**Research Summary**

**Continuous variables teleportation
**

In [31] I introduced a method for teleportation of quantum states of continuous variables. The method was implemented with squeezed light and became a basic theoretical and experimental tool in quantum information (close to one thousand citations in GS) complementing the teleportation protocol for discrete variables which was discovered one year before.

**Elitzur-Vaidman bomb tester
**

In [28] we introduced an * interaction-free measurement*: a method for finding an object without being near it. This bizarre effect (named once as one of the seven wonders of the quantum world) led to many experiments, theoretical analyses, improvements and applications. In [87] I clarified the subtle meaning of its "interaction-free" property.

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The impossibility of counterfactual communication
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The interaction-free measurement [28] led to a counterfactual key distribution and to a limited counterfactual computation. In [101], [132] and [139] I argued against counterfactuality of the unconstrained "counterfactual" computation and several "counterfactual" communication protocols which were proposed and implemented recently.

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The Aharonov-Albert-Vaidman Effect
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In [8] the concepts of *weak measurements* and *weak values* were introduced and their use for amplification of the precision of measurements was suggested. A few years ago it was implemented in several experiments with unprecedented precision. Currently, there is a very extensive discussion regarding the efficiency of the AAV effect. In [R17] I explained in which situations it definetely provides a huge advantage.

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Weak value
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The impact of [8] goes far beyond the precision measurement method. *Weak value* is a universal concept describing pre- and post-selected quantum systems which explains many of their surprising properties. Numerous analyses, applications and controversies regarding the meaning of the weak value continue to appear. (More than one thousand citations in GS). In [13], [18], [29], [51], [56], and [103] we continued the analysis and currently I arrived at a novel understanding of weak value: it is a property of the system and not a conditional average, as it usually presented [R18].

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The past of a photon
**

While Bohr preached that one should not talk about quantum particles between detections, Wheeler proposed that the trajectory of the photon in the past is known when there is only one allowed path connecting the source and the detector. In [120] I contested this claim showing that this view does not explain the weak trace which the photon leaves. I argue that a consistent way of describing the past of the photon is to say that it was in the overlap of the forward and backward evolving quantum states. This is where it leaves a weak trace. This approach created a large controversy because it suggests that one cannot talk about photon trajectories and because in this picture the photon can be in places through which it cannot pass. To support my approach, I, together with a team of experimentalists at Tel-Aviv university, performed a highly discussed experiment "Asking photons where have they been" [125] which continues to get much attention.

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The two-state vector formalism
**

This is the framework for a large part of my research. It is developed in [13], [18], [56], [100], [111] and reviewed in [103]. It led to the concept of weak measurements and weak values, to the 3-box paradox, to the conceptual design of a quantum time-translation machine [15], [19], to a novel approach to the past of particles inside interferometers [120], and much more.

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Defence of time-symmetric quantum mechanics
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A large part of my research relies on time-symmetric approach to quantum theory originated by Aharonov at 1964.
The approach attracted extensive criticism to which I tried to reply in several (published and unpublished) works [R10], [45], [64], [70], [R14].

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The mean king problem
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You give an electron to the king who measures spin x, spin y or spin z and returns it to you. The task is to find a method such that you can tell the king, without mistakes, what was the result of his measurement for these three possible choices, otherwise, you will be executed. Surprisingly, it is possible to meet the challenge of the king. In [4] we explained how to do this. We found that this task is possible after formulating the concept of a * generalized two-state vector *, which was published later in [18]. The mean king problem continues to generate novel quantum information protocols and extensive mathematical analyses.

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The 3-box paradox
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In [18] we introduced a quantum protocol in which a single particle can be found with certainty in box A and also in box B. Numerous works analyzed various aspects of this strange behavior based on different interpretations ranging from classical physics to the consistent histories interpretation of quantum mechanics. In [99] and [104] the paradox was transformed into the * Aharon-Vaidman game*, an example of a *local* classical task, in which quantum technology has an advantage in performing a classical task.

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The failure of the product rule
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If A=a and B=b with certainty, then AB=ab. This is a true statement for quantum observables. In [23] I realized that it is not true in general for pre- and post-selected quantum systems. This, in turn, explains various quantum paradoxes.

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More paradoxical features of pre-and post-selected quantum systems
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Quantum systems behave very differently than classical systems, and pre- and post-selected quantum systems behave in an even more bizarre way, as seen in some of the above examples. But there are more paradoxes. One shutter can close any number of slits for a single classical or quantum particle [88]. We can simulate Popescu-Rohrlich boxes exhibiting stronger than quantum correlations [98]. We can put three particles in two boxes such that no pair is present in one box [118]. If we look at "weak measurement reality" [56], things become even more surprising: the particle can act as if it is in one place, but its magnetic moment appears in another place [118].

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Modular value
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In [113] we introduced a *modular value*, a novel concept of a pre- and post-selected quantum system which fully characterizes its coupling to a projection operator variable. It is related to a weak value, but it can be measured in ordinary, strong, measurements.

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Time-symmetric quantum counterfactuals
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The common approach to counterfactuals is that they are asymmetric relative to the arrow of time.
In [R10], [70], [73], [74] and [R14] I introduced and defended the concept of time-symmetric quantum counterfactuals which is important for building a philosophical basis for the two-state vector formalism.

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Protective measurements
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In [27] we introduced a method which allows measurement of a (protected) quantum state of a single quantum system. This was surprising, because the common belief is that we need an ensemble for measuring a quantum state. In [52] and [133] I clarified the meaning of this proposal. Protective measurements continue to generate a conceptual debate.

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The physical origin of topological phases
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In [6] and [16] we showed how a naive argument explaining the Aharonov-Casher effect using classical lag fails due to a subtle concept of the "hidden momentum". The neutron feels the force but does not accelerate. However, recently I came to the conclusion that Aharonov-Casher and Aharonov-Bohm effects *can* be explained by the lag of the wave packets, when we take into account the source of the electromagnetic field which becomes entangled with the particle during the process [116].

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Quantum nonlocality
**

I firmly believe that there is no action at a distance and that everything can be explained in local terms and entanglement which is the only nonlocal ingredient of Nature. In [47], [75] ,[80] we analyzed the nonlocal features of the quantum formalism. In [116] I found a local explanation for the Aharonov-Bohm effect which opens the way to undo the half-century old revolution in understanding the role of potentials in physics. The general applicability of my result is, in my view, an important open problem [138].

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Measurements of nonlocal variables
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A and B are variables of two systems separated in space. The question is: which variables f(A,B) can, and which cannot be measured instantaneously in a nondemolition way? In [1] and [32] we showed that essentially, the only measurable variables are A+B and (A+B)modC. Thus, the product of local variables at different places is unmeasurable, while the Bell operator with maximally entangled states is measurable. In [86] I found that measurement of * all * nonlocal variables is possible, when we allow to destroy the eigenstates of the measured variable. This surprising result has a far reaching consequences in quantum information, including novel quantum cryptographic protocols.

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Pseudo-telepathy quantum games
**

In [71] I proposed a game with an advantage of a quantum team of the type later called pseudo-telepathy. In [84] I proposed another game of this type. In [79] I argued that pseudo-telepathy games are the way to rule out local realistic theories.

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Advantages of qubits versus bits
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Although we need infinitely more information to specify a qubit relative to a bit, Holevo has shown that using a qubit we cannot transfer more information than using a bit. Using the idea of dense coding, with the help of an ancilla, we can gain a factor of 2. However, we found some specific tasks [93] and [95] in which qubits are exponentially more powerful than bits.

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Error prevention method
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Shor's error correction method requires 9 physical qubits to encode one qubit of information. More complicated schemes reduce the number of physical qubits to 5. In [55] we applied the Zeno effect to simplify the method. We use only 4 physical qubits and we do not need the correction process: the act of frequent measurements prevents the errors.

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Cryptography with orthogonal states
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In [46] we introduced a quantum method for secure key distribution which, unlike previous methods, does not rely on the no-cloning theorem. This became the basis for conceptually new protocols. It was apparently the first direct communication protocol, as well as the first relativistic quantum protocol.

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Quantum Gambling
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Alice and Bob are equipped with a quantum channel. They want to gamble, but they do not trust each other. In [72] we showed how they can do this. It opened a new direction for quantum protocols. In particular, it was apparently the first quantum protocol for coin tossing.

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Practically secure quantum bit commitment
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After the celebrated discovery of the quantum secure key distribution method, it was claimed that one can make an unconditionally secure bit commitment. Entanglement allowed to break the security of this protoclol. In [114] we introduced a protocol whose security is based on current experimental limitations on nondemolition measurements and on the storage of quantum information. Each one of the limitations is enough for the security of our protocol. The protocol is very simple: any high-fidelity one-way quantum key distribution device can be used to implement it.

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Experimental work
**

The bit commitment protocol [114] is the first work in which I took part in performing a real experiment. (The fidelity was hardly satisfactory and it should be repeated with a better equipment.) In [14] we were the first to suggest a concrete design of the weak measurement procedure. Recently, I joined a Chinese experimental team for analyzing their state of the art weak measurement experiment [121]. But my first significant experimental work is the demonstration of the past of the photon [125], where I invented a new techniques: tagging the location of photons by adding disturbances with various frequencies, followed by a spectral analysis of the signal. The method proved to be very robust: we could see the effect which otherwise was screened by noise.

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Development of the Many-Worlds Interpretation of Quantum Mechanics
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I am arguably the strongest proponent of the many-worlds interpretation. In my view, it is by far the best approach to foundations of quantum theory because it removes randomness and action at a distance from physics. In [36] I provided the motivation, and in [66] , [83] , [112] , [129], and [131] I developed my version of the MWI. I am still working on it, in the hope of making it the leading interpretation of quantum mechanics.

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Measure of existence of a world
**

In [66] I presented my main contribution to the MWI. I introduced the measure of existence of a world and the Born-Vaidman rule which specifies the probability of self location in a particular world. I refined it in [117] and [131]. This solution of the probability problem of the MWI has been used in [126] for resolving the * Sleeping Beauty problem*, a hotly debated philosophical probability puzzle.

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Analysis of the Ghirardi-Riminy-Weber-Pearle collapse proposal
**

In [12] we constructed a Stern-Gerlach experiment in which the GRWP model does not lead to a single outcome even when a macroscopic number of atoms evolve into a superposition of macroscopically different states, showing a weakness of their approach. On the other hand, in [131] I resolved the "tails problem" of the GRWP collapse model, by pointing out that "tail worlds" of GRWP are not only "tiny" in some sense, but also do not resemble the world we know.

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Conditions for "surrealistic" trajectories in Bohmian mechanics
**

In [58], [94], and [115] we analyzed Bohmian trajectories of in various situations. We found that position detectors, characterized by slow robust recordings of the results, record trajectories which are very different from actual Bohmian trajectories.

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Duality relation between path distinguishability and fringe visibility D ^{2} +V^{2}≤1
**

In [39] we derived a tight duality relation for particles in an interferometer. It provides a useful bound for practical applications. The relation also led to several conceptual claims regarding the particle-wave duality (which I do not necessarily support).

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Aharonov-Vaidman Formula A Ψ = <A> Ψ + ΔAΨ _{⊥}
**

In [13] we introduced this formula to simplify the calculation of the result of a weak measurement. In [54] and [20] it was applied for the simplest (to my knowledge) derivations of uncertainty relations and bounds on the speed of evolution of a quantum state.