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Secure continuous variable quantum teleportation and Einstein-Podolsky-Rosen steering
Qiongyi He, Laura Rosales-Zarate, Gerardo Adesso, Margaret Reid
TL;DR
The paper asks which entangled resources enable secure continuous-variable teleportation of coherent states, where fidelity must exceed the no-cloning threshold. It extends teleportation to arbitrary Gaussian gains with pre-amplification or post-attenuation, and proves that two-way steering is required for the relevant protocols and resources. Heralded noiseless pre-amplification is proposed as a route to high-fidelity heralded teleportation with minimally entangled steerable resources.
Problem
Secure coherent-state teleportation requires fidelity F > 2/3, but the entangled-state requirement for excluding a non-degraded copy by an eavesdropper was unresolved.
Method
The paper analyzes arbitrary-gain continuous-variable teleportation, including local pre-amplification and post-attenuation, for asymmetric Gaussian resources and steering criteria.
Results
For the lsatt, esa, and BK protocols, achieving F > 2/3 requires two-way steering; all Gaussian entangled resources with Ent < 1/3 are useful for secure teleportation.
Takeaways & Limitations
Heralded noiseless pre-amplification may enable high-fidelity heralded teleportation using minimally entangled yet steerable resources.
Abstract
from arXiv · showhide
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.
I. PROOFS
The proofs establish entanglement and steering requirements for secure teleportation, including sufficient thresholds for two-way steering and the no-cloning fidelity.
- Result (1a): Entanglement is bounded below by the asymmetry parameter in Result (1a).The bound is derived generally for two-mode states and simplifies in the Gaussian case using the fidelity limit for amplified teleportation.
- Result (1b): Two-way steerable states require a minimum symmetry, expressed as gsym < √2.This symmetry condition is used to certify two-way steering through the entanglement threshold in Result (1b).
- Result (1b): Ent < 0.5 is sufficient for two-way steering when gsym = 1, while Ent < 1/3 suffices for all values of gsym.For 1 ≤ gsym < √2, Ent < 1/3 is sufficient; for gsym ≥ √2, Ent < 1/3 is impossible under the preceding bound.
- Result (3): F > 2/3 requires resource entanglement Ent < 1/2, but this condition alone is insufficient.For the lsatt, esa, and BK protocols, achieving the no-cloning fidelity necessarily requires a two-way steerable resource.
II. FIDELITY FOR QUANTUM TELE-AMPLIFICATION (QAT)
Quantum tele-amplification is optimized by matching the classical gain to resource asymmetry, while highly entangled symmetric resources are not always best at higher gains.
- Gain optimization: Choosing ḡ = gsym optimizes amplified-teleportation fidelity relative to the benchmark.For symmetric resources, BK is optimal; for asymmetric resources, BK is not optimal.
- TMSS resources: For quantum tele-amplification with ḡ > 1, lower squeezing can optimize relative fidelity for a two-mode squeezed resource.The TMSS cannot provide quantum-amplified teleportation at high gains when highly entangled, whereas lower-r resources perform better.
- TMSS resources: For pure symmetric TMSS resources, gsym = 1 and the BK protocol optimizes relative fidelity.The TMSS enables quantum teleportation under BK for all r satisfying the Duan entanglement condition, and QT extends to gains up to coth(r/2).
- Absolute fidelity: The maximum absolute fidelity for amplification |α⟩→|ḡα⟩ is 1/ḡ^2, while the TMSS reaches its maximum absolute fidelity under BK when ḡ = 1.The optimal gain is ḡopt when it exceeds 1; otherwise the optimal gain is unity.
III. THE FIDELITY FOR ATTENUATION AS PART OF A PREAMPLIFICATION PROTOCOL
The attenuation stage optimizes fidelity for |α⟩→|¯gα⟩ with ¯g < 1, reaching unity for a TMSS at ¯gopt = tanh(r). This supports a preamplification strategy whose performance is set by the heralded preamplifier.
- Amplification: For amplified teleportation, the maximum absolute fidelity is bounded by 1/¯g2, while reducing TMSS squeezing improves fidelity at larger ¯g.The relevant process has |α⟩→|¯gα⟩ with ¯g > 1.
- Optimal attenuation: F¯gopt = 1 for TMSS attenuation when ¯gopt = tanh(r).The attenuation gain is chosen to minimize the fidelity denominator for a fixed EPR resource.
- Optimal attenuation: The TMSS optimum follows from ¯gopt = sinh(r)/cosh(r) = tanh(r) < 1.The covariance correlations determine an optimal gain below unity for the attenuation process.
- Figure interpretation: The attenuation optimum is represented by three circles on the fidelity curves in Fig. 3.These points mark the gains satisfying the TMSS optimality condition.
- Implementation: A practical recipe preamplifies with g = 1/¯g > 1 before attenuation; g ∼2 requires ¯g = 0.5 = tanh(r), giving r = 0.55.The attenuation stage can then reach fidelity one for arbitrary r, so heralded preamplification largely determines the final fidelity.
IV. FIDELITY USING BK AND LSATT PROTOCOLS
The paper evaluates BK and lsatt teleportation protocols using Gaussian resources, relating fidelity to entanglement, asymmetry, and steering. Secure teleportation beyond F > 2/3 requires two-way steerability under the stated protocol and resource conditions.
- Protocol evaluation: The BK and lsatt protocols are evaluated using fidelity expressions for two-mode squeezed resources with losses.Figure 4 presents optimal fidelities for BK and lsatt implementations with lossy TMSS resources.
- Secure teleportation: For BK, F > 2/3 with symmetric or asymmetric resources is achievable when ν ≡ ∆ent < 0.5, while the required squeezing increases as r approaches zero.For unit channel efficiencies, BK requires r > 0.347 for secure teleportation, a lower bound than for lsatt.
- Protocol evaluation: The non-unity-gain extension makes all (X −P)-balanced Gaussian entangled states useful for continuous-variable teleportation against the gain-dependent benchmark 1/(1 + ¯g2).The result concerns ordinary teleportation under arbitrary classical gains, not necessarily secure teleportation.
- Steering conditions: Two-way steering is certified when Ent < 1/(1 + gsym^2), with the tight condition including Ent < 1/3 for all (X −P)-balanced states.For symmetric resources, gsym = 1, so Ent < 1/2 suffices; two-way steerability also requires sufficient symmetry.
I III II
The paper expands secure coherent-state teleportation beyond symmetric BK protocols by optimizing local Gaussian operations and introducing late attenuation and early amplification. It shows that resources with Ent < 1/3 support high-fidelity secure teleportation through one of three protocols, while heralded pre-amplification may approach unit fidelity with steerable resources of low entanglement.
- I III II: Optimizing over local Gaussian operations yields fidelity bounds for a fixed resource entanglement, motivating asymmetric protocols beyond the standard BK scheme.The full optimization is difficult, but the MV bounds provide the relevant benchmark.
- I III II: All Gaussian entangled resources with Ent < 1/3 can achieve secure teleportation with F > 2/3 under an optimal protocol.This threshold matches the tight entanglement threshold for certifying two-way steering.
- I III II: The practical protocol set comprises late-stage attenuation, early-stage amplification, and the BK protocol, spanning the MV fidelity bounds for lossy TMSS resources.Late-stage attenuation locally attenuates Bob’s output when the teleportation gain exceeds one.
- I III II: Early-stage amplification pre-amplifies Alice’s coherent state and uses classical attenuation during teleportation, with an optimum constrained by the resource parameters.The protocol requires an entangled Gaussian resource satisfying EntB|A(ḡ) < 1 with ḡ < 1.
- I III II: For conventional teleportation, the paper generalizes the GG condition to asymmetric Gaussian resources and protocols while retaining two-way steering as the secure-teleportation requirement.The result applies to the lsatt, esa, and BK protocols for achieving F > 2/3.
- I III II: Heralded noiseless amplification may overcome the esa protocol’s 1/g^2 fidelity limit, allowing F → 1 without significant teleportation-resource entanglement.The remaining requirement is EPR steerability, which demands sufficient resource purity at low entanglement.