Quantifiers of Noise Reducibility Under Restricted Control

The gist The paper introduces quantifiers of noise reducibility for multitime quantum processes that satisfy monotonicity properties, showing they represent the largest amount of temporal mutual information that a process can possibly exhibit.

Framework

The framework centers on process tensors, which provide a complete characterisation of a quantum system’s observable statistics given any way that it can be accessed at a discrete number of times. A multitime process is described by the Choi state representation of the process tensor, which is obtained by letting it act on half of a maximally entangled state at each of the considered times. If the dynamics do not lead to correlations, then Markovian processes are represented by a product form where all intermediate maps between times are of the form Ttn. Non-Markovianity corresponds to the presence of inter-step correlations within process Tnˆ that cannot be written in the form of Eq. (2).

Control Combs

Experimental control is represented by a process tensor/quantum comb, which can transform process tensors with intermediate times nˆ to ones with mˆ ⊆ nˆ intermediate times. The action of a control comb Znˆmˆ on Tnˆ is denoted by T'm=JTn Zn hat K = JTn Zn hat nhatInhat mhat K. The IQI scenario, which is the focus, only allows for pre- and post-processing of each slot by means of independent quantum operations, potentially followed by coarse-graining. This restriction implies that the allowed superprocesses Znˆnˆ at the finest level of temporal access nˆ are themselves Markovian processes.

Resource Quantifiers

The paper introduces three temporal correlation quantifiers Im, Mm, and Nm that measure total, Markovian, and non-Markovian correlations of a process for sets of times mˆ ⊆ nˆ. These are defined as I(Tnˆ):=S (Tnˆ∥T marg nhat), M(Tnˆ):= S T mkv nhat∥T marg nhat, and N(Tnˆ):=S Tn h∥T mkv nhat. The crucial finding is that these functions are not monotonic under the constraints of IQI, as there exist superprocesses Znˆmˆ in IQI such that I(JTnˆZn hat mhat) ≥ I(Tnˆ). To ensure monotonicity, the paper defines resource quantifiers Im, Mm, Nm as suprema over allowed superprocesses Zn hat nhat in IQI under temporal coarse-graining.

Subadditivity of Resource Quantifiers

The introduced quantifiers satisfy a subadditivity relation: Imˆ (Tnˆ) ≤ Mmˆ (Tnˆ) + Nmˆ (Tnˆ). This is a manifestation of the fact that the distinct types of correlations cannot be simultaneously optimised in general, as optimally harnessing Markovian correlations typically requires sacrificing some non-Markovian correlations and vice versa. The trade-off is quantified by Theorem 2, which establishes that optimisation of the total information Imˆ will generally not also optimise the non-Markovianity Nmˆ and the Markovianity Mm.

Correspondence with Generalised Comb Divergences

The resource quantifiers are related to generalised comb divergences De, which is a mathematically well-behaved (pseudo-) distance measure between quantum combs that decreases monotonically under all superprocesses Zn hat mhat. The reachable comb divergence DreachTnˆ Rnˆ is shown to be upper bounded by the generalised comb divergence D Tnˆ Rn h. Specifically, the hierarchy I∅ = M∅ ≤ DreachTnˆ T marg nhat ≤ Dm (TnˆT marg nhat) is established.

Dynamical Decoupling as Temporal Resource Distillation

The resource quantifiers are directly related to the amenability of a quantum process to dynamical decoupling, which is interpreted as temporal resource distillation. The multitimescale optimal dynamical decoupling (MODD) scheme is shown to be an approximate optimisation method for quantifier I∅. Furthermore, the analysis shows that traditional DD superprocess Z DD nhat nhat does not extract all resources present in Tnˆ, and the potential for improvement is lower bounded by I(TnˆZ MODD nhat nhat) − I(TnˆZ DD nhat nhat).

Composition Structure

The paper investigates how resource quantifiers behave under sequential and parallel composition. For sequential composition, Proposition 1 shows that Imˆ is additive in IQI under sequential composition, Im (Sn h◦Tn h) = Im (Sn h) + Im (Tn). For parallel composition, Proposition 2 demonstrates superadditivity under parallel composition, where resources are superadditive: Imˆ is superadditive in IQI under parallel composition. The paper also shows that the resource quantifiers can be related to reachable comb divergences Dreach through Lemma 1.

Discussion

The results provide a methodology to construct operationally motivated resource quantifiers Im, Mm, and Nm tailored to the respective sets Zn hat mhat of available superprocesses. The monotonic behaviour of Im, Mm, and Nm under control operations is limited only to superprocesses in Znˆmˆ. This suggests that non-Markovianity is ‘expended’ during DD when optimising one resource type necessarily sacrifices the others. The paper concludes that these quantifiers provide a way to quantify the resources of complex quantum processes and devices given practical control available to an experimenter.

Acknowledgements

The authors thank Guilherme Zambon for bringing the error in Ref. [1] to their attention, and for fruitful discussions on this topic. SM acknowledges funding from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101068332. The paper cites numerous foundational works in quantum information, including those by Shor, Knill, and various others throughout the text.

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1. AI systems can utilize resource quantifiers like Imˆ, Mmˆ, and Nmˆ to determine the largest amount of temporal mutual information that a process can possibly exhibit, which allows for quantifying the amenability of a process to denoising under those constraints.

2. AI systems can perform noise reduction by finding the optimal superprocess Z∗nˆnˆ ∈ Z such that the transformed process T′∅:= JTnˆZ∗nˆnˆIn̂m̂K satisfies I∅(Tn̂) = I∅(T′∅) ≥ I∅(T∅), which quantifies the amenability of a process to denoising under the constraints of IQI.

3. AI systems can leverage sequential composition to achieve additive resource quantification, as Imˆ is additive in IQI under sequential composition and Imˆ (Snˆ′ ◦ Tn̂) = Imˆ′ (Snˆ′) + Imˆ (Tn̂).

4. AI systems can exploit parallel composition for superadditive resource extraction, as Imˆ is superadditive in IQI under parallel composition, meaning that the whole can be greater than the sum of its parts when multiple simultaneous processes are available.

5. AI systems can implement algorithms like the multitimescale optimal dynamical decoupling (MODD) scheme to find optimal decoupling sequences, which yields a numerical approximation for finding I∅, suggesting that non-Markovian processes tend to possess greater resource content than one would assume from the results of traditional DD alone.

6. AI systems can use reachable comb divergences Dreach to bound the resource quantifiers, as DreachTnˆ Rnˆ decreases monotonically under all superprocesses in Znˆmˆ, providing a measure of how much more noise reduction is possible compared to the generalised comb divergence De.