Tomáš Opatrný of Palacký University, Allison Brattley of Yale University, and Kunal K. Das of Stony Brook University have determined that benchmarks for assessing quantum teleportation may be too conservative. Their work establishes that fidelity thresholds needed to demonstrate a quantum advantage in teleportation can be higher than previously calculated when assuming a uniform distribution of qubits. The researchers determined the highest achievable mean fidelity for teleporting states drawn from a non-uniform distribution, specifically a von Mises – Fisher distribution, and modeled multi-qubit systems across a surprisingly broad range of particle numbers, N ∈ [1,∞).

Non-Uniform Prior Distributions Impact Teleportation Benchmarks

Benchmarks for assessing quantum teleportation may need recalibration, as new calculations reveal existing fidelity thresholds could underestimate the true bar for demonstrating a quantum advantage. Das of Stony Brook University determined that assuming a uniform distribution of input states leads to artificially low benchmarks. Their work focuses on the impact of prior knowledge about the teleported state, specifically when that knowledge skews the probability distribution away from uniformity.

The team investigated teleportation using a von Mises-Fisher distribution, a statistical model often used to represent distributions on spheres, as a more realistic representation of potential input states. This approach allowed them to derive new benchmark functions for single qubit teleportation using projective measurements along specific axes on the Bloch sphere, a geometrical representation of a qubit’s state. These benchmarks, derived through Bayesian methods, establish a higher fidelity requirement for demonstrating genuine quantum teleportation than previously understood.

Beyond single qubits, the researchers extended their analysis to multi-qubit systems, modeling them as spin coherent states. This modeling technique enabled benchmarking across a broad range of particle numbers, from N = 1 to infinity, demonstrating the scalability of their fidelity assessments. The calculations show that for systems with a finite number of qubits, the fidelity threshold needed to prove quantum advantage is lower than what the infinite particle limit would suggest.

This is significant because it suggests current assessments of quantum teleportation success may be too conservative, potentially overlooking genuine quantum effects in smaller systems. “We have shown here that teleporting a state taken from a non-uniform prior distribution will generally need to exceed a higher fidelity threshold to be considered genuine quantum teleportation, wherein entanglement provides a clear quantum advantage,” the researchers state in their published work.

von Mises-Fisher Distribution Models Qubit State Preparation

Employing a von Mises-Fisher distribution to model qubit state preparation reveals that fidelity thresholds for demonstrating quantum advantage in teleportation can exceed those calculated using uniform qubit distributions, a nuance previously unaddressed in benchmarking protocols. This analytical approach, detailed in recent work, considers scenarios where prior knowledge about the input state skews the probability distribution, leading to potentially higher fidelity requirements for establishing genuine quantum teleportation.

This baseline “do-nothing” strategy is a key reference point for evaluating the effectiveness of more sophisticated measurement techniques. The mean fidelity achieved with this optimized projective measurement approach is a benchmark against which to assess the performance of quantum teleportation protocols under non-uniform prior conditions. This provides flexibility in selecting measurement techniques, potentially simplifying implementation in certain experimental setups. The applicability of these benchmarking methods extends to collective spin coherent states, allowing for the assessment of multi-qubit teleportation performance.

The results indicate that the previously assumed fidelity threshold of (N + 1)/(2N + 1), which corresponds to teleporting a state drawn from a uniform distribution, actually represents the minimum achievable fidelity when using a non-uniform prior. The methods developed are based on general probabilistic considerations using Bayesian statistics, and can be applied to other relevant prior distributions beyond the von Mises-Fisher distribution explored in this study.

The ability to benchmark across the full range of particle number, from a single qubit to systems approaching infinity, highlights the scalability of these fidelity assessments and their potential relevance to increasingly complex quantum systems.

Projective Measurements on Bloch Sphere Axes Maximize Fidelity

These optimized measurements, directed along axes orthogonal to the peak of the von Mises-Fisher distribution, maximize information gain and yield mean fidelities exceeding those achievable with random projections. “We present evidence to support that this limit is very close to the mathematical limit based on projections on suitable bases,” the paper states. The analysis reveals that as N increases, the fidelity achievable with coherent spin POVM measurements converges with that of projective measurements using optimally chosen axes.

This convergence suggests a robust approach to benchmarking teleportation performance regardless of system size. The optimized projection scheme utilizes projective measurement on specific axes on the Bloch sphere, while a coherent spin positive operator-valued measure (POVM) can achieve similar results, based upon heterodyne detection.

For single qubit teleportation, the POVM fidelities closely follow those computed using projective measurements with axes maximally separated from the peak of the prior distribution. The resulting mean fidelity is an average of fidelities based on projections on random directions around the Bloch sphere.

Coherent Spin POVMs Approach Optimal Fidelity Limits

Fidelity benchmarks for quantum teleportation may need recalibration, as coherent spin positive operator-valued measures (POVMs) demonstrate performance approaching theoretical limits even when input states are not uniformly distributed. Traditional assessments assume a uniform distribution of qubits or coherent states, a simplification that doesn’t reflect scenarios where prior knowledge of the input state exists and skews that distribution. This work establishes that achieving quantum advantage in teleportation may require exceeding previously established thresholds when using a non-uniform prior, specifically a von Mises-Fisher distribution.

As quantum systems scale beyond single qubits, projective measurements, the standard method for assessing teleportation fidelity, become computationally impractical. Projecting onto basis states within a Hilbert space, where N represents the number of particles, presents a significant challenge for both experiments and calculations.

Researchers have found that coherent spin POVMs offer a viable alternative, achieving comparable fidelity to optimal projective strategies, particularly for multi-qubit systems modeled as spin coherent states. “The single qubit case is actually the worst case for the coherent spin POVM,” the paper notes. The effectiveness of coherent spin POVMs stems from their ability to prioritize measurements along directions that maximize fidelity. In the limit as N approaches infinity, the fidelity achieved with POVM measurements converges with that of optimal projective measurements.

The team derived analytical results for optimal POVM detections that reproduce the Q-function of the state, enabling accurate fidelity assessments for spin coherent states with N qubits. Determining the ultimate fidelity bounds, however, would necessitate projections onto a complex basis of N-particle states.

While these results were obtained using spin coherent states drawn from a von Mises-Fisher prior distribution, the methods can be extended to benchmark teleportation of other state types, such as spin squeezed states or rotated Dicke states. Further research will focus on refining these POVM schemes and defining optimal detection strategies to push the boundaries of quantum teleportation fidelity.

Multi-Qubit Teleportation Benchmarks with Spin Coherent States

The work establishes a new analytical expression for fidelity limits, accounting for non-uniform distributions that reflect prior knowledge of the teleported state, a factor often overlooked in standard benchmarking procedures. This means current thresholds used to declare a successful demonstration of quantum teleportation may be unnecessarily conservative. As the number of qubits increases, standard projective measurements become computationally impractical due to the exponential growth of the Hilbert space dimension, requiring projections onto a space.

The analysis reveals that for larger ensembles, where the average particle number is significantly greater than one, the fidelity approaches one-half, but for smaller ensembles, fidelity can be substantially higher. “There is therefore a necessity to generate reliable benchmarks for quantum teleportation of states of a qubit or of collective spins which are drawn from a nonuniform distribution, which defines the purpose of this paper,” the paper states.

The team’s findings suggest that the fidelity threshold for claiming genuine quantum advantage in a teleportation scheme could be substantially higher when the input state is selected from a non-uniform distribution. The work also acknowledges that while POVM detection schemes are expected to be very close to the ultimate fidelity bounds, defining optimal detection schemes remains an open area of investigation.

Fidelity Thresholds Exceed Classical Limits with Prior Knowledge

The work reveals that assuming prior knowledge of the input state’s distribution allows for significantly higher fidelity in teleportation than benchmarks predicated on uniform distributions would suggest. Calculations focused on single qubit teleportation with a prior distribution peaked at a specific point on the Bloch sphere show that projective measurements tailored to that prior can surpass standard fidelity limits.

Specifically, the team determined that projecting spins along the direction of prior knowledge yields no benefit if the average particle number is less than a critical value, defined as one over (κ + 2), where κ represents the degree of polarization. Only with smoother, more dispersed prior distributions and a higher average particle number does incorporating measurement results improve mean fidelity.

This suggests that current methods for evaluating teleportation success may underestimate performance in scenarios where the initial state is not entirely random. The team conjectures that the positive operator-valued measure (POVM) fidelity limits they derived will closely approach the absolute maximum achievable, particularly as the number of qubits increases. This is supported by the observation that these limits converge even for a single qubit, the most challenging case for POVM application, and coincide with uniform distribution results when no prior knowledge is available.

These findings have implications for benchmarking diverse quantum teleportation schemes, including those using entangled collective spins. The ability to accurately assess performance with non-uniform prior distributions is essential for evaluating the true potential of these systems.

👉 More information
🗞 Benchmark for quantum teleportation with nonuniform prior distributions
✍️ Tomáš Opatrný, Allison Brattley and Kunal K. Das
🧠 DOI: http://link.aps.org/doi/10.1103/64hf-p413

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