Polish physicists construct entangled states with prime power dimensions

Palash Pandya for Theoretical Physics, Polish Academy of Sciences, and colleagues have constructed explicit families of absolutely maximally entangled (AME) states in dimensions twice an odd prime power. The work demonstrates AME(13,2q) states are achievable for any prime power q greater than or equal to 17, expanding the possibilities for highly entangled systems. For every odd N greater than or equal to 3, the same coupling gives AME(N,2q) under the sufficient condition q ≥ maxN,2^(N-2) on the odd prime power q, and within this coupling, the smallest such q must grow exponentially with N. AME State Construction Using Qubit and Q-Level Coupling This approach circumvents a common obstacle in entanglement construction by using q-level states to remove interference between different branches of the system, allowing qubit states to achieve maximally mixed reductions only on average. The researchers detail how this method provides explicit, verifiable constructions for AME states, moving beyond theoretical possibilities to concrete implementations. The core of this construction relies on superposing products of qubit and q-level states, a technique that effectively cancels interference and simplifies the requirements for qubit entanglement. A key step involves projecting one extra party from an existing AME state, a standard procedure in quantum coding, to create a “parent” state used in the coupling process. This parent state, combined with a carefully chosen qubit mixture, forms the basis for generating the new AME states.

The team extended existing constructions, such as those found in the Atlas Lean library and work by Kitamura, to establish concrete thresholds and specifications for these entangled states. Exact arithmetic was used to verify the inputs for smaller field sizes, ensuring the accuracy of the construction. The versatility of this coupling mechanism extends to any odd number of parties, offering a scalable approach to building complex entangled systems. “The sharper thresholds of Theorem II.3 use fewer branches,” the researchers note, indicating an ongoing effort to optimize the efficiency of the construction. The extended Cauchy construction provides the necessary parent state when q is greater than or equal to N, simplifying the process for certain configurations. The capacity of the q-level systems to supply branches satisfying a specific equation is limited by the integer q, as proven in Appendix F.1, while the qubit side is constrained by the rank of the qubit mixture. This interplay between the qubit and q-level components dictates the overall efficiency of the construction.

The team also demonstrated the creation of AME(15,2p) states using prime numbers p, achieved through a cyclic shift of the parties and compensation with local unitaries. This focus on practical construction represents a step toward realizing the potential of highly entangled states for quantum technologies. The researchers emphasize that their contribution lies in providing explicit infinite families of AME states with defined thresholds and concise specifications, rather than simply proving their existence. Explicit AME(N,2q) Families for Odd Prime Power q This finding establishes a specific lower bound for achieving high degrees of entanglement within systems of this size building on established methods that couple qubits with systems operating at q-levels. The construction of these states relies on a coupling that maintains fixed qubit mixtures while varying the field size, allowing for the exploration of different prime power values. This flexibility is further demonstrated by the ability to combine known inputs and products to fully classify nine-party AME states, establishing that an AME(9,d) state exists if and only if the local dimension d is greater than or equal to 3. Thresholds for AME(N,2q) Existence with Growing q This means scaling up the number of entangled parties demands a rapidly increasing local dimension, q, to maintain the desired entanglement properties. Specifically, the work details achievable configurations for several party numbers. Larger systems, like AME(17,2*q*), necessitate even higher prime powers, with a minimum q of 67. An alternative construction method for AME(15,2*p*) states further expands the possibilities, with a cyclic shift of the parties compensated by local unitaries, for every prime p. “The sharper thresholds of Theorem II. Nine-Party AME(9,d) Classification and Cyclic AME(15,2p) The construction of absolutely maximally entangled (AME) states with nine parties is now complete, with AME(9,d) states existing for integer dimensions d greater than or equal to three. This conclusion stems from newly constructed families of entangled states, combined with known examples and product rules, solidifying the classification for this specific number of parties. The work addresses a long-standing question regarding the feasibility of achieving maximal entanglement across nine quantum systems with varying local dimensions. The analysis also reveals that the constructed families are not stabilizer states in the cyclic Weyl convention, and that canonical graph-state AME constructions are excluded in even local dimensions when the number of parties is divisible by four. These findings contribute to a deeper understanding of the constraints and possibilities within the realm of AME state construction. The detailed constructions and proofs are available in the accompanying repository, allowing for independent verification and further exploration of these entangled states. Construction Details: Interference Cancellation and Branching Specifically, the incorporation of q-level systems alongside qubits allows for maximally mixed reductions on the qubits only on average, a departure from requiring them for every possible subset of parties. Further refinement allows for AME(13,2q) states when q is 17 or greater, AME(15,2q) with q at 9 or greater, and AME(17,2q) when q reaches 67 or more. The resulting superposition avoids the need for full entanglement across all qubits, simplifying the requirements for achieving maximal entanglement. The analytical proofs of these infinite families are supported by exact arithmetic verification for smaller fields, ensuring the accuracy of the construction. The process requires both a qubit mixture and a parent state. 👉 More information🗞 Explicit Absolutely Maximally Entangled states in dimensions twice a prime power✍️ Palash Pandya (Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, Poland)🧠 ArXiv: https://arxiv.org/abs/2611.17216v1 More like thisQuantum PhysicsQuantum operations guide shows how physics limits information processingQuantum PhysicsAnderson orthogonality catastrophe is seen in quantum dot experimentsQuantum Research NewsPhysicists demonstrate a spin-state memory for quantum informationPhysicsResidual coherence dominates entanglement in neutrino oscillation, study findsStay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: