arXiv:1411.3040v2 [quant-ph] 9 Dec 2015 xs nage ttsb hc teigcni nyone only there in can that steering shown which been by has states entangled it exist example, For potential and extensions. quantification properties, unique steering’s al. et [9]. not (1SDI-QKD) can QKD device- parties one-sided independent i.e., the apparatus, of measurement their one trust quantum when to (QKD) application Fur- distribution has key [6–8]. effect reported steering been the have thermore, steering of EPR steerability of demon- strations the experimental Several detect steering systems. to quantum and bipartite introduced quantum are [4] verify [5] wit- inequalities to measures entanglement steering used EPR and widely correlations, been inequalities have Bell which nesses, like not- worth that, is Bell It ing between entanglement. hierarchy and for- strict of steering new non-locality, a one This illustrates of also untrusted). devices mulation (or uncharacterized measurement en- are share the them can if parties two even that EPR tanglement information- showing of a [3] of concept task terms the theoretic in Recently, reformulated been for pair. has chosen steering the settings of measurement re- particle between the This shared each and pair Bob the [1]. one and of entanglement measurement Alice of the party, of both ability remote on choice another the lies her of is state through steering the Bob, affect the Such to to Alice, party, response [2]. in paradox Schr¨odinger [1] EPR by introduced nally ∗ [email protected] ic h eomlto fERsern yWiseman by steering EPR of reformulation the Since origi- was steering (EPR) Einstein-Podolsky-Rosen 3,teehsbe ag fivsiain into investigations of range a been has there [3], etfigsnl-ytmsern o unu informatio quantum for steering single-system Certifying ermtl rprdb esrn n atceo nentangl an of particle one for measuring steering by prepared remotely be h teaiiyteen hc r plcbebt osing to both applicable are which therein, steerability the a en fmmcigsern.Rln u ‘false-steerin out Ruling u steering. the mimicking enable of conditions means steering cal our case single-system the ASnmes 36.d 36.d 03.67.Lx 03.67.Dd, 03.65.Ud, numbers: PACS descri mu we genuine fai Finally, identifying and for communication, e tools size. quantum an as entanglement-based arbitrary as function serve also of conditions can gates these they logic that quantum the in that of computation, show quantum also and We in attack cations qudits. using individual distribution cloning-based key quantum both against channels h-igLi Che-Ming isenPdlk-oe ER teigdsrbshwdiff how describes steering (EPR) Einstein-Podolsky-Rosen 1 eateto niern cec,Ntoa hn ugUni Kung Cheng National Science, Engineering of Department .INTRODUCTION I. 5 eateto hsc,Uiest fMcia,AnAbr M Arbor, Ann Michigan, of University Physics, of Department 3 eateto hsc,Ntoa hn ugUiest,Tai University, Kung Cheng National Physics, of Department single 1 4 , ainlCne o hoeia cecs snh 0,Tai 300, Hsinchu Sciences, Theoretical for Center National 2 , ∗ quantum uhNnChen Yueh-Nan 2 ES IE,Wk-h,Siaa3109,Japan 351-0198, Saitama Wako-shi, RIKEN, CEMS, d dmninlsses(uis n eieecetconditio efficient devise and (qudits) systems -dimensional 3 , Dtd ue2,2021) 23, June (Dated: 4 el Lambert Neill , n n-a unu optn 1] ial,w dis- we Finally, [19]. computing quan- quantum [18] validate model one-way circuit to quantum and used the both then for computation and tum standard we setting the in Moreover, EPR applied be non-local cases. can the qudit conditions extend these to how and show [9] [17] qubit inequality tempo- from steering the 1SDI-QKD the violating of of meaning analog strict ral a give implementation. results experimental Our for settings measurement u teigta a eue ocriytereliability the certify to used be ex- quan- can of for there that criteria does efficient steering experimentally tum arises: and strict question a natural ist a steering, of tum bounds security the to [17]. connection QKD was a inequality an through such op- of found nontrivial violations a to and steering meaning [17], the erational introduced of been analog steer- has temporal multipartite ef- inequality a genuine steering Moreover, to bipartite Bob [13–16]. generalized ing original from been the not has addition, but fect In Bob to Alice. Alice to from [10–12], direction teigcniin edol the only Both need dimension). conditions arbitrary communica- of steering to (systems quantum applied qudits be and using can tion computation which and quantum errors steer- of both presence quantum the single-system in genuine typical ing identify a of intro- to are illustration duced conditions schematic steering a novel Two for implementation. 1 Fig. See information quantum tasks. generic with steering quantum clear. nect not is any, quantum if of computation, role quantum the quan- in fact, generic steering In for steering processing. quantum information unified of tum no use is [9] the there 1SDI-QKD for However, that scheme shown steering. been EPR was from it benefits far, So tasks? tion ihti ral dacdudrtnigo quan- of understanding advanced greatly this With ee epeetasml u nfidpcuet con- to picture unified but simple a present we Here, both hu n-a unu optto,secure computation, quantum one-way thful 2 hn-iChiu Ching-Yi , ’seaishsipiain o securing for implications has scenarios g’ esse steering le-system abgosrln-u fgnrcclassi- generic of ruling-out nambiguous unu omncto n unu computa- quantum and communication quantum ehwtennlclERvratof variant EPR non-local the how be rn nebe fqatmsae can states quantum of ensembles erent dpi.Hr,w netgt quantum investigate we Here, pair. ed oeetatcswe implementing when attacks coherent esern odtosas aeappli- have also conditions steering se tprieERsteering. 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ρ Alice aˆ UaU † Bob s i ˆi Ai U Bu(i) 1 2 1 2
ai bu(i) 0,1,...,d-1 0,1,...,d-1
FIG. 1. Single-system steering for quantum information tasks. The statea ˆi is sent from Alice to Bob. Herea ˆi is a post- measurement state of a qudit ρs under the measurement Ai for i = 1, 2. 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. 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. To identify whether Alice can implement such steering, Bob can use the steering condition (7) or (9) to rule out the results mimicked by generic classical strategies. As illustrated, Bob performs measurements Bu(i) to implement these certifications. These steering conditions ensure secure quantum communication and faithful quantum computation (see table I). Here, it is allowed that Alice and Bob have no spatial separation but access the single system at different times.
cuss the implications for certifying genuine multipartite Bu(i) for i = 1, 2 are specified by the orthonormal bases EPR steering and implementing multipartite secret shar- b U b b = b v with the results { u(i) u(i) ≡ | iii | u(i) i ∈ } ing with partial uncharacterized measurement devices. b u(i) . { In an} ideal case, the state received by Bob is the same as the initial statea ˆi prepared by Alice under the trans- II. QUANTUM STEERING FOR SINGLE formation . In practical situations, however, noise from SYSTEMS the environmentU or other artificial effects introduce an unknown source of randomness. In order to explicitly In the scenario of single-system quantum steering, Al- qualify whether Alice can steer the states of the parti- ice’s ability to affect the quantum state Bob has access cles eventually held by Bob, and rule out either third- to is based on both her ability to prepare an arbitrary party eavesdropping, classical mimicry of the channel, or quantum state to send to Bob and her knowledge, if any, to qualify the quality of the channel itself, we consider about the state Bob finally receives (which may differ the following generic classical means of describing state from her prepared state, for various reasons) [20]. If Al- preparation, transitions between states, and the limits to ice has full information about the quantum system Bob which they can influence the measurement results of Bob. is holding, she is capable of steering this system into an First, we assume that the state of the particle sent arbitrary state. Alice can follow two steps to achieve this by Alice can be described by a classical realistic the- (Fig. 1). ory which predicts the particle is in a state described First, Alice prepares a specific state of a qudit with a by a fixed set (A1 = a1, A2 = a2). Suppose next that P (a1,a2) is the probability that, before the measure- given initial state state ρs generated from some quan- tum source, before sending it to Bob, by performing ments are performed, the particle is in a state (a1,a2). Under this assumption the marginal probability P (ai) complementary measurements Ai for i = 1, 2. Once and the conditional probability P (ai aj ) for i, j = 1, 2 the particle is measured with a chosen Ai, ρs becomes | aˆ a a for a v = 0, 1, ..., d 1 , where the d and i = j should follow the relation i ≡ | iiiih i| i ∈ { − } 6 states constitutes an orthonormal basis ai i [20]. The P (a1,a2)= P (a1)P (a2 a1)= P (a2)P (a1 a2). (1) set of states a is complementary{| to thei } state set | | {| 2i2} d 1 a2a1 Second, we assume that the particle state can change, a1 by defining a2 =1/√d − ω a1 , with {| i1} | i2 a1=0 | i1 while it is being transmitted from Alice to Bob, from ω = exp(i2π/d). P (a1,a2) to an unknown state ρλ with a transition prob- Second, the particle in the statea ˆ is then sent to Bob. i ability P [λ (a1,a2)]. Then, the state of the parti- Here Bob will not know the state of particlea ˆ sent from | i cle changes to P (a1,a2) P [λ (a1,a2)]ρλ. To Alice. To steer Bob’s statea ˆ into other quantum states a1,a2 λ | i connect this stateP with our steeringP scenario, where (ˆai) UaˆiU †, Alice can directly perform the unitary the state of the particle, and how it evolves, may de- Uoperation≡ U by herself before the particle transmission, pend on the choice to measure a1 or a2 individually, or publicly, via a classical channel, ask Bob to apply U on we rewrite the transition probability as P [λ (a1,a2)] = a . While the quantum operation is announced pub- | | iii U P (λ ai)P (aj λ, ai)/P (aj ai) [22]. From which, combined licly, the state (ˆai) is still unknown to Bob. It is clear with| the relation| (1), the| joint probability of finding that Alice hasU complete knowledge about the quantum (a1,a2) and observing λ as the final state can be explic- system held by Bob since the state ρs, the measurement itly represented by Ai and the subsequent operation are designed by Alice. U P [(a ,a ), λ]= P (a ,a )P [λ (a ,a )] When Bob performs measurements on his particle after 1 2 1 2 | 1 2 the operation , his two complementary measurements = P (a )P (λ a )P (a λ, a ). (2) U i | i j | i 3
As shown by (1) and (2), it does not matter what order The kernel of our first steering condition is Alice does a series of measurements, the joint probability 2 d 1 will always be the same. The state of the particle that − P (a ,b ). (6) Bob holds is then SdU ≡ i u(i) Xi=1 ai=0;Xbu(i) =ai d 1 − ρ = P (a ) P (λ a )ρ . (3) For ideal steering the maximum value for the kernel is B i | i λ aXi=0 Xλ dU = 2. Whereas, for the states described by Eq. (3), S we have αR = 1+1/√d. Thus the quantum steering When summing over all a1 and a2, Eq. (2) becomes condition reads 1 P (λ)= P (a1)P (λ a1)= P (a2)P (λ a2). (4) dU > 1+ . (7) | | S √d Xa1 Xa2
With the above classical realistic description of Alice’s For any unsteerable states the measured kernel will not violate this bound. To determine the maximum value of states, the state received by Bob becomes independent the kernel supported by realistic theories, we consider the of the measurement setting chosen by Alice, i.e., ρB = expectation value of the kernel dU for the state ρB (3). λ P (λ)ρλ, implying that Bob always has the same state S Then dU becomes Pwhatever measurement Ai and operation Alice designs. U S This means Alice cannot steer Bob’s states. We call the 2 d 1 − states with this feature unsteerable. The above proof = Tr[U a a U †ρ ]P (λ a )P (a ). SdU,R | iiii h i| λ | i i can be seen as equivalent to that used in the derivation Xi=1 aXi=0 Xλ of EPR steering inequalities and extended EPR steer- ing conditions, where Alice’s measurement results are as- This can be further manipulated to give sumed to be a classical distribution. See Appendix A for P (λ) Tr[ m m ρ ] + Tr[ n n ρ ] detailed discussions. SdU,R ≤ | i11h | λ | i22h | λ Xλ Finally, if Alice’s state and the unknown states ρλ are 1 described by a classical theory of realism, and thus only 1+ , classically correlated with Bob’s results, then the descrip- ≤ √d tions Eqs. (1), (2) and (4) are applicable to ρ as well. λ where m,n v. The first inequality is derived by using However, here Bob’s measurement results are assumed to the relation∈ (4) about P (λ), and the classical bound α = be based on measurements on a quantum particle. Thus R √ the expectation values of the two mutually-unbiased mea- 1+1/ d is then obtained by determining the maximum surements B and B with respect to the unknown eigenvalue of the operator m 11 m + n 22 n . u(1) u(2) Our second steering condition| i h is| based| i onh the| mutual quantum states ρλ obey the quantum uncertainty rela- tion in the entropic form [23] information between Alice and Bob. From the point of view of information shared between sender and receiver, the ability for Alice to steer Bob’s state is confirmed if the H(Bu(1) λ)+ H(Bu(2) λ) log (d), (5) | | ≥ 2 mutual dependence between the measurement results of d 1 Alice and Bob is stronger than the dependence of Bob’s where H(Bu(i) λ)= b− =0 P (bu(i) λ) log2 P (bu(i) λ). | − u(i) | | measurement outcomes on the unknown states ρλ and P ρB. This condition of steerability can be represented in terms of the mutual information as follows, III. QUANTUM STEERING CONDITIONS 2 2 I(B ; A ) > I(B ; λ ). (8) A. Steering conditions u(i) i u(i) { } Xi=1 Xi=1 In order to distinguish steerability from the results From the basic definition of mutual information, Eq. (8) mimicked by the methods based on the classical theories implies that considered above, in what follows we will introduce two 2 d 1 2 novel quantum steering conditions of the from > αR, − S P (a )H(B a ) < P (λ)H(B λ). where is the kernel of the criterion and αR is the max- i u(i)| i u(i)| imum valueS of the kernel supported by classical theories. Xi=1 aXi=0 Xi=1 Xλ For ideal steering, will be maximized. Since ruling Imposing the relation (5) on the state ρ , we obtain the out classical mimicryS is equivalent to excluding unsteer- λ second steering condition of the form able states (3), exceeding the αR will deny, or rule out, processes (e.g., noisy channels) that make once steerable 2 d 1 − 1 states unsteerable and thus assist in confirming genuine U = P (a ) H(B a ) > log . (9) Sent − i u(i)| i 2 d quantum steering. Xi=1 aXi=0 4
In addition to the steering conditions devised here, vi- channel real, the value of the kernel dU is olating the temporal steering inequality [17] can serve as U S 2 d 1 an indicator of single-system steering. In Appendix B, we − = P (a ) F (a ,u(i)), show that this inequality can be derived from the classical SdU i i conditions (1) and (3), which provides a strict meaning of Xi=1 aXi=0 violating that inequality. As shown therein, the steering where the probabilities P (a ) = Tr[ρ aˆ ] and the state conditions are related to practical quantum information i s i fidelities [18] F (a ,u(i)) = Tr[ (ˆa )ˆa ]. Let us as- tasks and then more useful than the temporal steering i real i u(i) sume that an error is introducedU by a quantum cloning inequality from a practical point of view. See Appendix machine [24] which copies equally well the states of both C for a concrete demonstration of the sensitivity of these bases, F (a ,u(i)) = F , for all a v [25]. If Al- conditions. i ice wants to demonstrate steering of∈ Bob’s particle in In particular, one of the main advantages of the steer- the presence of such eavesdropping, they have to find ing criteria is that they can be efficiently implemented in =2F > 1+1/√d, or alternatively the state fidelity minimum dU experiments. The two measurement settings Smust satisfy the condition: are sufficient to measure the kernels and U. In SdU Sent addition, they are robust against noise. See Appendix D 1 1 F > (1 + ). for demonstrations of the robustness of our steering con- 2 √d ditions and the EPR steering inequality for single sys- tems. It is equivalent to saying that the disturbance, D =1 F , or error rate, has to be lower than a certain upper bound− Dind = (1 1/√d)/2. This bound is exactly the same as B. Implications of the steering conditions the well known− security threshold [24]. For the second steering condition (9), we derive a sec- We use the generic classical means of describing state ond criterion on the state fidelity F [25]: preparation and transitions between states to consider ˜ 1 the threshold αR for the steering conditions. Such condi- F > log2(d), tions certify quantum steering (EPR steering and single- −2 system steering) when the measurement apparatus of Al- where F˜ F log (F ) + (1 F ) log [(1 F )/(d 1)]. ice is uncharacterized or both of the Alice’s measurement ≡ 2 − 2 − − This provides the upper bound, Dcoh, on D under co- devices and the operation U are untrusted. herent attacks. If D
A , one can obtain the lower bound of the process fi- 2 TABLE I. A summary of the steering conditions for quantum delity Fprocess Tr[ real ] by F process Faˆ (ˆa ) + ≡ U U ≥ 1→U 1 information processing. The criteria derived from steering Faˆ (ˆa ) 1 [28]. Hence, using the steering condition 2→U 2 − conditions for secure quantum communications and faithful together with the above relation, we obtain a condition quantum computations are represented in terms of the state for a faithful quantum process in terms of process fidelity: fidelity F and the process fidelity Fprocess, respectively. 1 Condition Communication Computation Fprocess > . 1 1 1 1 SdU > 1+ F > 1+ F > d √d 2 √d process √d 1 1 SentU > log F˜ > − log (d) Fprocess > 1 − 2D Taking a two-qubit entangling gate for an example, 2 d 2 2 coh this indicator coincides with the well known criterion [28] in terms of the concurrence C [29]. Two qubits can be considered or recast as a single system with a level num- ber d = 22 = 4. The entanglement capability of a two- fact that, by changing the role of λ [31], both steering qubit entangling gate, like a controlled-not operation, conditions (7, 9) can be used to detect EPR steering for can be defined by the minimal amount of entanglement bipartite d-level systems shared between Alice and Bob. that can be generated by the real operation rel. In See Appendix A.1.d. However, the converse is also true, terms of the concurrence C, a measure of quantumU en- such that EPR steering inequalities, for example, the in- tanglement, it is found that C 2Fprocess 1 [28]. Then, equalities used in the experiments [6, 7], can serve as for a nontrivial gate, one requires≥ C > 0,− which implies criteria for single-system steering (see Ref. [17] and Ap- that Fprocess > 1/2. Our condition on Fprocess derived pendix B). from the steering condition (7) coincides with this crite- When using the bipartite counterpart of steering con- rion. Note that the condition derived from the second ditions (7, 9) for quantum communication, one obtains steering condition (9) is Fprocess > 62.14% and tighter security criteria for quantum channels that are the same than that resulted from the condition (7). as the single-system case, which can thus be considered The above results can be efficiently implemented with as a d-level extension of 1SDI-QKD [9]. Similarly, the the minimum two measurement settings. This is es- EPR steering conditions give criteria of computation per- pecially useful to evaluate experimental quantum logic formance for quantum gates realized in one-way modes gates of arbitrary size, for example, an experimental [19]. A quantum gate U can be encoded in a bipartite three-qubit Toffoli gate with trapped ions [30]. For a maximally-entangled state [32]: three-qubit gate (d = 23), the condition on the process fidelity is Fprocess > 1/√8 35.36%. The process fidelity ≈ d 1 of the experimental quantum Toffoli gate with trapped 1 − ions reported in [30] is F = 66.6(4)%, which can be U = ai Out(ai) , process | i √d | ii | i identified as being functional according to our proposed aXi=0 criterion. When the number of qubits N increases, the classical bound will decrease with √d = 2N/2 and ap- where Out(a ) U In(a ) , and In(a ) is the input proach zero when N is large. i i i state of| the quantumi ≡ | gate iU. A| readouti of the gate The second steering condition (9) can be used to eval- operation, Out(a ) , depends on the measurement result uate experimental quantum gates. When using the same i a , which is| just thei effect of EPR steering. See Appendix conditions as D to consider the quality of gate opera- i coh E for an application to a two-qubit gate realized in the tions under coherent attacks, one can obtain the condi- one-way mode. Hence our EPR steering conditions can tion on F in terms of D : process coh indicate reliable gate operations for experiments [33] in the presence of uncharacterized measurement devices. Fprocess > 1 2Dcoh, − The idea of bipartite steering conditions based on (7, which is tighter than the criterion derived from the first 9) can be straightforwardly generalized to genuine multi- condition (7). The relation F = Faˆ1 (ˆa1) = Faˆ2 (ˆa2) partite EPR steering. The main ingredient is to consider →U →U is used above. Alternatively, the gate can be also qual- a kernel, from either the joint probabilities like Eq. (6) ified if the average state fidelity satisfies F > 1 Dcoh. or the entropic conditions in Eq. (9), for a specific bipar- − Table I summarizes the above two conditions on Fprocess. tition of a multipartite system. Then a complete kernel of steering condition is composed of the joint probabili- ties, or entropic conditions, for all possible bipartitions of VI. EPR STEERING CONDITIONS AND the multipartite system. See [16] for concrete examples APPLICATIONS for steering conditions based on (7). In particular, the entropic condition for genuine multipartite EPR steering As discussed above, traditional EPR-steering and using (9) could be useful for multipartite quantum secret single-system-steering scenarios mirror each other. In sharing [34] when coherent attacks occur in the quantum the language we use, this can be understood from the network. 6
VII. CONCLUSION AND OUTLOOK Second, Alice keeps one particle of the entangled pair and sends the other particle to Bob. A subsequent uni- We investigated the concept of quantum steering for tary operator U is applied on the Bob’s subsystem ac- single quantum systems and pointed out its role in quan- cording to the instructions of Alice. This transformation tum information processing. We derived two novel steer- can be done either by Bob after receiving the particle, ing conditions to certify such steering. These conditions or by Alice herself before the transmission of the parti- ensure secure QKD using qudits and provide new criteria cle. After such transformation, the state vector of the for efficiently evaluating experimentally quantum logic bipartite system becomes gates of arbitrary computing size (see table I). Moreover, d 1 the bipartite counterparts of our steering conditions can 1 − (I U) Φ = a1 U b1 . detect EPR steerability of bipartite d-level systems, and ⊗ | i √ | iA1 ⊗ | iB1 d a =Xb =0 have practical uses for evaluating one-way quantum com- 1 1 puting and quantum communication with entangled qu- Then, depending on Alice’s measurement result a1, the dits and verifying genuine multipartite EPR steering. It state of the particle finally held by Bob can be steered may be interesting to investigate further the connection into a corresponding quantum state, Uaˆ1U †, which is the between single-system steering and other types of quan- same as the result derived from single-system steering. tum steering such as one-way steering [10–12]. When the state Φ is represented in the bases a2 A2 a a v and| i b b b v , we have{| i ≡ | 2i2 | 2 ∈ } {| 2iB2 ≡ | 2i2 | 2 ∈ } ACKNOWLEDGMENTS 1 Φ = a2 A2 b2 B2 , (A2) | i √d . | i ⊗ | i a2+Xb2=0 C.-M.L. acknowledges the partial support from the . Ministry of Science and Technology, Taiwan, under Grant where = denotes equality modulo d. Through the same No. MOST 101-2112-M-006-016-MY3 and MOST 104- method as that shown above, Alice can steer the state 2112-M-006 -016-MY3. Y.-N.C. is partially supported by of Bob’s particle into the quantum state, Uˆb2U †, by the the Ministry of Science and Technology, Taiwan, under measurement on her subsystem with a result a satisfying . 2 Grant No. MOST 103-2112-M-006-017-MY4. the correlation a2 + b2 = 0. We remark that, for an EPR source creating entangled states that are different from Φ , the transformation U | i Appendix A: Comparing single-system steering with could be implemented in other ways. For example, when EPR steering Alice and Bob share bipartite supersinglets [35], which are expressed as In this section we compare EPR steering with single- 1 ai system steering by discussing their basic assumptions and Ψ = ( 1) ai bi , (A3) | i √d − | iAi ⊗ | iBi the classical mimicries, or simulation, of steering effects ai+Xbi=d 1 − (Fig. 2). This provides a clear connection between EPR and single-system steering and the steering conditions for for i = 1, 2, Alice can steer the state of Bob by di- both cases discussed in our work. From this comparison, rectly measuring her qudit in a basis featured in U. we show that classical mimicry or simulation can in both Since supersinglets are rotationally invariant [35], i.e., cases be considered as equivalent. (R R) Ψ = Ψ , where R is a rotation operator, Al- ice’s⊗ measurement| i | i in the basis R a will steer the { | iii} state of Bob’s qudit into a corresponding state, R bi i, for a + b = d 1. For d = 2, supersinglets become| uni-i 1. EPR steering for quantum information i i − processing (QIP) tary invarient and provide a resource for implementing any unitary transformations U to Bob’s qubit. Compared with the single-system steering [Fig. 2(a)], the scenario of EPR steering also consists of two steps: 2. Steering conditions First, Alice generates a bipartite entangled system from an entanglement source (or called EPR source) [Fig. For both ideal single-system and EPR steering sce- 2(b)]. To have a concrete comparison, let us assume that ˆ this entangled state is of the form narios, the state received by Bob, bi, is the same as or perfectly correlated with the initial statea ˆi prepared by d 1 Alice under the transformation U. However, for Bob’s 1 − Φ = a1 b1 (A1) limited knowledge about the measurements used or the | i √d | iA1 ⊗ | iB1 a1=Xb1=0 particle prepared by Alice, her measurement results be- come untrusted to Bob. He is uncertain whether these v where a1 A1 a1 1 a1 and b1 B1 b1 1 b1 measurements and state preparation are qualified. In the v . {| i ≡ | i | ∈ } {| i ≡ | i | ∈ worst case where Alice’s measurement outcomes may be } 7
(a) (b) ρ Aliceaˆ UaU † Bob Alice Bob s A i ˆi B A B i U u(i) i EPR U u(i) 1 2 1 2 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 (c) (d) Bob Bob (a ,a ) ρ ρ (a ,a ) ρi ρ 1 2 i λ B 1 2 λ B Alice U u(i) Alice U u(i) 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 (e) (f)
Alice Bob Alice ρi Bob ρi ρλ ρλ Bu(i) a a Bu(i) (a ,a ) U ( 1, 2 ) U 1 2 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1
FIG. 2. Comparison between single-system steering and EPR steering. We compare these two scenarios by, first, their basic concepts of ideal single-system steering (a) and ideal EPR steering (b), and, second, classical mimics of single-system steering (c), (e) and EPR steering (d), (f). For the ideal case, Alice can use the effect of EPR steering, by sharing the entangled states (EPR source), to implement the operation U on the state of Bob’s qudit. While the resources utilized for quantum steering are different, the state of the particle finally held by Bob can be steered into a corresponding quantum state, UaˆiU †, for both quantum steering scenarios. To distinguish classical mimicry from genuine quantum steering, the respective classical models based on realistic theories (c) and (d) are introduced. These “classical simulations” can be concretely represented in the practical descriptions of, for example, unqualified measurements of Alice and the unqualified operation U performed by Alice or Bob [(e) and (f)]. As shown in (e) and (f), these effective simulations are equivalent.
randomly generated from her apparatus, classical simula- output states ρi as the measurement setting i is chosen tions then can describe Alice’s measurement results. To by Alice. After the unqualified operation U, the state ρi show that Alice has true steerability in practical situa- becomes the unknown state ρλ which constitutes an un- tions, the steering conditions (7) and (9) have been in- steerable state ρB [Eq. (3)]. Here the joint probability of troduced to distinguish genuinely quantum steering from finding (a1,a2) and observing λ as the final state satisfies the classical mimicry. In what follows, we will detail the the classical relation (2). It is equivalent to say that Alice classical mimicry and their implication for practical ap- can consider the joint set (a1,a2), with the probability plications. With these examples, it will be clear that the of occurrence P (a1,a2), as describing predetermined in- proof for single-system conditions can be seen as equiva- structions for her to prepare and send a particle with lent to that used in the derivation of EPR steering con- final quantum state ρλ to Bob. See Fig. 2(e). ditions. It is also possible that the operation U is qualified but the measurement device of Alice is not. The two realism assumptions are applicable to this case as well. a. Mimicry of single-system steering The above classical mimicry scenario can be recast such that the output states ρi already correspond to the un- (0) In the case of single-system steering, as detailed in the known state ρλ , see Fig. 3(a). It does not matter what main text, the classical mimicry of steering is based on the subsequent qualified operation on the particle U is, the realistic assumptions that (1) the state of the parti- the final states held by Bob ρλ constitute an unsteerable state ρ . From a practical point of view, similarly, one cle sent by Alice can be described by a fixed set (a1,a2), B can think that Alice’s measurement apparatus randomly and (2) the state can change from (a1,a2) to another state λ which corresponds to a quantum state of the qu- generates outcomes with the probability of occurrence (0) dit ρλ finally held by Bob, see Fig. 2(c). In order to see P (a1,a2) that correspond to unknown output states ρλ this mimicry from a practical point of view, one can think [Fig. 3(c)]. that, for example, such a situation arises as a result of the unqualified measurement device and the states of parti- cles sent to Bob. For some reason, Alice’s measurement b. Mimicry of EPR steering apparatus does not properly output real measurement results ai but randomly generates outcomes with a dis- The above scheme for mimicking single-system steer- tribution P (a1,a2) [see Eq. (1)] that correspond to some ing can be readily mapped to the case of EPR steering. 8
(a) (b) (a ,a ) ρ(0) ρ Bob a a ρ(0) ρ Bob 1 2 λ λ B ( 1, 2 ) λ λ B Alice U u(i) Alice U u(i) 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 (c) (d) Alice (0) Bob Alice (0) Bob ρλ ρλ ρλ ρλ Bu(i) a a Bu(i) (a ,a ) U ( 1, 2 ) U 1 2 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1
FIG. 3. Steering mimicries where the operation U is qualified but the measurement device of Alice is not. The classical mimics of single-system steering (a) and EPR steering (b) are based on the realistic assumptions that (1) the state of the particle sent by Alice can be described by a fixed set (a1,a2), and (2) the state can change from (a1,a2) to another state λ which corresponds to a quantum state of the qudit ρλ finally held by Bob. One can concretely represent these scenarios in the practical descriptions of unqualified Alice’s apparatus (c) and (d), respectively. These concrete simulations are then shown to be equivalent.
Here, the mimicry of EPR steering depends on two simi- joint probability of finding (a1,a2) and observing λ as lar assumptions: (1) the state of the particle held by Alice the final state in this case satisfies the classical relation can be described by a fixed set obeying realism (a1,a2), (2). and (2) a given set (a1,a2) corresponds to some quantum state, ρλ, of the qudit finally held by Bob, see Fig. 2(d). The unqualified bipartite state shared between her and c. The equivalence between the steering mimicries Bob, and a subsequent unqualified operation, can result in such assumptions. For example, let us assume that the entanglement source does not create entangled pairs With the above concrete explanations of the classical but a qudit with state ρi for Bob and another separable mimicry for both the single-system steering and EPR particle for Alice instead. For the state ρi there is a cor- steering, one can interpret these two classical scenarios as responding measurement setting i chosen by Alice, for being equivalent to each other. See (e) and (f) in Fig. 2 which Alice’s measurement device creates an output of a and (c) and (d) in Fig. 3. Following the same approach random signal with a distribution described by the prob- based on the realistic assumptions and their practical sce- ability P (a1,a2) (1). The subsequent operation U takes narios, in what follows we will discuss two more cases to ρi to an unknown state ρλ, and then the final state held complete the proof of the equivalence between the steer- by Bob is unsteerable (3). The classical relation (2) is ing mimicries. again applicable to this transition between states. Here The case where Alice’s measurement apparatus is un- it is reasonable to incorporate the entanglement source qualified, while the EPR source functions as expected, into the measurement apparatus as a single unqualified can raise two other possible scenarios which again can experiment setup for Alice. See Fig. 2(f). Then it is be shown to be covered by ”realism” assumptions. Fig- effectively a scenario where Alice observes a set (a1,a2) ure 4(a) depicts one of the possibilities. As the oper- appearing with probability P (a1,a2) which creates a par- ation U is unqualified, one can practically think that ticle with a final quantum state ρλ for Bob. Alice’s measurement apparatus generates random out- comes with a distribution P (a1,a2), independent of the As discussed in the above mimicry of single-system entangled pair generated from the EPR source. The sub- steering, it is possible that the operation U is quali- sequent operation makes the state of the qudit of the fied but Alice’s measurement apparatus, including the entangled pair sent to Bob, say ρi, change to ρλ as illus- EPR source, is not. In this case one can effectively con- trated by Fig. 4(c). It is clear that such mimicry of EPR sider that the unqualified EPR source outputs a fixed set steering is equivalent to the simulation of single-system (a1,a2) for Alice’s particle and a qudit that is already in steering described by Fig. 2(e) [see also Fig. 2(c)]. (0) an unknown state ρi = ρλ for Bob [Fig. 3(b)]. For any Figure 4(b) illustrates the other situation where the en- (0) qualified operation U on the particle state ρλ , the fi- tanglement source and the operation U are qualified but nal state held by Bob is still unsteerable. From the same Alice’s measurement apparatus is not. One of concrete practical point of view as introduced above, we can think examples for this case is as the following. The unqual- that the joint set (a1,a2), with the probability of occur- ified measurement device of Alice always measures her rence P (a1,a2), resulting from the random outcomes of particle of the entangled pair, say Φ , in the first basis Alice’s device, corresponds to a particle with final quan- a a v intrinsically whatever| i measurement set- {| 1i1 | 1 ∈ } tum state ρλ for Bob, see Fig. 3(d). It is clear that the ting Alice chooses, and it announces random signals a1 or 9
(a) (b) Alice Bob Alice Bob ρλ ρλ a a Bu(i) a a Bu(i) ( 1, 2 ) EPR U ( 1, 2 ) EPR U 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 (c) (d) Alice Bob Alice (0) Bob ρi ρλ ρλ ρλ a a Bu(i) a a Bu(i) ( 1, 2 ) U ( 1, 2 ) U 1 2 1 2
ai bu(i) ai bu(i) 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1 0,1,...,d-1
FIG. 4. Mimicries of EPR steering where the EPR source is qualified but the measurement device of Alice is not. (a) the unqualified operation is used, and (b) the operation used is qualified in the mimicry. These two possible situations can be described by the two realism conditions and represented in practical descriptions (c) and (d), respectively. The demonstration (c) has the analogue of single-system steering described by Fig. 2(e). The concrete mimicry (d) is equivalent to that of single-system steering depicted in Fig. 3(c).
a2 as an outcome. Such Alice’s measurement and the ran- where, for Alice who implements quantum measure- dom signals announced make the state of the qudit sent ments, her measurement outcomes a result from { u(i)} to Bob unsteerable, i.e, ρ = ρ(0) belongs to the same set the measurement described by the basis a i λ { u(i) u(i) ≡ b1 b1 v whatever measurement setting chosen by v {| i1 | ∈ } R ai i au(i) = ai . Here Bob uses the same mea- Alice and as such then constitutes an unsteerable state surements| i | as that∈ used} by Alice. For the EPR steering ρB after the operation U. See Fig. 4(d). This is an ana- conditions represented in the entropic forms, we have logue of EPR-steering mimicry to that of single-system 2 d 1 steering described by Fig. 3(c) [see also Fig. 3(a)]. − 1 (EPR) P (a ) H(B a ) > log , SentUΦ ≡− i u(i)| i 2 d Xi=1 aXi=0 d. EPR steering conditions for QIP (A6) for the state Φ shared by Alice and Bob, and | i As shown above, the mimicry of single-system steering 2 d 1 − 1 is equivalent to that mimicking EPR steering. Then the (EPR) P (a ) H(B a ) > log , SentRΨ ≡− u(i) u(i)| i 2 d steering conditions (7) and (9) for single systems can be Xi=1 auX(i) =0 mapped to verifications of EPR steering for bipartite d- (A7) dimensional systems. All such EPR steering conditions for the supersinglets. can certify the reliability of QIP when entangled pairs As detailed above, the mimicry of single-system steer- are shared between Alice and Bob. For the EPR steering ing based on realistic theories is equivalent to that of conditions which correspond to the criterion (7), when EPR steering where Alice’s outcomes follows realist the- the state Φ is used to mediate steering the condition is | i ories but Bob performs quantum measurements. Hence of the form the proof for the conditions (7) and (9) can be readily d 1 applied to the above EPR steering conditions. In addi- (EPR) − tion, following the same analysis of quantum communica- P (a1,bu(1)) SdUΦ ≡ tion based on single-system steering as introduced in the a =Xb =0 1 u(1) main text, these bipartite counterpart of steering condi- 1 + P (a2,bu(2)) > 1+ . (A4) tions provide security criteria for quantum channels that . √d is equivalent to the single-system cases. a2+Xbu(2) =0
Similarly, with proper changes to the above joint proba- bilities, we have the following steering condition for su- Appendix B: EPR steering inequality for persinglets single-system steering
(EPR) P (a ,b ) The classical condition (1) and the results derived from SdRΨ ≡ u(1) u(1) a +Xb =d 1 which such as Eqs. (2) and (3) provide a strict meaning of u(1) u(1) − 1 violating the single-system analogue of the EPR steering + P (a ,b ) > 1+ ,(A5) inequality used in the experiment of Smith et al. [7], i.e., u(2) u(2) √ a +Xb =d 1 d the temporal steering inequality introduced in [17]. The u(2) u(2) − 10 kernel of such steering inequality reads what follows we will illustrate a simple example to show that, compared with the temporal steering inequality, the N 2 steering conditions can fulfil certain requirements so as SN E[ Bi,tB A ], (B1) ≡ h i i,tA to useful as checks for the reliability of QIP. Xi=1 Let us assume that a source generates particles in the where state ρs = 0 11 0 for Alice’s subsequent use for steer- ing. The task| i ofh Alice| and Bob is to perform an identity 1 operation I, or alternatively, to maintain the states of E[ B 2 ] = P (A = a) B 2 i,tB Ai,t i,tA i,tB Ai,t =a the particles during the particle transmission. For such h i A h i A Xa=0 an information task, the steering condition (7) for d =2 (B2) and U = I used by them to check the steerability can be and N = 2 or 3 is the number of measurement for Alice of the form and Bob. The probability of measuring Ai = a at the time tA is denoted by P (Ai,t = a). The expectation 1 1 A 1 value about Bob’s measurement at the time tB, condi- 2I P (a1,b1)+ P (a2,b2) > 1+ . tioned on the measurement result of Alice, is defined by S ≡ √2 a1=Xb1=0 a2=Xb2=0 1 b When the particles are transmitted without any distur- Bi,tB A =a = ( 1) P (Bi,tB = b Ai,tA = a). h i i,tA − | Xb=0 bance, they will have 2I = 2. To concretely show the undesired situation, e.g.,S a wrong gate operation in quan- To obtain the upper bound derived from generic classi- tum computation, or an unwanted interaction between cal means, we firstly introduce the final state of Bob’s the qubit and the quantum channel inquantum commu- particle (3) into the above equation and then have nication, we assume that there exists an effective opera- 1 tion X = 0 11 1 + 1 11 0 on the qubit such that the b | i h | | i h | final state of the qubit held by Bob is X bi i. Such an Bi,tB A =a = ( 1) P (Bi,tB = b λ)P (λ Ai,tA = a) h i i,tA − | | | i Xb=0 Xλ operation can make the qubit flip when the state is pre- pared in 0 1 or 1 1. Then the value of the kernel 2I = P (λ Ai,t = a) Bi,t . | i | i S | A h B iλ becomes 2I = 1, i.e., the reliability of maintaining qubit Xλ is not certifiedS by the steering condition (7). Then it is clear that Whereas, using the same number of measurement set- tings (N = 2), the temporal steering inequality is still E[ B 2 ] i,tB Ai,t violated by S = 2, and this can not reveal the real ef- h i A N 1 fect of a qubit flip on the particle during transmission. P (A = a) P (λ A = a) B 2 . Hence, the present form of the temporal steering inequal- ≤ i,tA | i,tA h i,tB iλ Xa=0 Xλ ity can not be used in practical quantum information tasks. However, after properly revising the kernel SN by Secondly, we use the result (4) derived from the criterion introducing a quantum operation U for quantum com- on state transition (2) in the main text to obtain munication or quantum computation, the revised version
1 of the temporal inequality also can serve the same role as the steering conditions. Its derivation and experimental P (λ)= P (A = a)P (λ A = a), i,tA | i,tA demonstrations will be detailed elsewhere. Xa=0 The above consideration is also true for the bipartite for all measurements i. The temporal inequality is non-local counterpart. When Alice and Bob share the 1 1 state Φ = = =0 a1 1 b1 1 to perform the N | i √2 a1 b1 | iA ⊗ | iB 2 same task as above,P they can certify the reliability by S P (λ) B P (λ)=1. N ≤ h i,tB iλ ≤ using the steering condition (A4) for d = 2 and U = I Xi=1 Xλ Xλ Thus S > 1 can be considered as a condition for single- 1 1 N (EPR) 1 system steering and deny processes that make states un- P (a1,b1)+ P (a2,b2) > 1+ . SdUΦ ≡ √2 steerable. a1=Xb1=0 a2=Xb2=0
If there is a bit flip error in the transmission of Bob’s Appendix C: Comparison between steering qubit, then the state suffering from such effect (I X) Φ conditions and the temporal steering inequality can not give results that satisfy the above condition⊗ to| acti as a reliability check ( (EPR) = 1) but still can violate SdUΦ One of the main difference between the steering con- the inequality (SN = 2 > 1). Then, the EPR steering ditions and the temporal steering inequality is in their inequality can not respond to the effect of a qubit flip in practical applications to quantum information tasks. In the bipartite non-local scenario. 11
0.40 SdU Appendix E: EPR steering for one-way quantum 0.37 SenpU computing SN 0.35 p 0.32 A cluster state can be represented by an array of vertices, where each vertex is initially in the state of 0.30 ( 0 + 1 ) /√2 where 0 and 1 constitutes an orthonor- | i | i | i | i 0.27 mal basis. Every connected line (edge) between ver- cphase 0.25 tices realises a controlled-phase ( ) gates acting Noise tolerance Noise tolerance as m n ωmn m n , where ω = exp(i2π/2) and 0.22 m,n| i⊗0 |, 1i →[19]. In| thei⊗ present | i illustration, we consider ∈{ } 0.20 a four-qubit chain-type cluster state of the form 2 3 4 5 6 7 8 9 10 Number of qudit dimension d 1 1 1 1 mn+nj+jk C4 = ω n j m k | i | iA1⊗| iA2⊗| iB1⊗| iB2 FIG. 5. Noise tolerance of steering conditions (7) and (9). If mX=0 nX=0 Xj=0 Xk=0 the probability of white nose pnoise
(a) (b)
A1i A1 B1 B1i 1 2 1 2 |mB i ⤫ H m m 1 0,1 0,1
A i B i 2 A2 B2 2 1 2 1 2 n | B2i H n n ⤫ 0,1 Alice Bob 0,1
FIG. 6. EPR steering for one-way quantum computing. (a) A genuine four-qubit chain-type cluster state shared by Alice and Bob is represented by a fully-connected horseshoe graph. Alice, who performs the measurements A1i and A2i for i = 1, 2 on her qubits A1 and A2, respectively, can reveal the EPR steering effect to realize the gate operation U on the qubits of Bob B1 and B2. (b) The state |mi ⊗ |ni is an input of the quantum gate U composed of one two-qubit cphase gate and two single- B1i B2i qubit Hadamard operations. For one-way quantum computing, the outcomes of Alice’s measurements, m and n, corresponding to the post measurement state |mi ⊗ |ni , determines the output state of the gate operation, U |mi ⊗ |ni . A1i A2i B1i B21i illustrated above, one can think of the quantum gate U as where b denotes the results obtained from Bob’s { u(i)} being encoded in a bipartite maximally-entangled state measurement specified by b U In(b ) b = { u(i) u(i) ≡ | i i | u(i) (EPR) 3 bi v . It is easy to find that the kernel and its 1 ∈ } SdUC4 U = a Out(a ) , (E5) condition for EPR steering are exactly the same as their | i 2 | iii ⊗ | i i aXi=0 single-system analogues (6) and (7). where a m n with a = m 21 + n 20 i i A1i A2i i and Out(| ia ≡) | Ui In(⊗a | )i, and In(a ) m× n× i i i B1i B2i is the| inputi≡ state| of thei quantum| gatei ≡U | .i Hence⊗| i the effect of EPR steering reveals that a readout of the gate operation, Out(a ) , depends on the measurement result | i i ai. Our EPR steering conditions serves an useful tool to It is worth noting that the idea of bipartite EPR steer- identify reliable gate operations for experiments in the ing effects and the steering condition (E6) for one-way presence of uncharacterized (or untrusted) measurement quantum computing is rather different from that based devices. For example, for the above concrete case, we on genuine multipartite EPR steering [16]. The present have the following EPR steering conditions steering condition detects EPR steering with respect to the fixed bipartite splitting of the four qubits A1, A2 and 3 (EPR) B1, B2. When certifying genuine four-partite EPR steer- P (a1,bu(1)) SdUC4 ≡ ing for one-way quantum computing, one needs the con- a1=Xbu(1) =0 cept and method introduced in [16] to consider and verify 3 quantum steering with respect to all bipartite splittings + P (a2,bu(2)) > 3/2. (E6) of the four qubits.
a2=Xbu(2)=0
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