Verifying Quantum Steering
for Quantum-information Tasks
DEPARTMENT OF ENGINEERING SCIENCE, NATIONAL CHENG KUNG UNIVERSITY
Einstein-Podolsky-Rosen (EPR) steering demonstrates that two parties share entanglement even if the measurement devices of one party are uncharacterized. Here, going beyond this bipartite concept, we describe two new types of quantum steering: genuine multipartite EPR steering and steering of single quantum systems. As concrete illustrations of these characterizations, we experimentally analyze how these steering effects can be verified in optical quantum systems and applied to quantum computing. Their role in preserving the security of quantum information is discussed as well. The concept and method presented may provide potential applications to future quantum-information tasks and quantum networks in the presence of untrusted measurement devices.
Einstein-Podolsky-Rosen (EPR) steering provides a new perspective of EPR non-locality . It determines which states can be remotely prepared at one location as a measurement is performed at another. With the operational definition introduced by Wiseman, Jones and Doherty , such spooky action at a distance becomes a subtle form of quantum correlation intermediate between entanglement and Bell non-locality. Recently, it has been used in quantum key distribution (QKD)  even if one-sided measurement devices are untrusted.
It is interesting to go a step further and consider the following question: what is the role of quantum steering in general quantum-information processing? Quantum information involves versatile applications, such as quantum computation , quantum simulation , and quantum communication , as well as quantum error-correction . Indeed, general quantum information tasks may inevitably need transmitting, sharing or processing quantum information between more than two spatially separated nodes of quantum systems(quantum nodes) via quantum channels in real-world quantum networks [8,9] as illustrated in Fig. 1. Quantum correlations of multipartite systems are thought to act as important resources, fuel that powers a wide range of quantum protocols for the networking tasks. How do we concretely show genuine multipartite steering in these multipartite systems and utilize this steering for quantum networks?
Compared with the scenarios of quantum networking tasks for multiple quantum nodes, quantum-information processing can be formulated by simple input-output schemes for single quantum systems. Quantum computing based on the quantum-circuit model  can be considered as a typical example. Quantum communication, which involves sender and receiver, fits this description as well. Do there exist strict and experimentally feasible (or even efficient) criteria for quantum steerability of input and output states of single systems that can be used for quantum-information protocols in terms of the input-output scenarios?
Here, to approach these issues, we describe two new types of quantum steering: genuine multipartite EPR steering and single-system steering, which are useful to satisfy the task-oriented purpose for quantum information. We analyze our recent photonic experiments on genuine four-partite EPR steering and show its applications to one-way quantum computing. Furthermore, we introduce a simple method to experimentally certify single-system steering, for dimensionality up to sixteen. Finally, to investigate their roles in quantum-information processing, we briefly discuss a no-cloning restriction on quantum steering to elucidate the physical reason why steering can be used to secure sources and channels against cloning-based attacks when implementing quantum information tasks.
Fig. 1: Quantum networking tasks and genuine multipartite EPR steering. Quantum networking protocols usually rely on both characterized measurement devices and genuine multipartite entanglement, such as implementing multiparty quantum communications [6,12,13] with graph states [14,15] of quantum multi-dimensional systems (qudits). Graph states of qudits constitute a large and highly significant class of genuine multipartite entangled states in physics. A N-qudit graph state can be represented by a fully-connected graph G(V,E). The graph G comprises the vertex set V with a cardinality |V |=N, representing the qudits, and the edge set E, each of which joins two vertices, representing interacting pairs of qudits. Genuine multipartite EPR steering provides users of quantum networks the ability to perform information tasks in the presence of uncharacterized (untrusted) measurement apparatus. As shown recently , such genuine high-order EPR steering can be efficiently verified by measuring the qudits with few measurement settings of the parties.
GENUINE MULTIPARTITE EPR STEERING
Genuine multipartite EPR steering can be defined from an operational interpretation as the distribution of genuine multipartite entanglement by uncharacterized (or untrusted) parties, as detailed in our recent report . Here we concretely consider a system composed of N parties and a source creating N particles ; see Fig. 1. Each party of the system can receive a particle from the source whenever an N-particle state is created. We divide the system into two groups, say As and Bs, and assume that As is responsible for sending particles from such a source to every party. Each time, after receiving particles, they measure their respective parts and communicate classically. Since Bs does not trust As, As's task is to convince Bs that the state shared between them is entangled. As will succeed in this task if and only if As can prepare different ensembles of quantum states for Bs by steering Bs's state. Here we say an N-particle state generated from the source to possess genuine N-partite EPR steerability if As succeeds in the task for all possible bipartitions As and Bs of the N-particle system.
Figure 1 depicts one possible scenario where As and Bs aim to share a N-qudit graph state for quantum networking tasks. Here, we assume that the measurement devices of Bs are trusted whereas those of As are not. In the worst case where As's measurement outcomes may be randomly generated from the untrusted apparatus, classical simulations then can describe As's measurement results, which results in a fail in the steering game.
With the specific correlations between qudits of graph states, one can construct quantum witnesses to verify genuine multipartite steerability for states close to
N-qudit graph states . These steering witnesses can be efficiently implemented with few measurement settings, which are like their analogues, to detect genuine multipartite entanglement [16,17]. The number of settings required for verifications is determined by the colorable property of a graph state.
Our recent experimental observations of genuine four-partite steering of a chain-type graph state (also called a cluster state) with steering witness confirm and demonstrate one possible type of high-order EPR steering for photons . We use a pulse (5 ps) of UV light with a central wavelength of 355 nm (an average power of 200 mW, repetition rate of 80 MHz) to double pass a two-crystal structured type-I-β barium borate (BBO). Such an arrangement can produce polarization entangled photon pairs either in the forward direction or in the backward direction. See Fig. 2.
Fig. 2: Experimental setup for creating and observing genuine four-partite EPR steering.
To create desired entangled pairs in mode RA, RB and in LA, LB, two quarter wave plates (QWPs) are properly tilted along their optic axes. Half wave plates (HWPs), polarizing beam splitters(PBSs), and eight single-photon detectors are used as polarization analyzers for the output states. Here 3-nm bandpass filters (IFs) with a central wavelength of 710 nm are placed in front of them.
Through temporal overlap of modes RA, and LA and of modes RB and LB, one can create a four-qubit (quantum two-dimensional system) state entangled both in spatial modes and horizontal and vertical polarizations and equivalent to the target chain-type cluster state (see Fig. 2a). This method has been used in our previous experiments on genuine multipartite entangled qubits .
Fig. 3: Implementing one-way quantum computing with partially untrusted measurements.
Our experimental states are highly similar to the target, and detected as genuine four-partite steering with 1.8829 ±0.0049 of the steering witness, which is clearly larger than the maximum value of ~1.7071 for the classical scenario can achieve by ~36 standard deviations.
One-way quantum computing
By performing proper measurements on qubits 2 and 3 of a cluster state, respectively (Fig. 3a), the rest of the qubits 1,4 together with the outcomes of measurements on qubits 2,3 would provide a read-out of a two-qubit gate together with two single-qubit gate operations (Fig. 3b) (see Refs. [14,19,20] for detailed discussions on the model of one-way quantum computing). Then, genuine four-partite EPR steering of the chain-type cluster state enables the two-qubit quantum gates to operate in the one-way mode for partially untrusted measurement apparatus. This method can be directly applied to quantum circuits for many qubits realized in the one-way mode. It is worth noting that the steerability can be verified by performing one-way computation and evaluating the performance of quantum gates (see supplementary material in ).
Fig. 4: Single-system steering for quantum-information tasks . By sharing certain information distributed via a classical communication channel (not shown), Alice can steer the state of Bob's particle by asking him to perform the quantum operation U. This is an analogue of EPR steerability of input and output states of U for single quantum systems. For example, by simply choosing U as an identity operator, Alice's steering enables them to realize QKD. When U is an arbitrary quantum logic gate, steering single systems is equivalent to performing quantum computation.
SINGLE-SYSTEM QUANTUM STEERING
EPR steering is the ability of one party, Alice, to affect the state of another remote party, Bob, through her choice of measurement. This relies on both the entanglement of the pair shared between Alice and Bob and the measurement settings chosen for each particle of the pair. Whereas, in the scenario of single-system quantum steering [21,22], Fig. 4, Alice's ability to affect the quantum state Bob has access to is based on both her ability to prepare an arbitrary quantum state to send to Bob and her knowledge, if any, about the state Bob finally receives. If Alice has full information about the quantum system Bob is holding, she is capable of steering this system into an arbitrary state.
Alice can follow two steps to achieve single-system steering. First, Alice prepares a specific state of a qudit with a given initial state generated from some quantum source, before sending it to Bob, by performing complementary measurements Ai for i= 1, 2. Once the particle is measured with a chosen Ai, the initial state is prepared in one of d possible orthonormal pure states, i.e., a d-dimensional quantum system. Second, the particle in the prepared state is sent to Bob. Here Bob does not know the state of particle sent from Alice. To steer Bob's state into other quantum states U, Alice can directly perform the unitary operation U by herself before the particle transmission, or publicly, via a classical channel, ask Bob to apply U on the received state. While the quantum operation U is announced publicly, the state operated by U is still unknown to Bob. It is clear that Alice has complete knowledge about the quantum system held by Bob.
Fig. 5: Experimental set-up for testing multidimensional quantum steering : (a) EPR steering, and (b) single-system steering. The 1-mm-thick C BBO and half-wave plate (HWP) are used for walk-off compensation. A HWP, a quarter-wave plate (QWP) and a polarizer on both sides of Alice and Bob are used to perform measurements on single-photon polarization states. All photons are filtered by interference filters (3 nm) and are measured by single-photon counting modules. A time-to-amplitude converter then records coincidences.
Experimental verification of multidimensional steering of single quantum systems
The steering features of the entangled states for EPR steering and the states of single quantum systems sent from Alice to Bob can be revealed by steering witnesses . Two different kinds of steering witnesses can be derived to certify both types of steering. These witnesses ensure secure QKD using qudits and provide criteria for efficiently evaluating experimentally quantum logic gates (U in Fig. 4) of arbitrary computing size. Furthermore, they have practical uses for evaluating one-way quantum computing and quantum communication with entangled qudits and verifying genuine multipartite EPR steering.
Fig. 6: Experimentally multidimensional EPR steering and single-system (SS) steering . Theoretical and experimental values of the ratio, Rd, are shown here.
The kernels of one specific kind of these steering witnesses have structures that are similar to those used for certifying genuine multipartite ERP steering . One of the important features of such steering witnesses is that the ratio between the measured witness and the maximum value achieved by classical mimicries, denoted by Rd, is monotonically increasing with the dimensionality , as a signature of quantum steering inequality with unbounded violation .
In our experimental investigation , we use an ultraviolet (UV) pulsed laser(an average power of 200 mW, repetition rate of 76 MHz) generated by second-harmonic generation with a Ti:Sapphire laser (780 nm, pulse duration of 120 fs) to pump a 2-mm-thick type-II BBO crystal to create polarization-entangled photon pairs by the spontaneous parametric down-conversion process (Fig. 5a). A d-dimensional entangled pairs where d=2n is composed of n pairs of polarization-entangled photons (qubits). Similarly, we assume that a d-dimensional single quantum system consists of n single photons therein (Fig. 5b). This strategy was used in our previous experiments to test multidimensional Bell inequalities .
Figure 6 depicts that all the experimental results indicate Rd >1, which implies the prepared sources are identified as steerable. In particular, the experimental ratios increase with d for both EPR and single-system steering, which are consistent with the theoretical predictions of multidimensional quantum steering.
NO-CLONING of QUANTUM STEERING
While one can utilize steering witnesses to verify quantum steering and confirm that all the parties involved in quantum-information tasks are trusted, this certification does not explicitly indicate how quantum steering preserves the security. To investigate this issue, we study how EPR steering behaves when a universal cloning machine clones one-half of a maximally entangled pair of qudits . The case where the qudit sent from Alice to Bob is cloned by a universal cloner is considered as well. See Fig. 7.
We use a criterion based on the mutual information between qudits to determine the steering between Alice and Bob and Alice and Charlie. We find that EPR steering and single-system steering as verified by this criterion can only be found in one of the copy subsystems but not both. Such no-cloning restriction shows the physical reason why steering can be used to secure quantum information against cloning-based attacks.
Quantum steering provides new resources and effective approaches to exploit quantum correlations and implement quantum-information protocols with partially untrusted measurement devices.
We investigated and described the characteristics of genuine multipartite EPR steering and steering of single quantum systems. These characterizations serve experimentally efficient quantum witnesses for verifying quantum steering. With such tools, we experimentally observed both types of quantum steering in photonic systems. In addition, we showed that it is impossible to observe quantum steering, as described by the mutual information criterion, in the two copies of a quantum cloner at the same time. Steering cannot be shared with a third party by using a universal cloning machine, which secures the related quantum-information tasks. These concepts and methods may play a role and find more applications in future quantum networks.
Fig. 7: Cloning quantum steering: (a) EPR steering, and (b) single-system steering . After cloning, the qudit B is sent to Bob and the qudit C, together with the ancilla C', is sent to Charlie. Each of the three parties has an apparatus to implement two complementary measurements mi for m=A, B, C and i=1, 2. These measurements help them to test the criteria for the no-cloning restriction on quantum steering.
Acknowledgement: The author acknowledges his collaborators and co-authors on the specific projects presented here, and expresses thanks for the support of the National Cheng Kung University and the Ministry of Science and Technology (MOST), Taiwan, under Grant No. MOST 104-2112-M-006-016-MY3.
 E. Schrödinger, Naturwissenschaften 23, 807 (1935); 23, 823(1935); 23, 844 (1935).
 H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett.98, 140402 (2007).
 C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, Phys. Rev. A 85, 010301(R) (2012).
 A.Montanaro, npj Quantum Information 2, 15023 (2016).
 I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014).
 N. Gisin and R. Thew, Nat. Photon. 1, 165 (2007).
 D. A. Lidar (Editor), T. A. Brun (Editor), Quantum Error Correction (Cambridge Univ. Press, 2013).
 H. J. Kimble, Nature 453 1023 (2008).
 L.-M. Duan and C. Monroe, Rev. Mod. Phys. 82, 1209 (2010).
 M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).
 C.-M. Li, K. Chen, Y.-N. Chen, Q. Zhang, Y.-A. Chen, and J.-W. Pan, Phys. Rev. Lett. 115, 010402 (2015).
 M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. Lett. 59, 1829 (1999).
 Y.-A. Chen, A.-N. Zhang, Z. Zhao, X.-Q. Zhou, C.-Y. Lu, C.-Z. Peng, T. Yang, and J.-W. Pan, Phys. Rev. Lett. 95, 200502 (2005).
 R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003).
 M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A 69, 062311 (2004).
 O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
 C.-M. Li, K. Chen, A.Reingruber, J.-W. Pan, andY.-N. Chen, Phys. Rev. Lett. 105, 210504 (2010).
 K. Chen, C.-M. Li, Q. Zhang, Y-A. Chen, A. Goebel, S.Chen, A. Mair, and J-W. Pan, Phys. Rev. Lett. 99, 120503(2007).
 R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
 H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Nat. Phys. 5, 19 (2009).
 Y.-N. Chen, C.-M. Li, N. Lambert, S.-L. Chen, Y. Ota, G.-Y.Chen, and F. Nori, Phys. Rev. A 89, 032112 (2014).
 C.-M. Li, Y.-N. Chen, N. Lambert, C.-Y. Chiu, and F. Nori, Phys. Rev. A 92, 062310 (2015).
 C.-M. Li, H.-P.Lo, L.-Y. Chen, and A.Yabushita,arXiv:1602.07139 (2016).
 A.Rutkowski, A.Buraczewski, P.Horodecki, and M.Stobińska, arXiv:1603.07861 (2016).
 H.-P. Lo, C.-M. Li, A.Yabushita, Y.-N. Chen, C.-W. Luo, T. Kobayashi, Sci. Rep. 6, 22088 (2016).
 C.-Y. Chiu, N. Lambert, T.-L. Liao, F. Nori, and C.-M. Li, npj Quantum Information 2, 16020 (2016).
Che-Ming Li is an Associate Professor in the Department of Engineering Science (DOES), National Cheng Kung University (NCKU). After receiving a Ph.D. from the National Chiao Tung University, he worked at the Department of Physics, NCKU as a postdoctoral research associate from 2008 to 2009. Then he had the position of postdoctoral research scholar with appointment of National Science Council(NSC) from 2009 to 2011, and became an assistant research scholar of NSC from 2011 to 2012 before joining DOES, NCKU. He was a guest scientist at the Physikalisches Institute, Ruprecht-Karls-Universität Heidelberg, Germany from 2008 to 2010. His research field is quantum optics and quantum information science.