Two commuting channels yield the same composite channel in either fixed order, yet their quantum switch can alter information preservation. We study how this effect depends on the input state, retaining the joint output of the control and target. Exact Kraus commutator identities determine overlap and squared fidelity gaps. Fixed order preserves product inputs at least as well as the switch in fidelity. The reverse inequality holds for pure states of two qubits whose control Schmidt basis can be chosen to coincide with the order basis. A commutator matrix determines the target basis that maximizes the squared fidelity gap at fixed entanglement within this aligned family. For Pauli $X$ and $Y$ channels with equal error probabilities, the switch replaces a logical phase flip with an operator that stabilizes a joint code. At fixed noise, the switch preserves at least as much distinguishability and quantum Fisher information as fixed order for every state family in this code. For the pure encoded family studied here, exact formulas connect the phase information gain to the squared fidelity gap and quantify the reduction in information about the noise probability. Optimizing the probes yields equal maximal quantum Fisher information for noise estimation under both joint channels. These results connect commutator geometry and logical error structure to state preservation and the retention of encoded information.
Two commuting channels yield the same composite channel in either fixed order, yet their quantum switch can alter information preservation. We study how this effect depends on the input state, retaining the joint output of the control and target. Exact Kraus commutator identities determine overlap and squared fidelity gaps. Fixed order preserves product inputs at least as well as the switch in fidelity. The reverse inequality holds for pure states of two qubits whose control Schmidt basis can be chosen to coincide with the order basis. A commutator matrix determines the target basis that maximizes the squared fidelity gap at fixed entanglement within this aligned family. For Pauli X and Y channels with equal error probabilities, the switch replaces a logical phase flip with an operator that stabilizes a joint code. At fixed noise, the switch preserves at least as much distinguishability and quantum Fisher information as fixed order for every state family in this code. For the pure encoded family studied here, exact formulas connect the phase information gain to the squared fidelity gap and quantify the reduction in information about the noise probability. Optimizing the probes yields equal maximal quantum Fisher information for noise estimation under both joint channels. These results connect commutator geometry and logical error structure to state preservation and the retention of encoded information.
For the quantum switch of two commuting channels, we extend the commutator identities of Chen and Ma (arXiv:2609.21310). For arbitrary inputs, the difference between the switch and fixed-order outputs is a single off-diagonal operator built from Kraus commutators; it gives the overlap gap, trace distance and purity change exactly, and contains Eq. (11), Theorem 2 and Proposition 4 of that paper as special cases. For a pure control, Theorem 1 of that paper becomes quantitative, and for inputs of rank at most two the root-fidelity gap is exact. We also give bounds for mixed controls, a criterion for mixed aligned states, and a qubit counterexample. A seeded Python reproduction package is included. This is a working note and has not been peer reviewed.
Abstract The quantum switch is known to remove noise probabilistically, but whether this constitutes an advantage over causally ordered processing has remained contentious: postselection, weak classical benchmarks, and task-dependent comparisons have all drawn criticism. Here we give an exact analysis of the question for a qubit under Pauli noise. For any two qubit Pauli channels, measuring the switch’s control qubit implements a commutation syndrome : it sorts each noise realization by whether the two Kraus errors commuted, syndrome information that an unencoded qubit does not otherwise possess, and the optimal measure-and-feed-forward protocol has a closed form. The analytical results have two levels of generality. For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis, we prove a complete ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations and receiver-side post-processing, deterministic or heralded, has entanglement fidelity at most ma x P c P , and the switch exceeds this ceiling deterministically , by p q / 3 over most of parameter space. For arbitrary qubit Pauli pairs, the ceiling is proven against arbitrary receiver-side post-processing, and the gain is strictly positive if and only if the two syndrome branches call for different Pauli corrections, confining the exceptions to error-dominated noise. Side systems are covered whenever they are non-signaling about the message. Beyond these theorems the evidence is numerical: strict gain in 99.8 % of 5000 random Pauli pairs, and no case, within a 200 -pair sample, in which an optimized midpoint echo overturns the gain. The state-resolved gain obeys an exact geometric law, Δ F = 2 p q / 3 | r ^ ⋅ n ^ | 2 : alignment is the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation survives control dephasing down to a threshold visibility that falls with noise strength, and at strong noise it outperforms coherent control of the two orderings even with twice the channel uses.
Fisher information appears in quantum theory in two mathematically distinct settings: as a classical probability functional in variational reconstructions of Schrodinger dynamics and as the quantum Fisher metric of parameterized density operators. Here we examine how far these roles can be connected without identifying the two objects. Starting from a Hamilton-Jacobi ensemble action, we recover the Fisher term that generates the quantum potential and isolate its contribution to the kinetic-energy expectation. We then analyze the quantum Fisher information matrix (QFIM) of bipartite states. For any pure two-qubit state, the optimized magnitude of the off-diagonal QFIM element for normalized local generators equals the concurrence, while a collective phase generator yields FQ = 4 C^2. Extending the optimized pure-state quantity by a convex roof gives an exact identity with Wootters concurrence for arbitrary mixed two-qubit states. This differs from evaluating the QFIM directly on a mixed density operator: for Werner states the raw optimized cross-QFIM is 2 p^2/(1 + p) and remains nonzero inside the separable region. The gap between the raw and convex-roof quantities therefore isolates, for this family, correlation geometry not captured by entanglement. Finally, in a two-path interferometer with a which-way marker, the optimal phase Fisher information available from local output statistics is I_phi^opt = V^2. These results place probability-gradient energy, bipartite correlation geometry, entanglement, and decoherence-induced loss of local phase sensitivity in a common mathematical setting while preserving the distinctions between classical Fisher information, QFIM geometry, Bell nonlocality, and global phase quantization.
Abstract The teleportation of Quantum Fisher Information (QFI) is central to the precision limits of distributed quantum metrology, yet it is inevitably degraded by environmental decoherence. Here we investigate QFI teleportation under composite Pauli channels and show that a quantum switch can improve the outcome-resolved and success-probability-weighted QFI relative to the corresponding fixed-order protocol. The enhancement arises from a coherent reorganization of Pauli error trajectories rather than from a reduction of the physical error probabilities. For the three binary pairs of anti-commuting Pauli channels studied here, we identify a heralded branch whose normalized state has the ideal QFI whenever that branch has nonzero probability. These findings highlight the potential of utilizing quantum switches to mitigate Pauli-noise-induced QFI degradation in quantum communication and sensing networks.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
Abstract Superdense coding is a fundamental protocol in quantum information theory that facilitates the transmission of classical information at capacities exceeding classical limits by exploiting quantum entanglement. However, the presence of noise in communication channels, particularly in the case of Pauli channels, poses a significant challenge to the efficiency of superdense coding. In this paper, we explore an active protocol designed for scenarios involving two distinct, addressable noisy channels. We investigate how a quantum switch, by actively composing these two channels in an indefinite causal order, can construct a new composite channel with significantly enhanced capacity. Our findings indicate that, when confronted with the same type of Pauli channels which are commutative, the quantum switch offers no advantage over classical strategies constrained by definite causal order. Remarkably, when two different channels are composed, the quantum switch can exploit the non-commutativity of their underlying operators to coherently cancel their noise effects. This vividly illustrates the ‘different is more’ principle: using the quantum switch to actively compose two different non-commuting channels yields a significant capacity gain over composing two identical commutative channels. This underscores the potential of the quantum switch as a tool for active noise-aware protocol design, where the difference itself is a resource, rather than as a tool for passive noise mitigation.