We investigate state estimation in discrete-time quantum walks with a single absorbing boundary. Using a spectral approach, we obtain closed expressions for the escape probability as a function of the coin state and the boundary position, and their corresponding classical Fisher information for a simple absorption readout. Comparing with the single-copy quantum Fisher information shows a clear complementarity: near boundaries carry broad information about the population angle of the coin, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields a full-rank Fisher matrix and tight joint Cramér--Rao bounds, while retaining a binary, tomography-free measurement. We outline an integrated-photonics implementation in which an on-chip sink realizes the absorber and estimate a substantial reduction in configuration count compared to mode-resolved qubit tomography. These results identify absorption in quantum walks as a simple and scalable primitive for coin-state metrology.
We investigate state estimation in discrete-time quantum walks with a single absorbing boundary. Using a spectral approach, we obtain closed expressions for the escape probability as a function of the coin state and the boundary position, and their corresponding classical Fisher information for a simple absorption readout. Comparing with the single-copy quantum Fisher information shows a clear complementarity: near boundaries carry broad information about the population angle of the coin, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields a full-rank Fisher matrix and tight joint Cramér--Rao bounds, while retaining a binary, tomography-free measurement. We outline an integrated-photonics implementation in which an on-chip sink realizes the absorber and estimate a substantial reduction in configuration count compared to mode-resolved qubit tomography. These results identify absorption in quantum walks as a simple and scalable primitive for coin-state metrology.
In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\ensuremath{\theta}$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space ${H}_{w}(\ensuremath{\theta})$ increases with $\ensuremath{\theta}$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare ${H}_{w}(\ensuremath{\theta})$ to the full QFI ${H}_{f}(\ensuremath{\theta})$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\ensuremath{\theta}$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.
We address multiparameter quantum estimation for one-dimensional discrete-time quantum walks and its applications to quantum metrology. We use the quantum walker as a probe for the unknown parameters encoded in its coin degrees of freedom. We find an analytic expression of the quantum Fisher information matrix for the most general coin operator, and show that only two out of the three coin parameters can be accessed. We also prove that the resulting two-parameter coin model is asymptotically classical, i.e., the Uhlmann curvature vanishes. Finally, we apply our findings to relevant case studies, including the simultaneous estimation of charge and mass in the discretized Dirac model.
We address metrological problems where the parameter of interest is encoded in the internal degree of freedom of a discrete-time quantum walker, and provide evidence that coin dimensionality is a potential resource to enhance precision. In particular, we consider estimation problems where the coin parameter governs rotations around a given axis and show that the corresponding quantum Fisher information (QFI) may increase with the dimension of the coin. We determine the optimal initial state of the walker to maximize the QFI and discuss whether, and to what extent, precision enhancement may be achieved by measuring only the position of the walker. Finally, we consider Grover-like encoding of the parameter and compare results with those obtained from rotation encoding.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length η. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For N copies, inverse-Fisher A loss has the exact value 2/[Nf(η)], where $
f(η)=
\frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely A- and D-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have A-loss coefficient 4/η2, whereas collective measurements attain 2(1+η)/η2. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.
We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a "pattern counting'' estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramer-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of \cite{GiGoGu} this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of \cite{DayouCounting}.