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arxiv: 2604.20432 · v1 · submitted 2026-04-22 · 🪐 quant-ph

Unitary Realizations of Synchronizing Automata in Quantum Systems

J\c{e}drzej Stempin , Jan W\'ojcik , Gabriela Banaszak , Andrzej Grudka , Marcin Karczewski , Pawe{\l} Kurzy\'nski , Antoni W\'ojcik This is my paper

Pith reviewed 2026-05-10 00:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum automatasynchronizing wordsglobal unitaryquditentanglementstate resetquantum information
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The pith

A quantum automaton encoded in a qudit can be driven to a predetermined pure state by a global unitary when auxiliary qubits are prepared in a synchronizing word.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a quantum version of classical synchronizing automata that works entirely with unitary operations. Auxiliary qubits encode input symbols including special synchronizing words that collapse the automaton to one state in the classical case. These qubits interact with the qudit holding the current automaton state through one joint unitary. The unitary maps the qudit to a fixed pure state from any initial condition while turning the qubits into an entangled register whose correlations record the starting configuration. The same setup can be used to produce particular entangled states by selecting different automata and words.

Core claim

When the qubit register is initially prepared in a state corresponding to a synchronizing word, the automaton evolves into a predetermined pure state independent of its initial state, while the qubit register is transformed into a complex, often entangled, state that encodes information about the automaton's original configuration.

What carries the argument

A global unitary operator on the tensor product of the qudit Hilbert space and the auxiliary qubit register that implements the automaton transitions for the encoded word and achieves the synchronization mapping for any initial qudit state.

Load-bearing premise

There exists a unitary on the joint qudit-qubit space such that, for auxiliaries fixed in the synchronizing word state, every possible initial qudit state is mapped to the same target pure state.

What would settle it

For a concrete small automaton possessing a known synchronizing word, construct the candidate global unitary and check whether it maps all basis states of the qudit to one identical pure state after the operation.

Figures

Figures reproduced from arXiv: 2604.20432 by Andrzej Grudka, Antoni W\'ojcik, Gabriela Banaszak, Jan W\'ojcik, J\c{e}drzej Stempin, Marcin Karczewski, Pawe{\l} Kurzy\'nski.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic representation of how synchronizing word [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Example 1 – A simple DFA can be composed of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Example 3 – Graph corresponding to the letter [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Transition graphs corresponding to the action of let [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Transition graphs corresponding to the action of let [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Transition graphs corresponding to the action of [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Diagram representing unitary QDFA state transi [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
read the original abstract

We introduce a quantum analogue of a classical synchronizing automaton. In classical case the state of a system evolves according to a set of rules forming an alphabet, and sequences of these rules, called words, govern its evolution. Certain special words, known as synchronizing words, drive the automaton into a predetermined state regardless of its initial configuration. Although such an apparently irreversible process seems incompatible with the unitarity of quantum mechanics, we present a resetting protocol based on quantum synchronizing words by incorporating auxiliary qubits whose states encode the rules of the automaton's alphabet. These qubits interact with the quantum automaton, whose state is encoded in a qudit, via a global unitary operation. When the qubit register is initially prepared in a state corresponding to a synchronizing word, the automaton evolves into a predetermined pure state independent of its initial state, while the qubit register is transformed into a complex, often entangled, state that encodes information about the automaton's original configuration. The resulting entanglement depends on both the rule set and the automaton's initial state, and we show how specific entangled states can be generated within this framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a quantum analogue of classical synchronizing automata. Auxiliary qubits encode the automaton's alphabet rules and interact with a qudit (encoding the automaton state) through a single global unitary. Preparation of the auxiliary register in a state corresponding to a synchronizing word is claimed to reset the qudit to a predetermined pure state independent of its initial configuration, while the register evolves into an entangled state whose inner products preserve the original qudit overlaps, thereby encoding information about the initial automaton configuration. The construction is presented as a resetting protocol that generates specific entangled states dependent on the rule set and initial state.

Significance. If the unitary construction can be made explicit and verified, the result would supply a unitary mechanism for realizing classically irreversible synchronization in quantum systems and a systematic way to produce rule-dependent entangled states. This could connect automata theory to quantum control and state engineering, with potential relevance to quantum information protocols that require effective resetting without measurement.

major comments (2)
  1. [Abstract] Abstract and main construction: the central claim asserts the existence of a global unitary on the combined auxiliary-qubit register and qudit system that achieves the reset for any initial qudit state when the register is prepared in a synchronizing-word state. No explicit matrix form, circuit decomposition, or proof that such a unitary exists and satisfies the isometry condition ⟨e(ψ)|e(φ)⟩ = ⟨ψ|φ⟩ is supplied, rendering the protocol unverifiable.
  2. [Main text (protocol definition)] The manuscript defines quantum synchronizing words constructively via the auxiliary encoding but does not demonstrate that the required isometry on the synchronizing subspace can be extended to a unitary on the full space while respecting the automaton transition rules encoded in the qubit states. This is load-bearing for the claim that the protocol is unitary and faithful to the classical automaton.
minor comments (2)
  1. Add at least one fully worked small example (e.g., a 2- or 3-state automaton with explicit alphabet encoding) showing the initial and final states and the resulting entangled register state.
  2. Clarify the minimal dimension of the auxiliary register needed to embed the isometry and state whether the construction works for arbitrary automaton sizes or only when the register dimension is at least the qudit dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas where the presentation of our quantum synchronizing automata construction can be strengthened. We address each major comment below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main construction: the central claim asserts the existence of a global unitary on the combined auxiliary-qubit register and qudit system that achieves the reset for any initial qudit state when the register is prepared in a synchronizing-word state. No explicit matrix form, circuit decomposition, or proof that such a unitary exists and satisfies the isometry condition ⟨e(ψ)|e(φ)⟩ = ⟨ψ|φ⟩ is supplied, rendering the protocol unverifiable.

    Authors: We agree that an explicit construction and verification of the unitary would improve verifiability. The global unitary is defined to act according to the classical transition rules encoded in the auxiliary qubit states for every word in the alphabet, with the synchronizing word inducing the reset map on the qudit. In the revision we will add a dedicated subsection providing (i) the explicit action of the unitary on the computational basis of the combined space, (ii) a proof that the induced map on the synchronizing subspace is an isometry satisfying ⟨e(ψ)|e(φ)⟩ = ⟨ψ|φ⟩ by direct computation from the classical collapse property, and (iii) a concrete matrix representation and circuit decomposition for a minimal two-state, two-symbol automaton as an illustrative example. revision: yes

  2. Referee: [Main text (protocol definition)] The manuscript defines quantum synchronizing words constructively via the auxiliary encoding but does not demonstrate that the required isometry on the synchronizing subspace can be extended to a unitary on the full space while respecting the automaton transition rules encoded in the qubit states. This is load-bearing for the claim that the protocol is unitary and faithful to the classical automaton.

    Authors: We acknowledge that the extension step requires explicit justification. The isometry is constructed by letting the unitary implement the deterministic classical transitions on all auxiliary configurations; because a synchronizing word maps every initial qudit state to the same target state, the restriction to the corresponding subspace is isometric. Any isometry between finite-dimensional spaces of equal dimension admits a unitary extension. We will show in the revision that such an extension can be chosen so that the action on the orthogonal complement is consistent with the same transition-rule encoding (for instance by completing the partial isometry block-wise while leaving the auxiliary-controlled transitions intact). This construction preserves faithfulness to the classical automaton by design. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructive protocol

full rationale

The paper introduces quantum synchronizing words via an explicit construction: auxiliary qubits encode the automaton alphabet, and a global unitary on the combined system is defined such that preparation in a synchronizing-word state maps any initial qudit state to a fixed pure target while the register evolves to an isometry-preserving entangled state. This unitary exists by standard extension of an isometry on the relevant subspace (register synchronizing state tensor qudit space) to the full space, with no fitted parameters, no self-citation load-bearing the existence claim, and no reduction of the target state or entanglement to prior outputs by definition. The abstract and skeptic analysis confirm the protocol is presented as a definitional realization rather than a looped derivation, rendering the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The claim rests on the existence of certain global unitary operators that realize the synchronizing words; these operators are postulated rather than derived from more basic principles in the abstract.

axioms (1)
  • standard math All quantum evolution is implemented by unitary operators
    The resetting protocol is realized by a single global unitary on the combined system.
invented entities (2)
  • Quantum synchronizing word no independent evidence
    purpose: Initial state of the auxiliary qubit register that forces the qudit to a fixed pure state
    Defined as the quantum counterpart to classical synchronizing words; no independent evidence supplied.
  • Auxiliary qubit register encoding automaton rules no independent evidence
    purpose: To carry the input sequence and become entangled while enabling unitary resetting
    Introduced to circumvent the apparent irreversibility of classical synchronization.

pith-pipeline@v0.9.0 · 5525 in / 1375 out tokens · 53792 ms · 2026-05-10T00:11:11.549744+00:00 · methodology

discussion (0)

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