#semidefiniteprogramming

Positive Operator Value Measures

New quantum study disproves “most non-projective” measurement assumptions.

A recent study by Heinrich Heine University Düsseldorf and Lund University researchers Gabriele Cobucci, Raphael Brinster, and Shishir Khandelwal could revolutionise quantum measurements. Their findings disprove the long-held idea that Symmetric Informationally Complete (SIC) observations are the hardest to reproduce using conventional methods in quantum systems larger than basic qubits. The groundbreaking study “Maximally non-projective measurements are not always symmetric informationally complete” presents a novel method for identifying truly non-projective measurements and accurately calculating their simulability thresholds, shedding light on quantum information fundamentals.

Understanding Quantum Measurements: Beyond Projective

Standard quantum measurements are frequently characterised by entire orthogonal projector sets. Positive Operator-Valued Measures (POVMs) are the largest measurement concept. Importantly, some POVMs are “non-projective,” meaning they cannot be converted to projective measures. This type of non-projective measurement is perfectly shown by the Symmetric Informationally Complete (SIC) POVM. SIC-POVMs are important in quantum theory because they are the best tools for state tomography and randomness production. They even apply to quantum theory fundamentals.

Implementing these complex non-projective measurements in quantum devices is difficult. They require ancilla systems and entangling gates and are more expensive and complicated than projective measures. In light of this practical issue, how effectively can an investigator duplicate POVM effects using only projective measurements and standard processing?

Noise and Semidefinite Programming in Quantifying Non-Projectivity

The “projective simulability threshold” or “critical visibility” of a POVM demonstrates its non-projectivity. This threshold specifies the lowest depolarisation noise a POVM can tolerate before projective measurements can recreate it. More non-projective measurements have a lower visibility threshold. Due to its impartiality, mathematical elegance, and experimental value, depolarisation noise is a common benchmark. The study also examined a “worst-case” noise model, a more rigorous test that yields qualitatively equivalent results.

The semidefinite programming (SDP) criterion for projective simulability is a major invention of this study. Effective computational tools, SDPs have a good reputation. The researchers used this SDP criterion to accurately determine quantitative simulability conditions for generic POVMs and truly non-projective measurements. The implementation code is public. This necessary condition is also sufficient for projective simulation, which produces accurate results for qubit (d=2) and qutrit (d=3) POVMs.

Beyond Qubits, SIC-POVMs Are Not Always Strongest

The unique SIC-POVM is the most non-projective measurement for qubit systems (d=2), with a visibility threshold of 81.6%, according to previous studies. However, Cobucci, Brinster, Khandelwal, and colleagues' new study proves this is not true for dimensions larger than two.

The study found that the SIC-property alone cannot measure projective simulability for qutrit SIC-POVMs (d=3). This is because the SIC-property determines overlap magnitudes but not POVM element phases. The researchers' innovative SDP criterion shows that the Hesse SIC-POVM has the lowest visibility, at 79.3%. This overwhelming data suggests the Hesse SIC-POVM is the most non-projective qutrit measurement. Analytical simulation of the noisy Hesse SIC-POVM requires 72 equiprobable projective measurements.

Ququart Systems (d=4): Four-dimensional systems have one SIC-POVM till unitary and antiunitary transformations. The investigation astonishingly shows that this SIC-POVM is not the most non-projective measurement in d=4 with 82.6% visibility. Interestingly, all four-dimensional SIC-POVMs were less resilient than their lower-dimensional counterparts. This shows that SIC-POVMs are not always harder to replicate as dimension increases.

The New Non-Projectivity Benchmark: “Flagged” Measurements

The researchers utilised a systematic numerical search method that leverages the SDP criterion as a “oracle” to determine which observations beyond qubits are most non-projective.

The hunt for qutrits often led to the Hesse SIC-POVM, supporting its conjecture. However, it occasionally settled in a local extremum, revealing a new measurement type: the flagged SIC-POVM (E_fSIC2). An orthogonal projection (called a “flag”) is appended to the qubit SIC-POVM embedding to create this POVM. With 79.6% visibility, it is less visible than the qubit SIC-POVM but more visible than the Hesse. Contrary to expectations, an orthogonal projection “amplifies the non-projective features” of the qubit SIC-POVM.

The numerical search yielded a non-SIC-POVM for ququarts with ten outcomes. The highlighted Hesse SIC-POVM (E_fSIC3) is a new measurement that embeds it with an orthogonal projection. A conjectured visibility threshold of 78.2% was found compared to the SIC-POVM included in it. The researchers hypothesise this is the least projective ququart measurement. In contrast to traditional SIC-POVMs, this “flagged” measurement has a monotonically decreasing simulability threshold with increasing dimension when depolarising noise is considered.

The study confirmed that these fundamental findings are not specific to depolarisation as the quantifier of non-projective features by showing qualitatively identical results with a “worst-case” noise model.

Future implications and directions

These innovative results demonstrate that SIC-POVMs are not the most resilient to depolarisation noise in projective simulations, especially those beyond qubit systems. This paper clarifies quantum measurement restrictions and the essential simplicity-accuracy trade-offs. This affects quantum information processing and quantum technology advancement.