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Secure continuous variable quantum teleportation and Einstein-Podolsky-Rosen steering

We investigate the resources needed for secure teleportation of coherent states. We extend continuous variable teleportation to include quantum tele-amplification protocols, that allow non-unity classical gains and a pre-amplification or post-attenuation of the coherent state. We show that, for arbitrary Gaussian protocols and a significant class of Gaussian resources, two-way steering is required to achieve a teleportation fidelity beyond the no-cloning threshold. This provides an operational connection between Gaussian steerability and secure teleportation. We present practical recipes suggesting that heralded noiseless pre-amplification may enable high-fidelity heralded teleportation, using minimally entangled yet steerable resources.

Quantum teleportation is a process whereby Alice sends an unknown quantum state to Bob at a different location by communicating only classical information [1]. Quantum teleportation has inspired much interest, both as a fundamental challenge and as a tool for quantum information processing. To achieve quantum teleportation, Alice and Bob share an Einstein-Podolsky-Rosen (EPR) entangled state. Teleportation was first developed for the transfer of qubit (spin-1/2) states, and was extended to continuous variable (CV) spectra by Vaidman [2] and Braunstein and Kimble (BK) [3]. In the CV case, the EPR entanglement shared between Alice and Bob is modeled after the original EPR paradox where Alice and Bob share systems with perfectly correlated positions and anti-correlated momenta [4][5][6][7][8][9].
This paper addresses what type of EPR entanglement is useful for CV quantum teleportation. The issue was raised as early as 2001 [10]. There had emerged three types of EPR correlation [11][12][13][14][15][16]. Yet only one, where the observers Alice and Bob are treated symmetrically, was apparently relevant [8,17]. CV protocols for the teleportation of a coherent state have to date utilised only a subset of entangled states, where the entanglement can be certified by the Tan-Duan criterion ( = 2) [14,15] (∆X) 2 is the variance of X, and X A ,X B and P A ,P B are the positions and momenta of Alice and Bob's systems. It seems counter-intuitive that no other form of EPR entanglement can be useful, given that teleportation itself is asymmetric with respect to the roles of Alice and Bob: Alice transmits one unique state from her location to Bob. This is related to the fundamental question: "Can any entangled state provide better than classical fidelity of teleportation?", which has been posed [18] and addressed in the affirmative [19] for qubit teleportation. There appears as of yet no clear resolution for the CV scenario.
Here, motivated by these questions, and the recent interest in the asymmetric form of entanglement called EPR steering [20], we revisit CV quantum teleportation. EPR steering refers to the correlations of the original 1935 EPR paradox, which compares quantum mechanics with local realism [4,11]. Bob at his location can make seemingly paradoxical inferences about Alice's state, apparently adjusting ("steering") it remotely. Clearly, in this context Alice and Bob are not equivalent [21]. The role of steering in quantum teleportation was first explored by Grosshans and Grangier, who pointed out that EPR-paradox correlations are required of the resource in order to achieve no-cloning teleportation [22]. It is natural to expect that steering will play an important role in teleportation [23]. We resolve this question, by considering a whole class of protocols, that allow for arbitrary classical communication gain factorsḡ combined with local amplification or attenuation at Alice or Bob's station.
In this way, our work addresses not only the fundamental question, but also the practical question of whether the fidelity of CV teleportation can be optimised without increasing the squeezing of the EPR resource. CV protocols give the advantage of teleportation without postselection, because homodyne detection can be performed at Bob's location with exceptional efficiency [6][7][8][9]. The difficulty has been to achieve high fidelity, since with the BK protocol, the fidelity can only be enhanced by increasing the entanglement of the resource: this for the two-mode squeezed state ultimately requires infinite energy [24]. Our work suggests alternative routes could be possible, based on a heralded noiseless preamplification of the coherent state [25].
Our main result is that for the teleportation of a coherent state |α , the whole class of relevant CV entangled states certified by the well-known positive partial transpose (PPT) criterion of Peres and Simon [16,26] becomes useful for quantum teleportation. We show that the fidelity F amp g of the teleportation |α → |ḡα is a function of the steering parameter of the EPR resource: Here EP R B|A (ḡ) = ∆(X B −ḡX A )∆(P B +ḡP A ) < 1 verifies an EPR paradox, and an "EPR-steering" of Bob's system by measurements made by Alice. Quantum tele-amplification: We begin by generalising the Braunstein Figure 1. Quantum tele-amplification followed by a local operation. The coherent state |ψin = |α is sent from Alice (at A) to Bob (at B) by way of classical communication and the shared entanglement of the continuous-variable EPR beams. We simplify by takingḡx =ḡp =ḡ, but consider general amplificationḡ ≥ 1. We allow for non-ideal efficiencies (ηA,ηB ≤ 1) and for asymmetric efficiencies where ηA < ηB, which creates a resource with an asymmetric EPR entanglement characterised by a parameter g B|A sym. Quantum teleportation is achieved when we selectḡ = g B|A sym. We also consider deamplificationḡ < 1, in which case the coherent state is pre-amplified at Alice's station.
teleportation [3], as depicted in Fig. 1. Alice and Bob share an EPR entangled state, often modeled by the two-mode squeezed state (TMSS) |ψ = (1 − x 2 ) 1/2 ∞ n=0 x n |n A |n B [5,11]. The systems at Alice and Bob's locations are assumed to be single modes, with boson creation and destruction operators a, b and number eigenstates |n A , |n B , respectively. Here x = tanh r, where r is the squeezing parameter that determines the amount of entanglement shared between the two locations. The addition of losses to each of the modes means that the EPR entanglement is better described by a mixed Gaussian state [13,16,17]. Defining the position and momentum quadratures as A field V is prepared by Victor in the state |ψ in that is to be teleported to Bob. We restrict attention to the case of a coherent state: |ψ in = |α . The field V (amplitudes X V and P V ) interferes with Alice's EPR field (labelled A) on a beam splitter, to produce two fields at the output ports. This allows a simultaneous Bell measurement of the two combined quadratures X V − X A and P V + P A by homodyne detection, to give outcomes m x and m p respectively. The final stage of the teleportation is the displacement by Bob of the amplitudes of his EPR field (labelled B) by an amount given by Alice's readout values m x , m p , that are transmitted to him by Alice using classical communication.
Our generalisation of the BK protocol is to allow nonunity classical gain factorsḡ x ,ḡ p in the two classical channels. For simplicity, we consider equal gains,ḡ x =ḡ p =ḡ. This means that Bob's displacement is amplified/ deamplified to beḡm x andḡm p . In the BK protocol, g = 1. After feedback, Bob's field amplitudes are given by We can see there are two physical effects that will limit the fidelity of teleportation. First, the mean amplitude at Bob's station is scaled relative to Victor's, by the classical gainḡ. Ifḡ > 1, then teleportation amplifies the coherent state, and we know that the maximum fidelity is reduced to 1/ḡ 2 [27][28][29]. Second, the noise in the final reconstruction by Bob of Victor's amplitudes will be limited by the variances of the linear combinations X B −ḡX A , P B +ḡP A which are determined by the asymmetric EPR entanglement present in the EPR resource, as defined by the steering parameter (2).
The measure of success of the teleportation is given by the fidelity, defined as F = β tele |ρ out |β tele , where ρ out is the density operator of the output state at Bob's location, and |β tele is the state desired at Bob's output. The fidelity can be conveniently calculated using Wigner function techniques [3,8,30]. We have evaluated the Wigner expression for the output at Bob's location, including the gain factorsḡ and for a general input state with a Gaussian Wigner function. The result for the fidelity is F amp Here, σ Q = (1 + σ X )(1 + σ P ) and β out = x m + ip m where x m , p m and σ X , σ P are respectively the means and variances of the quadratures X f B , P f B of Bob's output field. Calculation shows that β out = X f B + i P f m = g( X V + i P V ) =ḡα, as expected. We evaluate the fidelity relative to the desired teleported state |β tele = |ḡα . Then we see that the exponential expressions vanish. The final fidelity for the protocol, called "quantum tele-amplification" (QAT) [29], is As we will see later, even though we may desire to teleport the state |α rather than the amplified one |ḡα , we can achieve this by a local attenuation at Bob's station, or else by a local pre-amplification at Alice's station. We now express the fidelity (3) in terms of the variances of the EPR resource. We find where σ in is the variance of X and P for the input state |α : i.e. σ in = (∆X) 2 = (∆P ) 2 = 1. We now restrict to the subclass of two-mode EPR resources symmetric in their position and momentum correlations [8], so that | X A , X B | = | P A , P B |, σ X = σ P and ∆(X B −ḡX A ) = ∆(P B +ḡP A ). We call this subclass X − P -symmetric. This subclass includes EPR resources such as the twomode squeezed state including all phase-insensitive losses and noise. This assumption is convenient but not necessary: The full generalisation to include non-X − P symmetric resources will require asymmetric gainsḡ x =ḡ p to optimise fidelity. On substitution, we see that the fidelity F amp g of the QAT protocol is given by Eq. (2). The fi- delity is determined by the gainḡ on the channel, and the EPR steering parameter EP R B|A (ḡ). The special case of Braunstein-Kimble reduces to F BK = 1 (1+∆ent) , where ∆ ent is the Tan-Duan entanglement parameter defined by equation (1).
Quantum benchmarks for quantum teleamplification: The threshold where one can rule out all classical "measure and regenerate" strategies and hence claim Quantum Teleportation was determined in Ref. [31]. For amplificationḡ > 1, the threshold for quantum teleportation is F amp g > 1 1+ḡ 2 [28]. On substituting into (2), the condition on the steering parameter of the EPR resource to obtain quantum teleportation is EP R B|A (ḡ) < 1 +ḡ 2 (providedḡ > 1). Importantly, this reduces to a condition on the entanglement of the resource: the quantum benchmark for teleportation using the QAT protocol is reached when The inequality Ent B|A (ḡ) < 1 if satisfied will certify entanglement between the two fields at A and B [32]. This is true for any value ofḡ, which is a real constant. The class of entangled states satisfying this condition is a strict superset of the class defined by the Tan-Duan inequality (1). In particular, we will show that the inequality Ent B|A (ḡ) < 1 for an optimally selectedḡ is necessary and sufficient for all Gaussian entangled states of the X − P -symmetric subclass. This means that the benchmark for quantum teleportation can be reached for all such entangled states, provided the gain factor and direction of teleportation is chosen appropriately.
We give more detail for the Gaussian EPR resources. Such resources are characterised in terms of the symplectic form of the covariance matrix C defined as C ij = X i , X j = (X i X j + X j X i ) /2 − X i X j where X ≡ (X A , P A , X B , P B ) is the vector of the field quadratures [16]. We use the notation and note that in this paper we restrict to the subclass of X − P -symmetric Gaussian fields, meaning that c 1 = −c 2 = c. To verify the entanglement shared between Alice and Bob, we can observe the EPR correlation by way of noise reduction in the position and momenta, as given by inequality (4). We choose theḡ that minimises Ent B|A (ḡ): this isḡ = g B|A sym where [33] The resulting inequality is equivalent to Simon's positive partial transpose (PPT) condition for entanglement which is necessary and sufficient for two-mode Gaussian states [33]. Thus, the quantum benchmark for the amplified teleportation |α → |ḡα from Alice to Bob can be reached for all X − P -symmetric Gaussian entangled fields that possessḡ = g B|A sym > 1. We simply match the classical gain factorḡ of the protocol to the valuē g = g B|A sym given by the EPR resource. It is easy to verify that g B|A sym = 1/g A|B sym . Thus, for Gaussian resources with g B|A sym < 1, quantum teleportation can be realised if the EPR channels A and B are exchanged, or else Bob teleports to Alice. Alternatively, we show below that a preamplification protocol is possible in this case. When moments are symmetrical under A ↔ B, n = m and g B|A sym = 1. Then, we use the BK protocol. The quantum benchmark is F BK > 1/2, and we recover the well-known result that the condition on the resource becomes that of Tan-Duan entanglement: ∆ ent < 1.
Clearly, the above strategies optimise the fidelity of the amplified teleportation protocol relative to the quantum benchmark. We see that only for symmetric resources does the BK protocol give the optimal relative fidelity: A second question is what resource will give the maximum fidelity for a fixed amplificationḡ. Calculation gives the surprising result that symmetric maximal entanglement is not always optimal (see Supplemental Materials [34]).
We can create an asymmetric EPR resource from the two-mode squeezed state by adding loss to each of the EPR channels (Fig. 2). If η A and η B are the resulting efficiencies at A and B respectively, the Gaussian covariances become n = η A cosh (2r) There is entanglement for all η A and η B , as confirmed experimentally [17,35]. This robustness translates to robustness of quantum teleportation to loss, provided the losses are directed away from Bob's channel. Late-stage attenuation (lsatt): To properly teleport the original coherent state |α → |α , Bob may decide to locally attenuate his output field after the QAT. We see that this will not change the requirement on his EPR resource. Bob can attenuate using a beam splitter which yields a new output variance of σ X = ησ T X + 1 − η where η = 1/ḡ 2 = g 2 < 1, and σ T X is the variance for the output after QAT. Then, the overall fidelity is F lsatt A,B (g) = 2 3−g 2 +EP R A|B (g) . The condition for Quantum Teleportation (QT) is F > 0.5. We see that this requires a resource with Ent B|A (ḡ) < 1 forḡ > 1. This is the same condition as for QAT, discussed above. We now consider how to maximise the overall fidelity, for a given resource. Where X B , X B > 1 (which will be the case for the X − Psymmetric fields), this leads to an optimal classical gain factorḡ opt = X B ,X B −1 for which the overall fidelity becomes F lsatt A,B = 2 3+ X A ,X A − X A ,X B 2 /( X B ,X B −1) , providedḡ opt > 1. Ifḡ opt ≤ 1, the maximum fidelity is given by the BK protocol whereḡ = 1.
Early-stage amplification (esa): Alternatively, to teleport the original state |α → |α , Alice may choose to amplify the input coherent state at her station by a factor g > 1, prior to a teleportation protocol that uses a classical attenuation factorḡ = 1/g < 1. Suppose Alice uses at her station a two-mode squeezed state amplifier [25]. Then the final amplified state at her station is described by a Gaussian Wigner function with mean gα and new variance σ in = 2g 2 − 1. The final Wigner output after teleportation to Bob has variance σ X =ḡ 2 σ in + EP R B|A (ḡ). Substitution reveals the fidelity F esa A,B (ḡ) for the overall teleportation process to be F esa A,B (ḡ) = F lsatt B,A (ḡ). QT requires F esa A,B (ḡ) > 0.5, which requires a resource satisfying the entanglement condition Ent B|A (ḡ) < 1 withḡ < 1. Hence, for an entangled Gaussian resource in the X − P -symmetric class with g B|A sym < 1, the choice of protocol withḡ = g B|A sym will give QT, by this early-stage amplification protocol. The final fidelity F esa A,B (ḡ) is maximised forḡ opt = X A ,X B X A ,X A −1 (providedḡ opt < 1), in which case the fidelity is given by F esa A,B = F lsatt B,A . Maximising the teleportation fidelity: Which protocol will teleport the coherent state |α → |α with maximum fidelity for a given resource? Figure 3 shows the optimal fidelity, for the lossy EPR resource of Figs. 1 and 2. All regions enable QT, via the lsatt-protocol in region I; or by the the esa-protocol in region II; or by the BK protocol as in region III. Teleportation is optimised by the BK protocol where efficiencies for A and B are reasonably symmetric, while optimised at the asymmetric protocols elsewhere.
Secure teleportation (ST): Quantum no-cloning tells us that when the fidelity is high enough (F > 2/3), there can be no non-degraded copy of the state that is teleported to Bob [22,27]. What is the requirement on the resource to achieve such "secure teleportation" (ST)? For late-stage attenuation protocol, ST leads to the steering condition EP R A|B (g) < g 2 < 1. On dividing through by g 2 , we see that this condition also requires steering of B by A i.e. EP R B|A (ḡ) < 1, whereḡ = 1/g. Thus, "two-way steering" [21] is required of the resource, with a stronger steering needed from Bob to Alice. For the BK protocolḡ = 1, this condition reduces to ∆ ent < 0.5.
Results for the lossy EPR resource are shown in Fig.  3. ST requires η A , η B > 0.5, consistent with the fact that steering of A by B is destroyed when η B ≤ 0.5 (Fig.  2) [17,35]. ST can be carried out via the BK protocol; or , for larger r (where r > 0.8814) via the asymmetric protocol. For some regimes (I and i of lower Fig. 3), an asymmetric protocol is required for ST, i.e., it cannot be achieved with the BK protocol. The white region indicates no ST, despite there being two-way steering.
Discussion: We conclude by suggesting a further application of steering to enhance the fidelity. The early-stage amplification protocol relies on pre-amplification of a coherent state by a factor g > 1, which has a limited maximum fidelity of 1/g 2 . Recent methods use heralding to overcome this limitation: heralded noiseless amplification of |α to |gα potentially allows fidelities approaching 1 [25]. The teleportation deamplification |gα → |α has fidelity F = 2 1+ḡ 2 +EP R B|A (ḡ) given by (2) but wherē g = 1/g < 1. We show in the Supplemental Material that for the two-mode squeezed state resource,ḡ opt = tanh(r) and F = 1. This is valid for all r and hence does not depend on achieving a large entanglement of the resource.
The fidelity for the overall teleportation is then limited by the fidelity of the heralded amplification at Alice's site, suggesting a promising route for achieving high teleportation fidelities. Finally, we leave as an open question whether all Gaussian entangled states are useful for quantum teleportation, once one allows for adjusting both gainsḡ x andḡ p independently.
We thank the Australian Research Council for funding via Discovery and DECRA grants. QYH thanks the support by the National Natural Science Foundation of China under grants No.11274025, 61475006, and 11121091. MR is grateful to G. Pryde for stimulating discussions.