No-Cloning of Quantum Steering Suppose That Alice Has an Entanglement Source to Create Pairs of Qudits Φ

Home , Quantum steering

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Figure 1. Cloning quantum steering. (a) Einstein-Podolsky-Rosen (EPR) steering. Alice creates maximally entangled pairs Φ (1) from an EPR source. She keeps one qudit (A) and sends the other qudit of the pair into a universal cloning machine. The| i cloning machine, assisted by ancilla qudits (not shown), creates a four-partite composite state (3). After cloning, the qudit B is sent to Bob and the qudit C, together with the ancilla C′, are 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. Their measurement results ni 0,1,...,d 1 for n = a,b,c are then used to certify EPR steering of the subsystems (A,B) and (A,C) using a steering criteria∈{ (2). (b)− Single-system} (SS) steering: A qudit with the state s is sent from Alice to a cloning machine. Here s is | iAi | iAi a post-measurement state of some initial qudit (not shown) under the measurement Ai for i = 1,2. A tripartite composite system is then created by the cloning machine, and the qudit B is sent to Bob and the qudit C, together with the ancilla C′, are sent to Charlie. The measurement apparatus used by each party are the same as the devices used in the case of EPR steering. They can also use a steering criterion (2) to identify the SS steering of the subsystems (A,B) and (A,C).

[often called an EPR source, see Fig. 1(a)]. To have a concrete illustration, let us assume that this entangled state is of the form

d 1 Φ 1 − = ∑ s A1 s B1 (1) | i √d s=0 | i ⊗ | i where s = s = s , where s s = 0,1,...,d 1 is an orthonormal basis that corresponds to bases of Alice’s mea- | iA1 | iB1 | i1 {| i1 | − } surement A1 and Bob’s measurement B1. Second, Alice keeps one qudit of the entangled pair and sends the other qudit to Bob. Then, depending on Alice’s measurement result a1 = s, the state of the qudit finally held by Bob can be steered into a corresponding quantum state, s , for the result b1 = s. Such remote preparation of Bob’s states can be also be seen in | iB1 other bases. For example, suppose that Alice and Bob’s measurements A2 and B2 correspond to another orthonormal basis √ ∑d 1 2π Φ s 2 s = 0,1,...,d 1 , where s 2 = 1/ d k=−0 exp(i d sk) k 1, the state vector of represented in this basis is of the {| i | −d }1 | i | i | i form Φ = 1/√d ∑ − s d s , where s = s = s . It is clear that Bob’s outcome b2 will respond to Al- | i s=0 | iA2 ⊗ | − iB2 . | iA2 . | iB2 | i2 ice’s outcome a2, which satisfies a2 + b2 = 0, where = denotes equal module d. Such dependence can be made manifest d 1 d 1 by the conditional entropy H(B1 A1)= H(B2 A2)= 0, where H(Bi Ai) ∑ − P(ai)∑ − P(bi ai)log P(bi ai). In practi- | | | ≡− ai=0 bi=0 | 2 | cal experiments, the marginal probabilities P(ai) and the conditional probabilities P(bi ai) can, in principle, be measured to explicitly consider this dependence. | This description of EPR steering can be directly mapped to single-system or temporal steering and vice-versa (see9 for detailed discussions). As depicted in Fig. 1(b), first, Alice prepares a qudit with the state s by performing complementary | iAi measurements A1 or A2 on an initial state. Second, Alice sends the prepared qudit to Bob. Then she can steer the state Bob holds s ( s = s for the ideal case) into other quantum state by, for example, asking Bob, via a classical channel, to | iBi | iBi | iAi perform a unitary transformation on s Bi. In practical situations, demonstrations| i of both EPR steering and SS steering are imperfect. Environmental noise, or Φ randomness introduced by an eavesdropper, can affect both the quantum source for creating and s Ai and the properties of the state during its transmission from Alice to Bob. In addition, in its information task formulation,| i | Bobi also does not trust Alice nor her measurement apparatus, and wishes to verify whether she is truly steering his state. Hence, it is important to have an objective tool that can certify the ability of Alice to steer the states of the particles eventually held by Bob. Here, we describe and verify quantum steering in terms of the mutual information between measurement results of Alice and Bob IA B = H(Bi) H(Bi Ai). Earlier works showed that if the mutual dependence between Alice and Bob’s measurement results i i − |

2/6 violates the bound9

∑ IAiBi > log2 d, (2) i=1 their dependence is stronger than the correlation between Bob’s outcomes and the results derived from unsteerable states alone, verifying Alice’s ability to steer Bob’s state. As shown in,9 it is worth noting that the entropic steering criteria (2) are applicable to both EPR steering and SS steering. One difference between them is that P(bi ai) for SS steering are derived from | measurements on single systems where ai and bi are taken at two different times.

No-cloning of quantum steering Suppose that Alice has an entanglement source to create pairs of qudits Φ . One qudit of the entangled pair is sent to a universal cloning machine and the other qudit (A) is kept by Alice. See Fig.| 1i(a). After passing through the cloning machine, two new qudits are created, and the state of the total system becomes

d 1 φ − λ φ φ ABCC = ∑ jk jk j,d k . (3) | i ′ q AB − CC′ j,k=0 The qudit B is sent to Bob whereas the qudits C and C are sent to a third party Charlie. The two-qudit state vectors φ ′ jk AB and φ j,d k are described by − CC′

φ jk = (I Uj k) Φ mn ⊗ , | i d 1 1 − 2π = ∑ exp(i sk) s m1 s + j n1 , (4) √d s=0 d | i | i ∑d 1 π for (m,n) = (A,B),(C,C′), where I denotes the identity operator, Uj,k = s=−0 exp(i2 sk/d) s + j n1n1 s , and s m1 = s n1 = s . The state of Alice’s and Bob’s qudits is | i h | | i | i | i1 d 1 − ρAB = ∑ λ φ φ , (5) jk jk ABAB jk j,k=0 where λ denotes the probability of observing φ . The mutual information of Alice’s and Bob’s measurement results jk jk AB derived from their measurements Ai and Bi on ρAB is

d 1 − t 1 IAiBi = log2 d ∑ qi log2 t , (6) − t=0 qi

t ∑d 1 λ t ∑d 1 λ t 10 where q1 = k−0 tk and q2 = j=−0 j,d t . The variables qi ﬁrstly introduced in are the probabilities of ﬁnding bi ai = t = − − or bi ai = t d for t = 0,1,..,d 1. The sum of mutual information under two measurement settings is then − − − 2 2 t ∑ IAiBi = 2log2 d ∑ H(qi). (7) i=1 − i=1

To determine the mutual information of Alice’s and Charlie’s measurement results IAiCi , we ﬁrst consider the mutual dependence between ai and the results derived from measurements on the subsystem composed of Charlie’s qudit C and the ancilla C by their mutual information I . It is clear that ′ Ai(CiCi′)

IA C I . (8) i i ≤ Ai(CiCi′) In addition, the mutual information I is constrained by the Holevo bound by Ai(CiCi′)

d 1 ρ − ρ IA (C C ) S( CC ) ∑ P(ai)S( CC ai ). (9) i i i′ ≤ ′ − ′| ai=0

3/6 ρ ρ ∑d 1 λ φ φ S( CC ) is the von-Neumann entropy of the state CC = j,−k=0 jk j,d k j,d k . It can be explicitly represented by ′ ′ − CC′CC′ −

d 1 S(ρ )= ∑− λ log λ H(λ). (10) CC′ jk 2 jk j,k=0 − ≡

The state ρ is the reduced state conditioned when Alice obtains the result a . Now, we use the method presented in10 CC′ ai i | ρ ∑d 1 t 1 t to ﬁnd the upper bound in (9). The von-Neumann entropy of this state can be shown as S( CC ai )= t=−0 qi log2 t H(qi). ′| qi ≡ ρ ∑d 1 ρ In order to derive the upper bound of IA (C C ) by minimizing the difference between S( CC ) and a−=0 P(ai)S( CC ai ), we i i i′ ′ i ′| λ j t ∑d 1 λ ∑d 1 t ∑d 1 k substitute j,d k = f ( j,k)q1 into q2 = j=−0 j,d t , where k=−0 f ( j,k)= 1, and then obtain q2 = k=−0 f (t,k)q1. For each t, t− − all f (t,k)= q2 implies the minimum of the difference. Then we have λ t t t t H( )= H(q1)+ ∑q1H( f (t)) = H(q1)+ H(q2). (11) t

With Eqs. (8),(9)and (11), the upper bound of the mutual information IAiCi is shown as

t t t IA C H(q )+ H(q ) H(q ), (12) i i ≤ 1 2 − i which implies that

2 2 t ∑ IAiCi ∑ H(qi). (13) i=1 ≤ i=1 Hence, combining Eq. (7) with Eq. (13), we eventually derive the following relationship between the mutual information of Bob’s and Charlie’s systems with Alice’s

2 2

∑ IAiCi + ∑ IAiBi 2log2 d. (14) i=1 i=1 ≤ This criterion (14) provides a basis to investigate how EPR steering is shared between two copies of a qudit of a maximally entangled pair. When the correlation between the qudits shared by Alice and one of the two parties, say Charlie, is certiﬁed by the steering criteria (2), it is clear that the mutual dependence between qudits shared by Alice and Bob will not be stronger than an unsteerable state. Hence, EPR steering can be identiﬁed in only one of the copy subsystems. This analysis of the behavior of EPR steering subject to cloning can be directly applied to SS steering as well; see Methods section.

Securing quantum information processing The steerability of Alice over Bob or Charlie’s qudits, as certified by the steering criteria (2), implies that the mutual dependence between them is stronger than the mimicry that an unsteerable state can provide. In addition, such steering cannot be shared with a third party by using a universal cloning machine. Two direct applications to quantum information are illustrated as follows. (i) If a sender (Alice) and a receiver (Bob) confirm that their measurement results are classified as steerable, according to the criteria (2), they can be convinced that an eavesdropper (Charlie) who uses a cloning machine for coherent attacks cannot produce states that can be steered by the sender. This is because the mutual information between Alice and Bob is larger than the mutual information shared between Alice and the eavesdropper, Charlie. Thus they can use privacy-amplification techniques on their shared measurement outcomes to generate a secure key. Thus the no-cloning of quantum steering verified by (2) shows that ruling out false steering secures channels against cloning-based attacks when implementing QKD. (ii) As shown in9 , steering quantum systems is equivalent to performing quantum computation. No-cloning of steering provides a strict proof to show that the observation of quantum steering guarantees faithful implementation of a quantum computing implementation in the presence of uncharacteristic measurements and cloning-based attacks.

Discussion We investigated how quantum steering is cloned by a universal cloning machine and shared between two copy subsystems. We showed that it is impossible to observe quantum steering, as described by the mutual information criterion (2), in the two copies at the same time. This no-cloning of quantum steering ensures secure QKD and faithful quantum gate operations of arbitrary computing size against cloning-based attacks. Our results motivate several open questions. Is the no-cloning of

4/6 quantum steering applicable to the situation of genuine multipartite multidimensional EPR steering? If it is the case, then such high-order steering would serve as a source for reliable multipartite quantum information processing such as quantum secret sharing. In addition to high-order steering, does one-way steering possess this feature of no-cloning? Finally,if we use a steering measure instead of an entropic criteria (2), could the partial power of quantum steering in terms of the units of a steering quantiﬁer be copied by the cloning machine? Could the total quantity of steering be conserved after cloning? Finally, it is interesting to connect our results with other approaches, such as the principle of monogamyof certain quantum correlations.11, 12 In particular, the principle of the monogamy of temporal steering, shown in the work12 [see Eq. (5) therein], is consistent with our results, and suggests our criteria can also be interpreted as a monogamy relation in the entropic form. However, whether such a result can provide a relation in the form of Coffman-Kundu-Woottersmonogamy inequality [see, for example, Eq. (1) in11] still needs further investigation. In addition, He et al.13 have shown that two-way steering is required to overcome the no-cloning threshold for secure teleportation. This relationship, between no-cloning and EPR steering, also suggests a principle of no-cloning for the correlations utilized for teleportation. Their quantum-information-task-oriented method, to investigate the relationship between the no-cloning theorem and steering, indicates that it may be interesting, in future work, to consider the security threshold for secure quantum teleportation derived from our input-output scenario for cloning quantum steering, and to compare this condition on ﬁdelity with their criterion.13

Methods No-cloning of SS steering As illustrated in Fig. 1(b), after operating the cloning machine on a single system sent from Alice, the state s becomes | iAi d 1 φ − λ φ φ BCC = ∑ jk jk j,d k , (15) | i ′ q B − CC′ j,k=0 where

φ jk = Uj k s . (16) B , | iBi

d 1 (note that φ00 = s ). The state of Bob’s qudit is then ρB = ∑ − λ jk φ jk φ jk . With this reduced state, we obtain the | iB | iAi j,k=0 BB mutual information IAiBi (6). When considering the mutual information IA iCi , it is easy to ﬁnd that the connection between A and CC here can be mapped to the case of EPR steering. There are no differences between the states S(ρ ) together with ′ CC′ ai S(ρ ) in these two steering cases. Then we arrive again at the result of a constraint on mutual information| for subsystems CC′ (14). Hence the SS steering can be observed in only one of the copy subsystems.

References 1. Schr¨odinger, E. Discussion of probability relations between separated systems. Math. Proc. Cambridge Philos. Soc. 31, 555-563 (1935). 2. Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 0777-0780 (1935). 3. Wiseman, H. M., Jones, S. J. & Doherty, A. C. Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007). 4. Reid, M. D. et al. Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications. Rev. Mod. Phys. 81, 1727-1748 (2009). 5. Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802-803 (1982). 6. Dieks, D. Communication by EPR devices. Phys. Lett. A 92, 271-272 (1982). 7. Buˇzek, V. & Hillery, M. Quantum copying: Beyond the no-cloning theorem. Phys. Rev. A 54, 1844-1852 (1996). 8. Scarani, V., Iblisdir, S., Gisin, N. & Ac´ın, A. Quantum cloning. Rev. Mod. Phys. 81, 1225-1256 (2005). 9. Li, C.-M., Chen, Y.-N., Lambert, N., Chiu, C.-Y. & Nori, F. Certifying single-system steering for quantum-information processing. Phys. Rev. A 92, 062310 (2015). 10. Sheridan, L. & Scarani, V. Security proof for quantum key distribution using qudit systems. Phys. Rev. A 82, 030301 (2010).

5/6 11. Kay, A., Kaszlikowski, D. & Ramanathan, R. Optimal cloning and singlet monogamy. Phys. Rev. Lett. 103, 050501 (2009). 12. Bartkiewicz, K., Cernoch,ˇ A., Lemr, K., Miranowicz, A. & Nori F. Temporal steering and security of quantum key distribution with mutually-unbiased bases against individual attacks. arXiv:1503.00612. 13. He, Q., Rosales-Z´arate, L., Adesso, G., & Reid, M. D. Secure continuous variable teleportation and Einstein-Podolsky- Rosen steering. Phys. Rev. Lett. 115, 180502 (2015).

Acknowledgements This work is partially supported by the Ministry of Science and Technology, Taiwan, under Grant Numbers MOST 104-2112- M-006-016-MY3 and No. MOST-104-2221-E-006-132-MY2. This work is partially supported by the RIKEN iTHES Project, the MURI Center for Dynamic Magneto-Optics via the AFOSR award number FA9550-14-1-0040, the IMPACT program of JST and a Grant-in-Aid for Scientiﬁc Research (A). N.L. is partially supported by the FY2015 Incentive Research Project. N.L. and F.N. acknowledge the support of a grant from the John Templeton Foundation.

Author Contributions C.-Y.C. and C.-M.L. devised the basic model. C.-Y.C., N.L., T.-L.L. and C.-M.L. established the ﬁnal framework. N.L., F.N. and C.-M.L. wrote the paper with contributions from all authors.

Additional Information Competing ﬁnancial interests: The authors declare no competing ﬁnancial interests.

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