investigate EPR steering, in which Bob measures along several directions when steered by Alice,^{20,25} our work exhibits a particular feature that the measurement performed on Bob’s qubit is restricted to a single direction, which is z in this work. This feature implies that the SD task offers a convenient approach for identifying EPR steering. Another character of the SD task is the measurement sequence of Alice and Bob. In the previous works,^{4,21} considering that Alice steers Bob, Bob performs the measurements after receiving the measurement results from Alice. However, in the SD task, the sequence is reversed, which means Alice begins to measure her qubit after Bob’s measurements.
As EPR steering can be regarded as the oneside deviceindependent quantum information task,^{45} the steeringenhanced SD task, where Bob trusts his experimental device while Alice’s side is deviceindependent, shows the potential application in oneside deviceindependent quantum key distribution. Furthermore, in our work, the SD task on Bob’s side is implemented based on the oneway classical communication (from Bob to Alice). Considering the situation that Bell nonlocality relates to the twoside deviceindependent quantum information task,^{5,45} one might extend the SD task demonstrating EPR steering to investigate the Bell nonlocality. For instance, a similar quantum information task referring to a bipartite SD problem (SD tasks on both sides) with twoway classical communications, which relates to the communication complexity problem,^{54,55} might be used to characterize Bell nonlocality in an operational way.
Methods
The detailed expressions of A _{ j } and K _{ ij } in the twosetting case
Following the theoretical method to determine A_{ j } which is introduced in Section I of the Supplemental Material in detail (See the Supplemental Material.), we can obtain the expressions of A_{0}, A_{1} in the twosetting case as below
Considering the Kraus operators K_{ ij } which satisfy \(K_{ij} = \left i \right\rangle \left i \right\rangle \cdot A_j\) (i, j = 0 or 1), we can get
For multisetting cases, the corresponding expressions of A_{ j } and K_{ ij } can be obtained using the similar method which is shown in Section I of the Supplemental Material (See the Supplemental Material.).
Experimental implementation of the unitary operation U
We construct an inherently stable optical interferometer, namely, a Sagnaclike interferometer (SLI), to realize this operation U (see Fig. 2b). The path and polarization degree of freedom of the photons are used as the auxiliary qubit, which is initially in the state \(\left 0 \right\rangle\), and the probe qubit, respectively. A homemade beam splitter, of which one half is coated as a PBS and the other half is coated as a nonpolarized beam splitter (NBS), acts as the inputoutput coupling element of the interferometer. Each singlequbit gate evolution of the probe qubit (the polarization of photons), i.e., V_{1},V_{2}, and V_{3}, is realized through a combination of two HWPs. The operation E on the auxiliary qubit is realized by adjusting the ratio of the numbers of photons on the \(\left 0 \right\rangle\) and \(\left 1 \right\rangle\) paths, which is achieved by means of a continuously variable neutral density filter (CVF) crossing both paths. For the first CNOT gate, the path qubit is the control qubit, while the polarization is used as the target qubit. Thus, the polarization of photons on the \(\left 0 \right\rangle\) path remains the same, whereas the polarization on the \(\left 1 \right\rangle\) path reverses, meaning that the polarization \(\left {\mathrm{H}} \right\rangle\) is flipped to \(\left {\mathrm{V}} \right\rangle\) and \(\left {\mathrm{V}} \right\rangle\) is flipped to \(\left {\mathrm{H}} \right\rangle\). This process is realized by placing one HWP on each of the two paths; HWP0, located on the \(\left 0 \right\rangle\) path, is set at 0° for phase compensation, while HWP1 is set at 45° to reverse the polarizations of \(\left H \right\rangle\) and \(\left V \right\rangle\). The second CNOT gate is the inverse of the first CNOT gate; the polarization is treated as the control qubit affecting the target qubit, which is the qubit related to the path information. This gate is implemented in the PBS part of the homemade beam splitter. In detail, the \(\left {\mathrm{H}} \right\rangle\) polarization remains unchanged (retaining the same path information), while the \(\left {\mathrm{V}} \right\rangle\) polarization flips to the other path. The imperfect optical elements, especially the homemade beam splitter, would reduce the visibility of the interferometer and introduce system errors.
To realize {g_{ m }} in the multisetting cases, several wave plates including HWPs and quarterwave plates (QWPs) are employed. This part is explained in detail in Section II of the Supplemental Material (See the Supplemental Material.).
Preparation of the experimental states
To obtain the singlequbit upper bound and verify the setup, we perform the SD task using the following singlequbit state ρ(θ),
In this case, the photons on Bob’s side are prepared as the state expressed in Eq. (6), and the photons on Alice’s side are detected directly to provide coincidence signals. ρ(θ) are simply prepared with a halfwave plate (HWP) set at the angel θ/2 following a polarized beam splitter (PBS). The upper bound is then \(P^{\mathrm{s}} = {\mathrm{max}}_{\rho (\theta )}\left\{ {P_{\rho (\theta )}^{\mathrm{s}}} \right\}\).
The investigated ρ_{ AB } states are manufactured by combining the maximally entangled state \(\left \Phi \right\rangle\) and the maximally mixed state \({\mathbb{I}}{\mathrm{/}}4\). \(\left \Phi \right\rangle\) is prepared via the spontaneous parametric down conversion process where a χ^{(2)} nonlinear crystal of periodically poled KTiOPO_{4} (PPKTP) is pumped by an ultraviolet laser with a peak wavelength of 404.1 nm and a spectrum width of 0.05 nm. The crystal is placed in a polarization Sagnac interferometer,^{50} as illustrated in Fig. 2c. The dichroic mirror (DM) is designed to exhibit high transmission at 404 nm and high reflection at 808 nm. A dualwavelength polarization beam splitter (PBS) is employed as the inputoutput coupling element of the Sagnac interferometer, and a dualwavelength HWP set at 45° is used to change the vertically polarized component of the ultraviolate photon to the horizonal polarization to pump the PPKTP crystal. The crystal is placed in a thermoelectric oven with the temperature set at 28.5 ± 0.1 °C. The maximally entangled state \(\left {\mathrm{\Phi }} \right\rangle\) is prepared with a brightness of ~18,000 pairs s^{−1} mW^{−1}, which is filtered using 3 nm bandwidth filters, and the state fidelity is 95.5 ± 0.4%. As shown in Fig. 2c, a part of the input of unit M still remains as the maximally entangled state \(\left {\mathrm{\Phi }} \right\rangle \left\langle {\mathrm{\Phi }} \right\), and the other part is used to prepare the maximally mixed state \({\mathbb{I}}{\mathrm{/}}4\) with the dashed gray part in unit M. Two HWPs are set at 22.5° and a birefringent calcite crystal (BC) of 10 mm in length is employed to induce decoherence between the horizonal and vertical polarizations of the photons. The shutters are used to adjust the ratio between \(\left {\mathrm{\Phi }} \right\rangle \left\langle {\mathrm{\Phi }} \right\) and \({\mathbb{I}}{\mathrm{/}}4\) to control the parameter η.
Data availability
All relevant data and program codes are available from the corresponding author upon the reasonable request.
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Acknowledgements
This work was supported by National Key Research and Development Program of China (Grants Nos. 2016YFA0302700 and 2017YFA0304100), the National Natural Science Foundation of China (Grant Nos. 61327901, 61725504, 11274297, 61322506, 11325419, and 11774335), Anhui Initiative in Quantum Information Technologies (AHY060300 and AHY020100), the Key Research Program of Frontier Sciences, CAS (Grants No. QYZDYSSWSLH003), the Fundamental Research Funds for the Central Universities (Grant Nos. WK2030380015, WK2470000020 and WK2470000026) and the Youth Innovation Promotion Association and Excellent Young Scientist Program CAS. J.L.C. acknowledges the support by National Basic Research Program of China under Grant No. 2012CB921900 and NSF of China (Grant Nos. 11175089, 11475089).
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K.S. and X.J.Y. contributed equally to this work. J.S.X. and K.S. designed the experiment. K.S. performed the experiment with the help from Y.X. and X.Y.X. X.J.Y. took charge of the theoretical proof in assistance of J.L.C. and Y.C.W. K.S., X.J.Y. and J.S.X. wrote the paper. All authors read the manuscript and discussed the results. J.S.X., C.F.L. and G.C.G. supervised the project.
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Sun, K., Ye, XJ., Xiao, Y. et al. Demonstration of Einstein–Podolsky–Rosen steering with enhanced subchannel discrimination. npj Quantum Inf 4, 12 (2018). https://doi.org/10.1038/s4153401800671
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