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A New Paper Sets Lower Bounds on Quantum Measurement Memory
A study of exact quantum-measurement storage derives lower bounds on how quantum and classical memory can be divided. It also describes protocols whose memory requirements depend on how the measurement is accessed and on the target success
Quantum.world Team · Written with AI · fact-checked ·
Two inequalities constrain stored measurements
A universal processor that stores an unknown quantum measurement and later reproduces it exactly must satisfy two memory bounds: Q + C ≥ log₂ n and 2Q + C ≥ log₂((n−1)d² + 1). Here Q is the number of qubits in the stored quantum register, C is the number of classical bits, n is the number of measurement outcomes, and d is the dimension of the input system. The bounds apply to n-outcome positive operator-valued measures (POVMs) on a d-dimensional system, with d ≥ 2.
The paper studies exact heralded retrieval: the processor has a successful and a failure outcome, and the reproduced measurement is exact when retrieval succeeds. Its lower bounds apply to every universal retrieval processor, while its constructions depend on how the unknown measurement is made available during storage. The arXiv listing identifies version 2610.36972v1 as dated October 27, 2026.
“For n ≥ 2, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least d, at that dimension its classical alphabet must have at least n symbols, and both minima are attained by a single-query protocol with success 1/d².”
The authors of the paper, Filip Turoboś, Ludmiła Marcinkowska and Zbigniew Puchała, write.
Classical records cannot replace the quantum register
For at least two possible measurement outcomes, the paper proves that a finite classical record alone cannot support exact retrieval. The quantum program must have dimension at least d. If it has exactly that minimum dimension, its classical alphabet must still contain at least n symbols. A single-query protocol attains both minima, but succeeds with probability 1/d².
These results describe a trade-off, not a single required split between quantum and classical storage. The paper also proves that any processor retrieving every n-outcome POVM exactly with positive success needs program dimension at least n. In addition, its inequalities constrain how the two kinds of memory can be allocated, even when neither part is individually at its minimum.
Memory scaling depends on the target success probability
At fixed input dimension and a target success probability below one, the paper establishes total-memory scaling of Θ(log n) as the number of outcomes grows. At fixed d and n ≥ 2, total memory scales as Θ(log(1/(1−p))) as the uniform success probability p approaches one. These are asymptotic results; they do not say that the minimum-memory single-query protocol has a high success probability.
For direct black-box access to the measurement channel, the authors construct an exact compression of a port-based program. The compressed program consists of an outcome-type record and a quantum register, with total dimension polynomial in the number of queries. The paper counts persistent program memory held between storage and retrieval. It notes that decoding may temporarily recreate the full carrier, so the memory bounds concern storage rather than peak workspace or decoding efficiency.
Access models produce different constructions
The paper compares direct access to the measurement channel with two structured forms of access. A Naimark dilation represents a POVM using a larger quantum space and a projective readout. A branch decomposition presents the measurement as a probabilistic ensemble of simpler measurement branches. The paper treats these as distinct oracle models and does not address how to construct a dilation or branch decomposition from black-box access alone.
Under branch-decomposition access, the authors give an exact heralded protocol whose quantum register dimension is binomial in the query count and input dimension, but independent of the number of outcomes. Its classical record contains 2d outcome labels and is independent of both the query count and the number of branches. Other constructions and comparisons in the paper likewise depend on which representation is supplied; the models are not assumed to be interconvertible.
The paper presents its compression as a construction, not as proof that the resulting registers are optimal. Its central limits concern what universal exact-retrieval processors must store, while its protocols show how the allocation and total memory vary with access model and target success.