Researchers at ICFO-Institut de Ciències Fotòniques have determined that 2 log d bits of randomness can be generated device-dependently in any dimension where a symmetric informationally complete (SIC) measurement exists, partially solving a long-standing open problem in quantum physics. The work characterizes the randomness produced by unbiased extremal rank-one measurements on quantum states, revealing that the tetrahedral symmetric informationally complete (SIC) measurement generates the least intrinsic randomness within its class.
Four-outcome qubit measurements of this type are tomographic, extending these findings to scenarios with fully source-device-dependent randomness. This analysis advances understanding of how to maximize randomness generation from quantum measurements.
The newly presented family of skewed SIC measurements are derived from symmetric informationally complete (SIC) measurements and can produce this maximal randomness when applied to pure or isotropically noisy quantum states. The tetrahedral symmetric informationally complete (SIC) measurement has the least intrinsic randomness within this class.
Unbiased Extremal Measurements and Quantum Fidelity
The ability to generate truly unpredictable numbers is now demonstrably linked to the quantum fidelity of a measured state, a connection established through analysis of unbiased extremal measurements. This relationship provides a quantifiable metric for assessing the randomness achievable with a given quantum state and measurement scheme, moving beyond purely theoretical bounds. This work extends to scenarios where the source of the quantum state is untrusted, a critical consideration for secure applications.
Because skewed symmetric informationally complete (SIC) measurements are informationally complete, the generated randomness remains secure even when the state’s origin is unknown. These measurements are analogous to higher-order counterparts of previously studied two-dimensional systems known to generate maximal device-independent randomness, suggesting potential applications in advanced cryptographic protocols. This finding challenges the intuitive expectation that a complete measurement, one capable of fully characterizing a quantum state, would necessarily maximize randomness generation.
The researchers found that the lower randomness stems from the specific geometric properties of the tetrahedral SIC measurement, highlighting a subtle interplay between completeness and unpredictability. “The most famous MICs are undoubtedly the symmetric informationally complete (SIC) measurements,” the paper states, acknowledging their prominence while simultaneously revealing their relative weakness in this specific context. The team’s findings also demonstrate a connection between randomness generation and the broader landscape of quantum measurements.
The conditional min-entropy generated by measuring any state is lower bounded by log n/d bits, where n is the number of measurement outcomes and d is the dimension of the system. This lower bound is state-independent, meaning it holds true regardless of the specific input state being measured.
The researchers note that if a state ρ is within the convex hull of the measurement projectors, the fidelity is one, while any state outside this hull results in a strictly lower fidelity. The analysis extends to minimal informationally complete (MIC) measurements, which are unbiased and relevant for quantum tomography. These measurements, with elements, are necessarily MICs, bridging the concepts of unbiasedness and completeness.
The work’s implications are significant for developing robust and certified randomness sources, essential for applications demanding high levels of security and unpredictability. The results apply equally to source-device-dependent scenarios, as these measurements reconstruct the untrusted quantum state, further solidifying their practical relevance.
Intrinsic Randomness of Qubit Measurements in Two Dimensions
This capability extends the findings beyond ideal laboratory conditions and into scenarios where complete control over the input state is not possible, a condition important for practical applications of quantum randomness. The analysis demonstrates that these measurements can fully reconstruct the untrusted state directly from the measurement data, bolstering their utility in real-world settings. The parameterization of unbiased qubit minimal informationally complete (MIC) positive operator-valued measures (POVMs) allows for the calculation of the length, l, which dictates the maximal guessing probability of the measurement.
The squared area of faces within the polytope formed by the measurement projectors is defined as sin^2δ(4 + cos^2α), where δ and α are parameters defining the measurement. These scissors measurements, previously used for device-independent randomness generation, offer a comparative framework for understanding the unique characteristics of the SIC measurement. Future research will focus on exploring the behavior of these measurements in the presence of noise, a critical consideration for practical implementation, and investigating whether similar distinctions between measurements hold in higher dimensions.
Conditional Min-Entropy Bounds Randomness Generation in Dimension d
The team’s analysis introduces the skewed SIC measurement family, derived from the well-known symmetric informationally complete (SIC) measurements, to explore the boundaries of randomness generation. In cases where the number of outcomes equals d squared, the team demonstrated the ability to reach at least log d bits of randomness. Further exploration of unbiased extremal rank-one measurements in dimensions two, three, and four revealed that these “scissors” measurements approach the limit.
Four-Outcome Qubit Measurements are Group Covariant
Unbiased extremal qubit measurements exhibit group covariance, meaning their measurement vectors can be generated by applying a specific set of unitary matrices to a fiducial vector. The analysis demonstrates that the measurement vectors, represented in Bloch space, can be systematically transformed using the functions f(α) and g(α), as defined in the published work. This finding has practical implications for designing and implementing quantum random number generators.
This independence is key for secure applications where an adversary could manipulate the input state to compromise the randomness. The researchers explicitly solved for the maximal intrinsic randomness achievable with these measurements in two dimensions, establishing a benchmark for evaluating other measurement strategies.
The ability to generate this maximum amount of randomness represents an advancement in the field of quantum randomness generation. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that 2 log d bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.
The analysis further reveals that the arrangement of measurement vectors forms a disphenoid, a tetrahedron with four congruent triangular faces. This geometric structure is not merely a visual curiosity. It directly relates to the properties of the measurement and its ability to generate randomness.
The team’s characterization of these measurements in terms of two angles provides a valuable tool for understanding and optimizing their performance. “The measurement vectors represented in (23) can be generated by acting on a suitable fiducial vector with the following basis of unitary matrices,” the researchers state, highlighting the mathematical framework underpinning their findings. The unpredictability of quantum physics gives rise to intrinsic randomness, which is exploited in the design of quantum random number generators.
Tetrahedral SIC Measurement Exhibits Least Randomness
This distinction emerges from an analysis characterizing randomness generated by unbiased extremal rank-one measurements, revealing that maximizing information retrieval does not necessarily maximize unpredictability. By adjusting parameters within the scissors family, the team found they could generate randomness arbitrarily close to this theoretical maximum, highlighting the potential for fine-tuning quantum measurements for optimal security applications. The analysis further reveals that the SIC measurement consistently yields less randomness than these adjustable configurations, even within the same dimensional space.
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