Input-Output Indefiniteness

• Input-output indefiniteness is a phenomenon where the causal direction between inputs and outputs is not fixed, appearing in both nonlinear systems and quantum channels.
• Mathematical frameworks like the Chen–Fliess series and shuffle algebra decomposition enable the factorization of input-output maps to address ambiguities and multiple zeroing inputs.
• Experimental approaches using photonic setups have validated quantum input-output indefiniteness, paving the way for protocols that exploit this resource in quantum information processing.

Input-output indefiniteness refers to phenomena—classically or quantum mechanically—in which the causal orientation between inputs and outputs of a device, system, or process cannot be consistently assigned a fixed direction. This concept has emerged independently in nonlinear systems theory, where systems can exhibit multiple distinct inputs that yield identical outputs (reflecting a loss of invertibility and “relative degree”), as well as in quantum information theory, where coherent superpositions of forward and backward signal propagation through a device result in physically observable scenarios without a definite input-output direction. Recent advances include mathematical frameworks for factorizing input-output maps in both settings and the experimental demonstration of quantum input-output indefiniteness, positioning it as a resource for information processing.

1. Mathematical Formulation in Systems Theory

In the analysis of nonlinear input-output systems, particularly those admitting a Chen–Fliess series representation, indefiniteness of input-output arises when the system’s generating series lacks a well-defined relative degree. For a given causal, analytic input-output map $F$, the output $y(t) = F[u](t)$ can be formally expanded as

$$y(t) = F_c[u](t) = \sum_{\eta \in X^*} (c, \eta)\, E_\eta[u](t)$$

where $c$ is the formal generating series over $X = \{ x_0, x_1 \}$, and $E_\eta$ denotes noncommutative iterated integrals of $u$.

The relative degree of $c$ is well-defined if its nontrivial part $c_F$ is supported on words beginning with $x_0^{r-1}$ and the coefficient of $x_0^{r-1} x_1$ is nonzero. If this fails, the system admits more than one formal input $u_*(t)$ driving the tracking error $e(t) = y(t) - y_\mathrm{ref}(t)$ to zero, i.e., multiple distinct $c_{u_*}$ such that $c \circ c_{u_*} = 0$. This reflects algebraic noninvertibility in the input-output map, and the failure of classic zero-dynamics techniques predicated on a unique solution (Gray et al., 2023).

2. Algebraic Decomposition and Shuffle Algebras

The resolution of input-output indefiniteness in analytic systems leverages the unique factorization properties of the shuffle algebra on generating polynomials. The shuffle product $\sha$ endows the space $\mathbb{R}\langle X \rangle$ with a commutative algebra structure, isomorphic to a symmetric algebra over the vector space of Lyndon words. Any polynomial $c$ can be uniquely (up to permutation) expressed as $c = c_1 \sha \cdots \sha c_n$, with each $c_j$ irreducible.

This decomposition offers a subsystem perspective: the original system is a parallel interconnection of systems corresponding to the shuffle factors. Input-output indefiniteness is realized when more than one shuffle-irreducible factor is primely-nullable—i.e., each admits a unique nulling input, but multiple such factors collectively yield multiple nulling solutions for the composite system. The algorithmic procedure for factorization is based on the Chen–Fox–Lyndon word decomposition, mapping the problem to multivariate polynomial factorization and back (Gray et al., 2023).

3. Quantum Theory: Bidirectionality and Coherent Superpositions

In quantum theory, bidirectional devices are those for which input and output ports may be physically exchanged, i.e., the device implements a bistochastic quantum channel that is both trace-preserving and unital. When such a device is employed in a coherent quantum superposition of forward and backward orientations, the resulting process cannot be assigned a definite causal direction.

Mathematically, this scenario is described using higher-order quantum operations (supermaps) acting on Choi operators. A supermap $S$ is input-output indefinite if it cannot be decomposed into a classical mixture of definite-direction maps,

$$S \neq p S_\mathrm{fwd} + (1-p) S_\mathrm{bwd}, \qquad p \in [0,1]$$

where $S_\mathrm{fwd}$ and $S_\mathrm{bwd}$ use the bidirectional device in fixed forward or backward mode, respectively. Theoretical and experimental approaches allow construction of witnesses $W$ such that $\mathrm{Tr}[WS] < 0$ if and only if $S$ is input-output indefinite, and such witnesses are measurable using informationally-complete local measurements on optical platforms (Guo et al., 2022).

4. Hierarchies of Higher-Order Quantum Maps

The hierarchy of input-output indefiniteness in quantum processes is captured by stratifying higher-order maps in terms of both local input-output orientation and global causal order. Level-1 (supermaps) correspond to quantum time-flip operations, e.g., $\mathcal{F}$, which use a bistochastic channel in a coherent superposition of directions, producing outputs indistinguishable from any fixed-direction composition.

At higher levels, "bi-slot $n$-combs" and "bistochastic process matrices" encode scenarios where multiple bistochastic channels are composed in sequences with indefinite local orientation and/or indefinite global causal order. The characterization is given by the structure of their admissible Choi operators lying in cones and subspaces corresponding to such indefiniteness, with canonical examples such as the quantum time flip, flippable SWITCH, and Liu–Chiribella processes that surpass conventional process-matrix signaling bounds (Apadula et al., 31 Jan 2026).

5. Experimental Realization and Certification

Experimental evidence for quantum input-output indefiniteness centers on photonic platforms employing single-photon sources, polarization-encoded qubits, and optical interferometry to construct coherent superpositions of forward and backward device traversal. Certification relies on measuring witness observables $W^{\mathrm{opt}}$ with high statistical significance (violations exceeding $69\,\sigma$ reported), precluding explanations based on mixtures of definite-direction devices. Both optimal and simplified witnesses (using significantly reduced measurement settings) have yielded robust statistical rejection of definite-direction hypotheses (Guo et al., 2022).

Such photonic setups realize the full protocol: preparation of control and target qubits, coherent superposition of device traversals, recombination, and projective/POVM measurement of the control. The technology allows studies of both foundational phenomena (simulating indefinite time-arrows) and practical exploitation in quantum network links (Liu et al., 2022).

6. Operational Advantages and Resource Theory

Input-output indefiniteness constitutes a distinct resource within a formal resource-theoretic framework. The free operations comprise convex mixtures of definite-direction supermaps and composition with local bistochastic channels. A key monotone is the robustness $r(S)$—the minimal noise required to render a process definite. Applications include:

• Gate transposition/inversion: perfect realization of $U \mapsto U^T$ impossible in definite-direction circuits but achievable via indefinite direction protocols.
• Parameter estimation: coherent superposition of a phase gate with its transpose doubles generator range, improving precision.
• Discrimination games: unit-success discrimination of certain unitary pairs enabled by indefinite input-output direction, outperforming all definite-direction strategies (Guo et al., 2022, Liu et al., 2022).

In communication, the use of a noisy channel in an indefinite input-output direction enables reduction or complete elimination of noise under specific conditions (e.g., dephasing or depolarizing channels), surpassing the zero capacity of conventional fixed-direction use and realizing heralded noiseless quantum state transfer (Liu et al., 2022).

7. Broader Implications and Foundational Context

From a foundational perspective, input-output indefiniteness in quantum theory is intimately connected to considerations of time-reversal symmetry and generalized quantum operations unconstrained by a strict time arrow. The maximal hierarchy of higher-order maps constructed from bistochastic channels delineates the class of processes compatible with a time-symmetric, maximally mixed state-preparation paradigm. This perspective motivates new investigations into temporally indefinite processes, operational resource theories beyond standard circuit models, and the construction of noise-robust protocols in quantum information science (Apadula et al., 31 Jan 2026, Liu et al., 2022).

In nonlinear systems theory, the algebraic resolution of input-output indefiniteness—via shuffle algebras and Lyndon word factorization—illuminates the structure of multiple-input solutions and defines canonical decompositions into independently invertible subsystems (Gray et al., 2023).

The convergence of these lines of inquiry underlines the significance of input-output indefiniteness as both a conceptual and technical primitive in the analysis and manipulation of complex physical, mathematical, and informational systems.