Paris-Singapore-Tokyo Workshop

**Dates and Times:**

The workshop will take place via zoom on the 20th and 21st of May, between 09:30 - 12:30 (CEST) | 15:30 - 18:30 (SGT) | 16:30 - 19:30 (JST)

**Itinerary:**

Each talk, as well as the coffee breaks, is scheduled for 30 minutes (including questions).

*May 20th*

- Yingkai Ouyang -
*Tight Bounds on the Simultaneous Estimation of Incompatible Parameters* - Yuichiro Matsuzaki -
*Anonymous quantum sensing* - Coffee Break
- Jesús Rubio -
*Quantum sensing networks: a multi-parameter approach to the estimation of linear functions* - Ranjith Nair -
*Information-theoretic quantum limits in optical metrology* - Keith Ng -
*Sensing the shape of spacetime: detector response and entanglement harvesting in curved space*

- Nathan Shettell -
*A Cryptographic approach to Quantum Metrology* - Mankei Tsang -
*Quantum semiparametric estimation* - Coffee Break
- Matteo Lostaglio -
*Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response* - Giacomo Sorelli -
*Moment-based superresolution* - Xiao Yunlong -
*Uncertainty Principle for Quantum Networks*

This workshop was organized by Nathan Shettell (*nathan.shettell 'at' lip6.fr*) and Yingkai Ouyang.

**Title:** Tight Bounds on the Simultaneous Estimation of Incompatible Parameters

**Speaker:** Yingkai Ouyang, NUS (Singapore)

**Abstract:**

The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables greatly diminishes the applicability of estimation theory in many practical implementations. The Holevo Cramér-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multiparameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimization over multiple variables. In this work, we develop an analytic approach to solving the HCRB for two parameters. Our analysis reveals the role of the HCRB and its interplay with alternative bounds in estimation theory. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations that even numerically permits a quadratic speedup over previous state-of-the-art approaches. We apply our results to compare the performance of different probe states in magnetic field sensing and characterize the performance of state tomography on the code space of noisy bosonic error-correcting codes. The sensitivity of state tomography on noisy binomial code states can be improved by tuning two coding parameters that relate to the number of correctable phase and amplitude damping errors. Our work provides fundamental insights and makes significant progress toward the estimation of multiple incompatible observables.

**Title:** Anonymous quantum sensing

**Speaker:** Yuichiro Matsuzaki, AIST (Tokyo, Japan)

**Abstract:**

Magnetic field sensing is an important application in biology and material science. Here, we propose an anonymous quantum sensing with security inbuilt. More specifically, in our scheme, an information of positions having non-zero magnetic fields is hidden after measuring magnetic fields with a quantum-sensing network. Importantly, even if the readout results of the quantum sensors are stolen by an eavesdropper, information of the positions with non-zero magnetic fields is still protected. Our results pave the way for new applications of quantum-sensing network.

**Title:** Quantum sensing networks: a multi-parameter approach to the estimation of linear functions

**Speaker:** Jesús Rubio, University of Exeter (Exeter, England)

**Abstract:**

Quantum sensing networks promise to revolutionise the acquisition of information that is spatially distributed. Since many physical properties can often be modelled linearly, the construction of efficient sensing networks needs a reliable approach to the simultaneous estimation of multiple linear functions-the latter are taken over a set of local parameters (the nodes) and can thus be seen as global properties (of the network). In this talk we will examine the role of inter-sensor correlations by linking them to the geometry of the vectors associated with the functions. Considering a specific qubit network prepared in a sensor-symmetric configuration, we will see that, if these vectors are clustered around two special subspaces, then the asymptotic optimum is achieved when the correlation strength approaches its extreme values. For any other geometry, a monotonic transition between such extremes can be identified. The potential impact of a finite amount of information in the performance of the network will further be discussed by means of a Bayesian analysis. We will conclude with a brief survey of recent progress, including a perspective on the role of Bayesian techniques in the theoretical development of networked quantum sensing.

**Title:** Information-theoretic quantum limits in optical metrology

**Speaker:** Ranjith Nair, NTU (Singapore)

**Abstract:**

We derive fundamental bounds on the optimal performance of optical metrological and communication systems from information-theoretic arguments. The derivation uses ideas from rate-distortion theory in conjunction with bounds on the classical capacity of various optical channels specified by the system design. The bounds are expressed in terms of the system parameters, the prior probability distribution of the parameter, and the average energy, i.e. number of photons E in the probe state. In the absence of loss, we consider phase estimation using multimode probes, estimation of an arbitrary parameter using a single mode, and arbitrary single-mode communication systems. We also consider estimating a phase parameter under loss using a signal-ancilla mode pair. Our bounds show Heisenberg-limit scaling with E in lossless phase estimation and standard-quantum-limit scaling in the presence of any finite amount of loss.

**Title:** Sensing the shape of spacetime: detector response and entanglement harvesting in curved space

**Speaker:** Keith Ng, NTU (Singapore)

**Abstract:**

The Unruh-Dewitt (UDW) detector is a simplified model of a particle detector, extensively used in the nascent field of Relativistic Quantum Information to study the interaction of detectors, quantum fields, and gravitational curvature. By interacting with a background quantum field, these UDW detectors can extract information about their surroundings, beyond what a naive application of the equivalence principle might predict. We thus wish to explore the capabilities of the UDW detector in characterizing the shape of spacetime. I will present a number of examples of these capabilities: First, a detector can determine whether it has been placed within a massive shell, despite no change in local gravitational field; second, a detector can probe the existence of spacetime features behind the event horizon of a black hole, e.g. in an RP3 geon; and finally, I will present work done to characterize the entanglement structure of a model spacetime, AdS4.

**Title:** A Cryptographic approach to Quantum Metrology

**Speaker:** Nathan Shettell, LIP6 (Paris, France)

**Abstract:**

Quantum metrology is widely accepted as one of the most advanced pillars of quantum information, where quantum effects lead to enhanced precision measurements of unknown quantities. On the other hand, quantum cryptography uses quantum systems to detect and avoid the effects of malicious behaviour. In the future, where quantum devices are more accessible, and quantum tasks are delegated to third parties, it is natural to want to combine these fields, to ensure that any quantum metrology process occurs in a secure manner. In this talk, I will discuss the framework of quantum metrology enhanced with quantum cryptography. In particular, I will show that the performance from a metrology perspective can be directly linked to the soundness of a cryptographic protocol. Additionally, I discuss the cryptographic protocols developed for quantum metrology in the presence of a malicious adversary.

**Title:** Quantum semiparametric estimation

**Speaker:** Mankei Tsang, NTU (Singapore)

**Abstract:**

I propose a formalism that can give simple analytic expressions for the quantum Cramer-Rao bound for a large class of high-dimensional problems [Tsang, Albarelli, Datta, Phys. Rev. X 10, 031023 (2020)]. The formalism is especially suited to semiparametric estimation problems, also called shadow tomography by the quantum information community. The theory originates from classical statistics [Bickel et al., Efficient and Adaptive Estimation for Semiparametric Models, Springer, New York (1993)] and relies on an underappreciated geometric approach to the quantum Cramer-Rao bound.

**Title:** Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response

**Speaker:** Matteo Lostaglio, University of Amsterdam (Amsterdam, Netherlands)

**Abstract:**

I will prove a fundamental difference between classical and quantum dynamics in the linear response regime, by showing that the latter is in general contextual. Given the ubiquity of linear response theory, I anticipate that these tools will allow one to certify the nonclassicality of a wide array of quantum phenomena. For example, one can devise quantum engines whose favorable power output scaling unavoidably require nonclassical effects in the form of contextuality. Here, I will use the result to show that phase estimation is fundamentally contextual.

**Title:** Moment-based superresolution

**Speaker:** Giacomo Sorelli, LKB (Paris, France)

**Abstract:**

Sensitivity limits are usually determined using the Cramér-Rao bound. Recently this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits, usually require the full measurement statistics, which can be challenging to access. Here, we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this technique may be constructed for arbitrary thermal sources, and we study their sensitivities when one has access to spatially resolved intensity measurements (direct imaging) and photon counting after spatial mode demultiplexing. For demultiplexing, our method is optimal, i.e. it saturates the quantum Cramér-Rao bound. We also investigate the impact of noise on the optimal observables, their measurement sensitivity and on the scaling with the number of detected photons of the smallest resolvable separation. For low signals in the image plane, we demonstrate that our method saturates the Cramér-Rao bound even in the presence of noise.

**Title:** Uncertainty Principle for Quantum Networks

**Speaker:** Xiao Yunlong, NTU (Singapore)

**Abstract:**

As a central concept of quantum mechanics, Heisenberg’s uncertainty principle barricades what we can learn from a quantum system under incompatible measurements, clearly separating quantum theory apart from the classical world. Unlike the uncertainty principle of quantum states, we have very little knowledge about the uncertainty associated with dynamical quantum processes, which might possess temporal quantum memory described as quantum or classical feedback controls over time. The lack of such a fundamental limit for dynamical processes has hampered our understanding of the capacity of nature. In this work, we fill the gap by establishing the uncertainty principle in the presence of temporal quantum memory, which is expressed by quantum combs. In particular, this framework allows us to bound the uncertainties associated with non-Markovian quantum processes, since these processes can be characterized as a special case of comb called process tensor. Using this insight, we also provide bounds for temporal uncertainty principle in terms of entropies and majorization. As by-products, we introduce the one- shot dynamical entropies for combs, which generalize the well-known max-relative entropy, and justify their operational significance in optimal universal uncertainty relations.

This workshop was made possible through the support of the ANR through Grant No. ANR-17-CE24-0035 VanQuTe.