Resource theory of coherence in continuous position basis from measurement-induced dephasing

Abstract: We develop a resource-theoretic framework for quantum coherence directly in continuous basis, with emphasis on the position representation. Since position eigenstates are non-normalizable generalized eigenstates, the standard finite-dimensional dephasing map cannot be transferred directly to normal states. We therefore introduce a physically motivated dephasing channel based on random momentum kicks, equivalently described as the unconditional back-action of a finite-resolution position measurement. This yields a fixed-point notion of incoherence and a natural class of dephasing-covariant free operations. For physically relevant kernels, however, the fixed-point set contains no normal states, showing that continuous-basis coherence is tied to dephasing disturbance rather than to distance from a nonempty set of diagonal states. We study two quantifiers built from the channel action: a relative-entropy dephasing loss and a Hilbert-Schmidt dephasing loss. The former satisfies the main resource-theoretic properties under the free operations considered, while the latter is convex and experimentally transparent but fails monotonicity and strong monotonicity. We also formulate threshold witnesses for certifying coherence above a finite value and connect them, in a two-path setting, with interference visibility. Finally, we illustrate the framework with a Gaussian wavepacket evolving in a gravitational potential. The resulting theory provides a mathematically consistent and physically motivated treatment of coherence in continuous-variable systems.

• The paper introduces a resource-theoretic framework for continuous-variable coherence using physically motivated, random momentum kick dephasing channels.
• It formulates new coherence measures, including relative-entropy and Hilbert–Schmidt dephasing losses, which address challenges from non-normalizable position eigenstates.
• The framework links theoretical coherence quantifiers with experimental observables in interferometry, enabling practical certification through threshold witnesses.

Resource Theory of Coherence in Continuous Position Basis via Measurement-Induced Dephasing

Introduction and Motivation

This work develops a rigorous resource-theoretic framework for quantum coherence directly in the continuous position basis, centering on the operational consequences of measurement-induced dephasing. In contrast to the well-understood discrete setting, where the notions of coherence, incoherent states, and free operations are mathematically straightforward, the continuous-variable domain exhibits a range of nontrivial obstacles due to the non-normalizability of position eigenstates, the absence of a bona fide dephasing projector, and the ill-definedness of diagonal states. The authors construct a consistent framework based on completely positive trace-preserving (CPTP) channels characterized by random momentum kicks—operationally equivalent to finite-resolution position measurements—thereby connecting physically realizable decoherence processes with the quantification of spatial coherence.

Limitations of Discrete-Basis Resource Theories in the Continuum

Traditional resource theories of coherence, as formalized in finite dimensions, are based on three key ingredients: the set of diagonal (incoherent) states, the class of free operations (MIO, IO, SIO, DIO), and the coherence monotones satisfying strict axioms. The extension to infinite-dimensional, continuous-variable systems poses serious challenges:

• Non-normalizability of position "eigenstates": $\{|x\rangle\}$ are generalized functions and do not reside in the Hilbert space.
• Ill-defined basis dephasing: Naive generalization of projective dephasing $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$ leads to divergences ($\delta(0)$ terms) and does not yield a valid quantum channel on trace-class states.
• Absence of normal incoherent states in the fixed-point set: For physically relevant dephasing kernels, the only fixed point in the space of normal states is the zero operator.
• Breakdown of idempotency: Physically motivated position dephasing channels produced by random momentum kicks are not projectors; repeated application further suppresses off-diagonal components rather than leaving the state invariant after one action.

These properties fundamentally distinguish the continuous-variable coherence resource theory from its discrete analogs.

Measurement-Induced Dephasing: Random Momentum Kick Channel

The work proposes a physically motivated dephasing map, $\Delta_g$, arising from the unconditional back-action of a finite-resolution position measurement. Operationally, such measurements are modeled by the von Neumann interaction Hamiltonian $H = \lambda \hat{X}_S \otimes \hat{P}_M$, with the system and pointer prepared in normalizable wavepackets. The unconditional output (after averaging over readout) has the form:

$$\Delta_g(\rho) = \int dp\, \tilde{g}(p) \, e^{ip\hat{X}/\hbar} \rho\, e^{-ip\hat{X}/\hbar},$$

where $\tilde{g}(p)$ is the probability distribution describing random momentum kicks imparted to the system (determined by the meter's momentum spread), and its characteristic function $g(\xi) = \int dp\, e^{ip\xi/\hbar} \tilde{g}(p)$ encodes the kernel. In position representation:

$[\Delta_g(\rho)](x, y) = g(x-y)\, \rho(x, y)\,,$

where $|g(\xi)| < 1$ for $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$0 for smooth pointer (meter) wavefunctions.

Figure 1: Time dependence of the lower-bound expression for the relative-entropy dephasing loss, shown relative to its initial value for a Gaussian wavepacket evolving in a gravitational potential.

The dephasing channel thus leaves diagonal elements invariant and continuously suppresses spatial coherence at scales set by the resolution (spread of $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$1). Key consequences include:

• The set of fixed points, $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$2, contains no nonzero trace-class operators for physically admissible (regular) kernels.
• Resource-theoretic coherence, therefore, is not a notion of proximity to diagonal states, but rather quantifies the disturbance induced by physically realizable dephasing.

Coherence Quantifiers: Relative-Entropy and Hilbert–Schmidt Dephasing Loss

Relative-Entropy Dephasing Loss

Defined by

$$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$3

where $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$4 is the quantum relative entropy, provided support conditions are met. This measure satisfies:

• Faithfulness: strictly positive for all physical states due to the empty fixed-point set.
• Monotonicity and strong monotonicity under the class of dephasing-covariant CPTP maps ($$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$5).
• Convexity and additivity on product systems.
• For pure states,

$$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$6

but, crucially, does not reduce to the entropy of the dephased state since $$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$7 is not idempotent.

Hilbert–Schmidt ($$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$8) Dephasing Loss

Defined as

$$\rho \mapsto \int dx\,|x\rangle\langle x|\rho|x\rangle\langle x|$$9

• Convex and experimentally transparent (computable from the kernel), but fails monotonicity and strong monotonicity under $\delta(0)$0.
• For two-mode superpositions (e.g., two-path interference), this measure quantifies the coherence decay due to the channel at a given separation.

Threshold Witnesses and Interference Visibility

Because the fixed-point set is empty, standard witness operators lose operational meaning. The paper introduces threshold witnesses for finite coherence—bounded Hermitian operators $\delta(0)$1 separating states with $\delta(0)$2 from those with $\delta(0)$3. Using the two-mode sector and the Hilbert–Schmidt quantifier, the relevant threshold for interface visibility $\delta(0)$4 is

$\delta(0)$5

so sufficient visibility at separation $\delta(0)$6 certifies more than $\delta(0)$7 amount of $\delta(0)$8-coherence (where $\delta(0)$9 is path separation and $\Delta_g$0, the coherence kernel at that separation). This directly relates resource-theoretic coherence to operational observables in interference experiments.

Illustration: Gaussian Wavepacket in a Gravitational Potential

An explicit example is provided: a minimum-uncertainty Gaussian wavepacket of width $\Delta_g$1 and mass $\Delta_g$2 evolves under a Newtonian potential, subject to a Gaussian random-kick dephasing channel with spatial coherence scale $\Delta_g$3 set by the meter momentum spread. The resulting relative-entropy dephasing loss and Hilbert–Schmidt loss are computed as functions of the time-evolved wavepacket width $\Delta_g$4:

$\Delta_g$5

$\Delta_g$6

As the wavepacket spreads, the amount of position-basis $\Delta_g$7-coherence grows, quantifying the capability of continuous-variable systems to sustain coherence under physically motivated environmental monitoring. Figure 1 displays the temporal growth of relative-entropy dephasing loss for experimentally relevant parameters, directly demonstrating the dynamical interplay of coherence, spatial delocalization, and measurement-induced decoherence.

Theoretical and Practical Implications

• This framework solidifies the operational definition of coherence resources in continuous variables, justifying the focus on disturbance-based measures in the absence of nontrivial fixed points.
• The resource-theoretic structure remains valid in the continuum only when grounded in physically realizable channels.
• Relative-entropy dephasing loss emerges as the canonical quantifier, both for its strong monotonicity and faithfulness, and for its operational interpretability in interferometric settings.
• Threshold witnesses enable practical certification in experiments, especially in matter-wave interferometry, double-slit setups, and situations where spatial resolution of environmental monitoring can be controlled or inferred.

The mathematical architecture (e.g., absence of idempotent, trace-preserving dephasing projectors; the non-existence of normal incoherent states) will constrain generalizations to fields such as relativistic or non-Markovian open-system dynamics.

Conclusion

The study provides a robust and physically motivated resource theory of position-basis coherence for continuous-variable quantum systems, centered on CPTP dephasing channels realized by random-momentum (measurement-induced) disturbances. It systematically demonstrates the failure of finite-dimensional resource-theory analogs, constructs suitable coherence monotones, and links theoretical and operational aspects through experimental witnesses and explicit dynamical examples. This framework paves the way for further foundational, experimental, and theoretical analyses of quantum coherence in the continuum.