The measurement apparatus acts like an interface between the quantum and classical worlds. Surprisingly, even though the quantum behavior of matter or light has been investigated in increasingly nonclassical states, the quantum properties of measurements seemed less studied heretofore.

In my PhD Thesis [1], I explore the quantum behavior of measurement apparatus with illustrations in quantum optics. This is the first study of quantum properties of measurements performed by any kind of devices. I show that the quantum properties of a measurement, such as its projective or non-classical character, are revealed only by the quantum states of a less usual statistical approach of quantum physics: the retrodictive approach.

This approach involves retro-predictions about state preparations leading to a given measurement result, contrary to the predictive approach with which we usually make statistical predictions about the results of an experiment.

Figure 1. -- Quantum Predictions and Retrodictions. In quantum physics, any protocol is based on preparations, evolutions and measurements. The preparation of the system in a given state (represented by a red ball) is associated to a piece of information that we call the choice "m". The measurement provides another piece of information which is simply the result "n". In such a game, we can only make predictions about preparation choices and measurement results. Thus, two statistical approaches emerge from this fundamental game: the predictive approach, with which we make the usual statistical predictions about the measurement results "n", and the retrodictive approach allowing the less usual retro-predictions about the preparation choices "m". Each approach needs a quantum state (red and blue balls) and an exhaustive set of propositions (corresponding templates), as explained below.



By clarifying the mathematical foundations of the retrodictive approach (Fig. 2), I propose [1 , 2] a general procedure for reconstructing the quantum states of this approach: the retrodicted state.

Figure 2. -- This table illustrates and summarizes the complementarity between the preditive and retrodictive approaches. Thus, we can realize the reconstruction of the quantum state retrodicted from the result "n", by using the statistics of the preparation choices "m" leading to the result "n", contrary to usual methods in which we use the statistics of measurement results "n" from a given preparation choice "m".



I have realized [1, 4] these reconstructions for single-photon detectors (Fig. 3), widely used in quantum cryptography for instance. This is the first tomography of quantum states totally based on the retrodictive approach and preparation choices, contrary to usual reconstructions based on measurement results.

Figure 3. -- Results of the first quantum tomography of retrodicted states: Maximum-likelihood (MaxLike) estimation of the quantum state retrodicted from the "click" of an usual single-photon detector (time-multiplexed APD). Each bar represents the population corresponding to a given photon-number n, which is simply the probabilty that the light pulse contain exactly n photons. The reconstructed state is in very good agreement with the populations provided by the modeling of the detector (in blue), as highlighted by the remarkably high value (99.7%) of the fidelity (or overlap) between the two states.



This tomography enabled to study experimentally the noise influence on the quantum properties of measurements performed by these detectors, in particular their transition from a strongly quantum behavior into a more classical behavior (Fig. 4).

Figure 4. -- Experimental investigation of the quantum-to-classical transition of a single-photon detector. (a) Wigner representation of the quantum state retrodicted from the response "click" of a single-photon detector. Its minimum value is reached at the origin of the phase-space, which could be interpreted as the probability of having a strictly zero light field. However, this value may be negative and becomes the signature of the non-classical behavior of the light in such a state. Indeed, according to probability theory, an event which happens with probability zero happens almost never. A "negative probability" would describe an event that never happens. (b) Evolution of the negativity under the noise influence (horizontal axis : dark noise, vertical axis : efficiency). The quantum-to-classical transition occurs when the noise exceeds a certain threshold (black line). The experimental results are in a good agreement with the theory realized during my PhD [1,2,4].



Then, I propose [1, 5] a detector of "Schrödinger's Cat" states of light, which are superpositions of incompatible quasi-classical states of light. In a modern version of a thought experiment proposed by Eugene Wigner in 1961 (Fig. 5), such a device could allow the "Wigner's Friend" to detect a "Schrödinger's Cat", contrary to human eyes for which I specify some quantum properties (Fig. 6).

Figure 5. -- My version of the Thought Experiment proposed by Eugene Wigner in 1961. A laser diode allows the Wigner's Friend to monitor the state of the "Schrödinger's Cat". The behavior of this famous cat is described by a quantum superposition of the two classically incompatible states: "dead" and "alive". The state of the light produced by the monitoring diode gets entangled to the cat's state, and the global system (cat, diode) is equally in a superposition of two incompatible realities: "dead/light off" and "alive/light on". We can then investigate the effects of an observation performed by the Wigner's Friend. For this purpose, I described the human eye as a true optical detector, by using the mathematical tools introduced in my thesis and data from neurophysiology experiments. Thus, the human eyes are not "quantum" enough to observe "Schrödinger's Cat" states. Their retrodicted states are not characterized by "negative probabilities", contrary to single-photon detectors, as explained above. If we want to observe a "Schrödinger's Cat", we should make the observation with another kind of optical detector that is sufficiently sensitive to "Schrödinger's Cat" states of light (Fig. 7).


Figure 6. -- Quantum Properties of Human Eyes. The human eye can be seen as a true optical detector. Indeed, some neurophysiology experiments show that the human eyes are only sensitive to light pulses which contain at least a threshold number S of photons. For normal human beings, this threshold S usually ranges from 2 to 7 photons. Moreover, all the light pulses fulfilling this condition are not necessarily detected, since the human eyes have a detection efficiency around 10%. We can then characterize the quantum properties of measurements performed by the human eyes, as explained in my thesis [1]. Let us summarize here the main results of my study. (a) The Wigner representation of the quantum state retrodicted from the light sensation. (b) Evolution of this Wigner function with the threshold number S. We clearly see that this representation is always positive, and therefore, it can be used as a true probability distribution within the optical phase-space (x,p). Thus, we retrieve the classical behaviour of human eyes. If the light intensity is below a certain threshold, where the Wigner function is roughly zero, we do not have any light sensation. If the light intensity is above this threshold, the Wigner function reaches an uniform and positive value (red plateau on Fig. 6-a), corresponding to the light sensation. (c) The projective and non-classical characters of the measurement performed by human eyes are respectively characterized by the purity of its retrodicted state (projectivity) and the negativity of its Wigner representation. Thus, this measurement is neither projective nor non-classical. Therefore, the human eyes are not "quantum" enough for detecting "Schrödinger's Cat" states of light involved in the thought experiment described on Fig.5.




Finally, I generalize the use of the "Schrödinger's cat" states detector to an estimation protocol, totally based on the retrodictive approach and preparation choices. Such a procedure could enable optimal estimations, by reaching the quantum Cramér-Rao bound, which is a very topical issue of quantum metrology (Slides 1).


Figure 7. -- Wigner representations of "Schrödinger's Cat" states of light before and after a phase-space displacement. The interference pattern is the signature of the quantum superposition between the two quasi-classical states, which correspond to the red peaks in the Wigner representation. These quasi-classical states are classically incompatible, like the states "dead" and "alive" for a cat. The phase-space structure of this strongly non-classical state enables it to be extremely sensitive to any perturbation, even very small (slides 1). The detector that I have proposed is characterized by retrodicted states which are faithful to these "Schrödinger's Cat" states of light. Thus, my detector could enable optimal measurements by reaching the quantum Cramér-Rao bound, which is the ultimate precision that one can reach, whatever the method used.



Slides 1. -- Detector of "Schrödinger's Cat" States of Light for Quantum Metrology.

2.1 Mo






My CV is more detailed on LinkedIn. Feel free to connect with me through this network for professional purposes only.


You can find below the theses achieved during my studies in the Ecoles Normales Supérieures.




2008 - 2011 : Ph.D. in Quantum Physics, Laboratoire Kastler Brossel of the Ecole Normale Supérieure.

PhD Thesis [12.1 Mo]


2007-2008 : M.Sc. in Quantum and Statistical Physics, Ecole Normale Supérieure, Paris.

MSc Thesis [4.2 Mo]


2004-2008 : B.Sc. in Physics, Ecole Normale Supérieure, Cachan.

BSc Thesis [3.2 Mo]



PhD Thesis - New Chapter

In France, the New Chapter of the Thesis has for ambition to highlight the soft skills gained or sharped during this first professional experience. It also emphasizes the innovative nature of the Thesis, for which the EADS Corporate Foundation has awarded me the Best Thesis Prize.


The EADS Foundation's Best Thesis Prize rewards "significant progress by exploring options that are likely to lead to technological and conceptual breakthroughs". The jury of the Prize was chaired by Pr. Claudine Hermann, Former Professor at the Ecole Polytechnique.



Furthermore, the EADS Foundation has produced a short movie (in French) in order to popularize the subject of my PhD Thesis. 




Finally, in order to highlight some of my soft skills, you can find below some quotes (translated in English) from the Jury of my PhD, chaired by Pr. Serge Haroche, Professor at the Collège de France.


"The retrodictive approach has been little studied heretofore, and this thesis should fill this gap by demonstrating the relevance of this less usual approach [...] "

"The thesis is very well structured and written in a neat and precise style [...] We appreciate the original and clear presentation that demonstrates a very strong command of this tough subject."

"This thesis describes a highly innovative and comprehensive study of an unusual approach of quantum physics, which reveals promising perspectives [..] This is certainly a step forward."

- From an Examiner's Report.



"The author continually seeks to illustrate his approach, in a pedagogical way, with several concrete examples [...] The results are really impressive and demonstrate the high potential of this approach."

"The depth and the originality of this thesis are really outstanding. I am sure that it will be a landmark work [...]"

"I was really impressed by his ability to explain subtle points with a very fine analysis and to put his work in perspective [...] This is one of the best thesis I have ever had to examine."

- From an Examiner's Report.



"Taoufik Amri has defended his thesis about the retrodictive approach of quantum physics in a clear and concise way."

"During his defense, the jury asked him several questions about the meaning of his approach [...] He answered to all the questions in a thoughtful and thorough way, with a high scientific maturity and a strong command of a difficult subject, which deals with the most fundamental - and the most hotly debated - aspects of quantum physics."

"He demonstrated during his defense that he has all the qualities required to be a great explainer."

- Fom the Final Report of the Jury.





[1] T. Amri, Comportement Quantique des Appareils de Mesure : Illustrations en Optique Quantique. PhD Thesis. May 2011.

[2] T. Amri, Quantum behavior of measurement apparatus. January 2010.

[3] T. Amri et al. Characterizing quantum properties of a measurement apparatus: Insights from the retrodictive approach. Physical Review Letters 106, 020502. January 2011.

[4] T. Amri et al. Quantum Tomography of Retrodicted States from Single-Photon Detectors, in preparation.


[5] T. Amri et al. Detector of "Schrödinger's Cat" States of Light for Quantum Metrology, in preparation.




June 2011 : Detecting "Schrödinger's Cat" States of Light for Quantum Metrology

ICQI 2011 - International Conference on Quantum Information, Ottawa - Canada.


May 2011 : Quantum Behavior of Measurement Apparatus : Illustrations in Quantum Optics

Public Defense of my PhD Thesis.


October 2010 : "Dieu joue-t-Il aux dès avec l'Univers? Un survol des Lois du Hasard"

Popularization Lecture given in the French Cultural Institutes.


May 2010 : Characterizing Quantum Properties of Measurement Apparatus

Quantum 2010 - International Workshop on Advances in Foundations of Quantum Mechanics and Quantum Information with atoms and photons . Talk in plenary session. Turin, Italy.


May 2010 : Etats "Chats de Schrödinger" de la lumière avec des détecteurs supraconducteurs de photons uniques*

Interdisciplinary PhD Symposium of the Sorbonne University.

*Best Oral Communication Prize (1st).


October 2009 : "A l'aube d'une seconde révolution quantique : Introduction à l'Information Quantique