Encoding is one of the first and most consequential decisions in designing a quantum program. Just as classical computers must decide how to represent integers or characters as bits, quantum algorithms must map problem variables—whether they denote colors, routes, or cluster assignments—onto the quantum states of a register. Over the past decade, researchers have adapted classical schemes such as one-hot, binary, and Gray codes to qubits, and have invented quantum-native alternatives like angle encoding, amplitude encoding, and Schmidt decomposition. Each approach embodies a trade-off between the number of qubits consumed and the complexity of the circuits needed to operate on them. One-hot encoding, in which one bit is reserved for every possible value a variable can take and exactly one of those bits is set to 1, is particularly popular because it is simple to encode, decode, and check for validity—even though it is wasteful in qubit count. A variable with four possible values, for instance, needs four qubits under one-hot rules, yielding the four allowed states 0001, 0010, 0100, and 1000.

The insight behind Multi-Unary is disarmingly simple. Instead of requiring that exactly one bit be set to 1, the scheme requires only that at least one bit be set to 1, with the single forbidden state being the all-zero string. Under this relaxation, the four-qubit register above can represent not just four values but fifteen—the full set of non-zero bit strings. Because more than one value can live in the same group of qubits, the Spanish team describes Multi-Unary as a “dense” encoding scheme, and, to their knowledge, it is the first encoding in the quantum computing literature with this property. The consequence is profound for algorithms like Grover’s: a single measured bit string can now encode several valid assignments at once, so the quantum state effectively contains and delivers multiple solutions in a single shot.

Grover’s algorithm, first proposed in 1996, searches an unstructured space of size N containing M solutions with a complexity of O(√(N/M)), a quadratic speedup over any classical method. The algorithm begins by placing the input qubits into an equal superposition of all possible states. It then applies an oracle—a quantum subroutine that recognizes valid solutions and flips their phase by π radians—followed by a diffusion operation that inverts the amplitudes about their mean, amplifying the marked states. Each oracle-plus-diffusion cycle is one Grover iteration, and the probability of measuring a solution after i iterations is given by P = sin²((2i+1)·θg), where θg = arcsin(√(M/N)). The optimal number of iterations scales as R ≤ ⌈(π/4)√(N/M)⌉. Crucially, both the success probability and the required iteration count depend on the ratio M/N: the more states qualify as solutions, the better.

This is exactly where Multi-Unary shines. By enlarging the set of bit strings that count as legitimate solutions, the scheme increases M, which the team proves mathematically raises the success probability P and lowers the required number of Grover iterations R. They derive the partial derivatives of both quantities with respect to M, showing rigorously that P grows monotonically and R shrinks monotonically over the entire operating range. The practical payoff comes in three forms: a higher chance that any given measurement yields a solution, fewer oracle calls needed to maximize that chance, and measured outputs that can bundle multiple solutions into one value, reducing the total number of circuit executions—or “shots”—needed to recover a desired fraction of all solutions.

To demonstrate the idea concretely, the researchers applied Multi-Unary to the graph coloring problem, a canonical combinatorial challenge that asks for the assignment of k colors to the nodes of a graph such that no two adjacent nodes share a color. Graph coloring matters well beyond theory—it underpins scheduling, register allocation, and resource assignment tasks—and it has become a standard benchmark for quantum algorithms, tackled previously with quantum annealing, variational approaches, and Grover-based oracles using both binary and one-hot encodings. In the Spanish team’s demonstration, a six-node, six-edge graph is colored with three colors, the minimum possible.

Their oracle design rests on quantum counters, built from multi-controlled X gates, which tally constraint violations. The logic proceeds in three steps. First, for each node, the counter is incremented if all of that node’s color qubits are zero—penalizing strings that violate the Multi-Unary rule that every variable must carry at least one value. Second, for every edge and every color, the counter is incremented if both endpoints display that same color simultaneously, penalizing genuine coloring conflicts. Third, only strings for which the counter reads zero—strings satisfying every constraint—have their phases rotated by a multi-controlled Z gate, marking them for amplification. A single ancilla qubit acts as a trigger that fires each counter increment and is uncomputed immediately afterward, while the remaining ancillas store the running count. The design also incorporates a subtle optimization: because early counter increments only touch the least significant bits of the count, the number of controls on successive increment gates can be grown gradually, trimming expensive multi-controlled gates.

The resource analysis for the example circuit is instructive. The complete oracle comprises 367 gates, of which 284 are controlled X gates—and 188 of those, fully 66.2 percent, implement the counter alone, underscoring how central counting is to this style of oracle construction. The circuit depth is 287 layers, dominated by the trigger qubit through which every increment must pass. Simulating the full circuit with 36 Grover iterations—the number that maximizes success probability—yielded a 99.88 percent chance of measuring a solution, and a histogram of 1,000 shots in Qiskit’s Aer simulator recovered 126 distinct solution values, with only two spurious non-solutions appearing once each, a rate consistent with statistical expectations.

The headline result is the sheer expansion of the solution space. Under a conventional one-hot oracle, the example graph has exactly 48 valid colorings. With Multi-Unary, the number of solution values jumps to 126—78 additional bit strings that are also valid, thanks to nodes that can carry more than one compatible color. This flexibility arises naturally in real graphs: leaf nodes and nodes with low connectivity often admit several admissible colors, and Multi-Unary lets the quantum state express that freedom directly. Decoding is straightforward: any measured variable whose bits contain more than one 1 is expanded into as many one-hot candidates as there are set bits, each assigning a single value while preserving the rest. In one worked example, the output string 011 010 001 100 010 101—where two nodes each hold two colors—unpacks into four distinct, fully valid colorings of the graph. A single measurement delivered four answers.

The authors are careful to position their contribution honestly. Binary encoding still wins on raw qubit count—a 50-node graph with 10 colors needs 200 binary-encoded input qubits versus 500 for one-hot or Multi-Unary—but binary circuits must then explicitly reject the invalid states corresponding to out-of-range values, adding gates and ancillas that Multi-Unary avoids by construction. Indeed, validity checking is where Multi-Unary beats its closest cousin outright: the only forbidden state under Multi-Unary is the all-zero string, detectable with a simple NOR gate, whereas one-hot verification requires a counter to confirm that exactly one bit is set. Because encoding and decoding work identically in both schemes, the team argues that Multi-Unary could outright replace one-hot encoding, with migration requiring changes only to the oracle’s validity-check portion—and a simpler oracle as a reward. The advantage does taper off for densely connected graphs, where fewer nodes admit multiple colors and the scheme converges back toward plain one-hot behavior.

The implications reach beyond graph coloring. The researchers point to shortest-path, graph coverage, and clustering problems as natural targets, and note that their encoding could slot into emerging frameworks like the discrete quantum intermediate representation proposed for encoding-independent optimization. For today’s noisy, resource-starved quantum hardware, where every ancilla qubit and every additional Grover iteration carries a real cost, a scheme that squeezes multiple solutions into the same qubits—and trims the iteration count as a bonus—offers a rare kind of free lunch. If quantum computing’s near-term future depends on getting the most out of every qubit, Multi-Unary suggests that sometimes the smartest move is not a new algorithm, but a new way of writing the problem down.

The CQC conjecture concerns a tension at the heart of quantum mechanics. Two parties, commonly labelled Alice and Bob, may share a quantum state containing correlations that cannot be described entirely as ordinary classical information. When both measure their systems, however, the outcomes become classical data. The conjecture states that if Alice and Bob measure their systems in two mutually unbiased bases, the sum of the classical mutual information obtained from those two experiments cannot exceed the original quantum mutual information shared by the systems. In symbols, the conjecture is written as (I(Z^A:Z^B)+I(X^A:X^B)\leq I(A:B)). Here, (Z) and (X) represent incompatible measurements, while (I(A:B)) quantifies the total correlations in the original quantum state. The challenge is that measurement can reveal different aspects of a quantum state, raising the possibility that correlations extracted in separate experiments might collectively appear larger than the correlations present before measurement.

Mutually unbiased bases are central to the problem because they represent maximally complementary measurement choices. If a particle is prepared in one basis, a measurement in a mutually unbiased basis produces outcomes with equal probability. In a (d)-dimensional system, the computational basis can be paired with a Fourier basis, whose vectors are coherent superpositions of all computational states. Measuring in one basis can make the outcome of the other completely unpredictable. This complementarity underlies entropic uncertainty relations, which place lower bounds on the combined uncertainty associated with incompatible measurements. Unlike simple uncertainty statements about position and momentum, the CQC conjecture tracks mutual information between two systems, making it sensitive to both local randomness and shared correlations.

The conjecture was introduced more than a decade ago by researchers studying uncertainty relations for mutual information. It has already been established for several important families of states, including pure states, states with one maximally mixed subsystem and situations in which one of the measurements is minimally disturbing. It also has potential practical consequences. If the conjecture is correct, unusually large classical correlations observed in two incompatible measurement settings can serve as evidence of entanglement. The result may also strengthen uncertainty relations involving quantum memories and constrain how much information an eavesdropper can obtain in quantum key distribution. In cryptographic settings, Alice and Bob can use correlations between their measurement outcomes to establish a secret key, while the conjecture would help bound the information available to an adversary.

Iqbal’s first contribution is a sufficient condition derived from an information-exclusion result developed by researchers Patrick Coles and Marco Piani. That earlier result limits the combined mutual information Alice can obtain about Bob’s quantum system when she measures in two mutually unbiased bases. In simplified form, it states that (I(Z^A:B)+I(X^A:B)) cannot exceed (\log d-H(A|B)), where (d) is the system dimension and (H(A|B)) is the quantum conditional entropy. The new condition compares the loss of information caused by measuring Bob’s quantum system with the loss caused by converting it into classical outcomes. If the decrease from quantum memory to classical measurement is sufficiently large, the original CQC inequality follows automatically. This does not prove the conjecture for every state, but it identifies a measurable structural feature that guarantees its validity.

The condition is especially interesting because it applies beyond the examples already known to satisfy CQC. Iqbal reports numerical demonstrations using random mixed, separable two-qubit states that are neither pure nor equipped with a maximally mixed subsystem. These states were selected to satisfy the new mathematical criterion. For each state, the researchers compared the original quantum mutual information with the sum of the two classical mutual informations produced by incompatible measurements. The difference remained non-negative in the simulations, meaning the classical information extracted from the two measurement settings never exceeded the total quantum correlation. Such numerical evidence cannot replace an analytical proof, but it illustrates how the sufficient condition can identify previously inaccessible regions of the space of quantum states.

The study then advances a broader proposal called the extended CQC, or ECQC, conjecture. Instead of using only two mutually unbiased bases, the extension considers all (d+1) mutually unbiased bases available in prime dimensions and asks whether the original quantum mutual information is at least as large as the sum of the (d) smallest classical mutual informations generated by those measurements. Excluding the largest term is essential to the proposed formulation: summing every available basis can produce a quantity that is too strong to be universally plausible, whereas selecting all but the most informative measurement creates a more balanced comparison. Prime dimensions such as three and five are particularly useful because complete sets of (d+1) mutually unbiased bases are known to exist there.

The author also derives a sufficient condition for ECQC using a multiple-measurement entropic uncertainty relation. This relation connects the sum of conditional entropies from (d+1) measurements to both the dimension of the system and the quantum conditional entropy (H(A|B)). The resulting criterion compares the total information available when Alice measures while Bob retains a quantum memory with the information remaining after Bob also measures. If the reduction is large enough, the extended conjecture follows. The analysis further produces a bipartite generalization of the Maassen–Uffink uncertainty relation, placing a lower bound on the combined joint entropies of Alice’s and Bob’s measurement outcomes. In physical terms, the more complementary measurements the parties perform, the more uncertainty must appear in their combined classical records.

One of the most detailed tests involves isotropic states, a family that mixes a maximally entangled state with completely mixed noise. These states are described by (\rho{AB}=p|\Psi^+\rangle\langle\Psi^+|+(1-p)\mathbb{I}{AB}/d^2), where (p) controls the weight of the entangled component. Their individual subsystems remain maximally mixed for the full allowed range of (p), while their shared quantum mutual information changes continuously with the noise level. The calculations show that, in the chosen complete sets of mutually unbiased bases, two particular measurements retain nonzero classical mutual information, while the other measurements produce uniformly distributed joint outcomes and therefore zero mutual information. For prime dimensions, the author derives an explicit expression for the two nonzero contributions and shows that their sum remains below the original quantum mutual information. This establishes ECQC for isotropic states across the examined prime-dimensional family, including states that are entangled and states that are separable.

The numerical investigation extends beyond isotropic states. For dimension three, the researchers tested 100,000 random pure bipartite states and 100,000 random mixed bipartite states using four mutually unbiased bases, then compared the quantum mutual information with the sum of the three smallest classical mutual informations. No violation was observed. Additional tests examined the isotropic family across its entire physical parameter range. In dimension five, a more computationally demanding study used 10,000 random pure states and 10,000 random mixed states with six mutually unbiased bases. Again, removing the largest classical mutual information and summing the remaining five produced no value greater than the original quantum mutual information. These experiments are best understood as evidence supporting the conjecture rather than proof: random sampling cannot rule out rare counterexamples, and the structure of the chosen bases may influence the numerical outcome.

The findings arrive with important limitations and an open invitation to the quantum information community. The original CQC conjecture remains unresolved for arbitrary mixed states, and the extended version is even less established. The proposed sufficient conditions cover only states satisfying specific inequalities, while the numerical simulations explore finite samples and selected prime dimensions. Composite dimensions pose an additional obstacle because the maximum number of mutually unbiased bases is not known in general, and some available collections can behave differently. The next major goal is an analytical proof of ECQC for all pure states, followed by a formulation that works in both prime and composite dimensions. If those challenges can be overcome, the conjecture could become a powerful bridge between quantum correlations, uncertainty and communication security, turning incompatible measurements into a practical diagnostic for the hidden information structure of quantum matter.

Subject of Research: Quantum mutual information, entropic uncertainty relations, mutually unbiased bases, quantum correlations and entanglement

Article Title: On the CQC conjecture: a sufficient condition and an extension

Article References: Iqbal, H. “On the CQC conjecture: a sufficient condition and an extension.” Quantum Information Processing 25, Article 250 (2026). Foundational references include Schneeloch, Broadbent and Howell, “Uncertainty relation for mutual information,” Physical Review A 90, 062119 (2014); Coles and Piani, “Improved entropic uncertainty relations and information exclusion relations,” Physical Review A 89, 022112 (2014); and Berta et al., “The uncertainty principle in the presence of quantum memory,” Nature Physics 6, 659–662 (2010).

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05258-2

Keywords: quantum mutual information, CQC conjecture, ECQC conjecture, entropic uncertainty relations, mutually unbiased bases, quantum entanglement, quantum correlations, isotropic states, quantum information theory, quantum cryptography

When atoms or molecules interact with high-energy light, particularly in the extreme ultraviolet or X-ray range, a phenomenon known as the photoelectric effect occurs. This process results in the detachment of an electron, called a photoelectron, from the atom. The kinetic energy of these emitted electrons provides a wealth of information about the original atom being irradiated. This fundamental principle underlies the practice of photoelectron spectroscopy, a technique extensively used in the study of atomic and molecular structures. However, traditionally, the emitted electrons were treated as classical particles, neglecting their inherent quantum properties.

In reality, photoelectrons represent quantum objects that cannot be fully understood through classical physics. Their behavior necessitates a quantum mechanical approach, as these electrons exhibit properties akin to both particles and waves. The latest technique devised by the Lund University researchers, known as KRAKEN, allows them to address a critical question: to what extent can we characterize the quantum state of the emitted photoelectron? Drawing an analogy to medical imaging techniques like CT scans, the researchers employ a series of laser pulses to capture two-dimensional representations of the quantum state, which are then reconstructed into a comprehensive three-dimensional profile.

Through the application of this innovative measurement method, the researchers successfully measured the quantum states of photoelectrons emitted from helium and argon atoms. Importantly, their findings reveal that the quantum state of the emitted electrons is influenced by the type of material from which they originate. This discovery has far-reaching implications, especially given that the photoelectric effect was first elucidated by Albert Einstein more than a century ago, forming the bedrock for modern quantum mechanics.

The significance of the study extends beyond fundamental physics; it opens possibilities for practical applications in various fields. The researchers assert that while the current study focused on simple atomic systems such as helium and argon, future applications could include investigating complex systems found in molecular gases, liquids, and solids. An understanding of how ionized targets react to the sudden loss of an electron at a quantum level could prove crucial in areas such as atmospheric photochemistry and energy-harvesting technologies, including solar cells and the natural photosynthesis observed in plants.

Another key aspect of this groundbreaking work is its ability to connect diverse scientific disciplines. It bridges the realms of attosecond science and spectroscopy, areas where significant advancements have been made by researchers like Nobel Prize laureate Anne L’Huillier. At the same time, it intersects with emerging fields in quantum information and technology. This multi-disciplinary approach highlights the relevance of this study in the context of the ongoing second quantum revolution, which aims to manipulate individual quantum entities to harness their properties for practical applications.

Furthermore, the implications for societal advancements are profound. While the current measurement technique may not lead directly to the construction of new quantum computers, it offers unprecedented access to the quantum states of photoelectrons. This enhanced understanding could empower physicists and engineers to fully exploit these quantum properties, paving the way for innovations and applications that were previously unattainable.

In addition to its potential applications, the study sheds light on fundamental questions about the nature of quantum mechanics. For instance, why don’t we observe quantum effects in macroscopic objects? Researchers explain that at the microscopic scale, electrons, atoms, and molecules follow the laws of quantum mechanics, while larger objects adhere to classical physics. The phenomenon of decoherence plays a pivotal role here, as groups of quantum entities interact uncontrollably, effectively masking their individual quantum traits. The new technique not only aims to measure these traits but also seeks to track how quantum properties evolve over time, transitioning from quantum to classical behavior.

The researchers expressed their astonishment not only at the successful implementation of the new measurement method but also at how effectively it worked. Previous attempts to analyze the quantum state of photoelectrons had proven extremely challenging, requiring exceptionally stable conditions over extended periods. Achieving such stability represents a remarkable feat, highlighting the innovative nature of the KRAKEN technique.

This pioneering research is poised to transform the landscape of photoelectron spectroscopy, allowing scientists to glean deeper insights into the quantum world. By measuring the speed and emission direction of photoelectrons, researchers can unlock valuable information about material structures, enabling the exploration of novel materials and their properties. As a result, this work holds the promise of not only advancing our fundamental understanding but also catalyzing practical advancements across various scientific disciplines.

In summary, the groundbreaking measurement of photoelectron quantum states by researchers at Lund University marks a significant achievement, one that has potential ramifications in understanding quantum behavior and its applications in technology and materials science. The integration of quantum mechanics and spectroscopy in this innovative framework paves the way for future research that may unravel the complexities of light-matter interactions, opening doors to revolutionary technologies and scientific discoveries.

Subject of Research: Quantum state measurement of photoelectrons
Article Title: Measuring the quantum state of photoelectrons
News Publication Date: October 2023
Web References: Lund University
References: Busto, D. et al. (2023). Measuring the quantum state of photoelectrons. Nature Photonics.
Image Credits: Lund University

Keywords

Quantum mechanics, photoelectric effect, photoelectron spectroscopy, quantum state tomography, light-matter interactions, KRAKEN technique, helium, argon, decoherence, quantum properties, attosecond science, spectroscopy.