Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.
Symmetry, including the parity-time ($\mathcal{PT}$)-symmetry, is a striking topic, widely discussed and employed in many fields. It is well-known that quantum measurement can destroy or disturb quantum systems. However, can and how does quantum measurement destroy the symmetry of the measured system? To answer the pertinent question, we establish the correlation between the quantum measurement and Floquet $\mathcal{PT}$-symmetry and investigate for the first time how the measurement frequency and measurement strength affect the $\mathcal{PT}$-symmetry of the measured system using the $^{40}\m…
Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time …
We analyze quantum state estimation for finite samples based on symmetry information. The used measurement concept compares an unknown qubit to a reference state. We describe explicitly an adaptive strategy, that enhances the estimation fidelity of these measurements.
We describe a method to detect twin-beam multiphoton entanglement based on a beam splitter and weak nonlinearities. For the twin-beam four-photon entanglement, we explore a symmetry detector. It works not only for collecting two-pair entangled states directly from the spontaneous parametric down-conversion process, but also for purifying them by cascading these symmetry detectors. Surprisingly, by calculating the iterative coefficient and the success probability we show that with a few iterations the desired two-pair can be obtained from a class of four-photon entangled states. We then general…
By imposing system-observer symmetry on the von Neumann description of measurement, it is shown that the quantum measurement problem is structurally equivalent to a familiar reverse-engineering problem: that of describing the behavior of an arbitrary physical device as algorithm instantiation. It is suggested that this problem can at best be given a relational solution.
A definition of entanglement in terms of local measurements is discussed. Viz, the maximum entanglement corresponds to the states that cause the highest level of quantum fluctuations in all local measurements determined by the dynamic symmetry group of the system. A number of examples illustrating this definition is considered.
I propose a scheme for reconstructing the weak value of an observable without the need for weak measurements. The post-selection in weak measurements is replaced by an initial projector measurement. The observable can be measured using any form of interaction, including projective measurements. The reconstruction is effected by measuring the change in the expectation value of the observable due to the projector measurement. The weak value may take nonclassical values if the projector measurement disturbs the expectation value of the observable.
We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcom…
Some recent attempts at measuring non local weak values via local measurements are discussed and shown to be less robust than standard weak measurements. A method for measuring some non local weak values via non local measurements (non local weak measurements) is introduced. The meaning of non local weak values is discussed.
We consider a new approach to describe a quantum optical Bose-system with internal Gell-Mann symmetry by the SU(3)-symmetry polarization map in Hilbert space. The operational measurement in density (or coherency) matrix elements for the three mode optical field is discussed for the first time. We have introduced a set of operators that describes the quantum measurement procedure and the behavior of fluctuations for the amplitude and phase characteristics of three level system. The original twelve-port interferometer for parallel measurements of the Gell-Mann parameters is proposed. The quantum…
The role of measurement induced disturbance in weak measurements is of central importance for the interpretation of the weak value. Uncontrolled disturbance can interfere with the postselection process and make the weak value dependent on the details of the measurement process. Here we develop the concept of a generalized weak measurement for classical and quantum mechanics. The two cases appear remarkably similar, but we point out some important differences. A priori it is not clear what the correct notion of disturbance should be in the context of weak measurements. We consider three differe…
Methods for measuring the weak value of non local variables are investigated. We analyze local (indirect) measurement methods for obtaining the weak values. We also describe some new (direct) methods (Non local weak measurements) for measuring the weak values of some non local variables.
Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements' incompatibility, via backaction …
Non-statistical weak measurements yield weak values that are outside the range of eigenvalues and are not rare, suggesting that weak values are a property of every pre-and-post-selected ensemble. They also extend the applicability and valid regime of weak values.
A protocol of quantum dense coding protection of two qubits is proposed in amplitude damping (AD) channel using weak measurement and reversal measurement. It is found that the capacity of quantum dense coding under the weak measurement and reversal measurement is always greater than that without weak measurement and reversal measurement. When the protocol is applied, for the AD channels with different damping coefficient, the result reflects that quantum entanglement can be protected and quantum dense coding becomes successful.
Weak measurements with mixed probe states were investigated in the presence of noise in the probe system. We show that the completely mixed state can be used as a probe state for weak measurements of imaginary weak values. The completely mixed probe state has some advantages in terms of noise tolerance. We also experimentally demonstrated weak measurements of polarization rotation via unpolarized light, which corresponds to weak measurements with completely mixed probe states in a two-state system.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the pre-selection and the post-selection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and post-selections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and post-selections are exactl…
The article recapitulates the concept of weak measurement in its broader sense encapsulating the trade between asymptotically weak measurement precision and asymptotically large measurement statistics. Essential applications in time-continuous measurement and in postselected measurement are presented both in classical and in quantum contexts. We discuss the anomalous quantum weak value in postselected measurement. We concentrate on the general mathematical and physical aspects of weak measurements and we do not expand on their interpretation. Particular applications, even most familiar ones, a…
It is well known that any projective measurement can be decomposed into a sequence of weak measurements, which cause only small changes to the state. Similar constructions for generalized measurements, however, have relied on the use of an ancilla system. We show that any generalized measurement can be decomposed into a sequence of weak measurements without the use of an ancilla, and give an explicit construction for these weak measurements. The measurement procedure has the structure of a random walk along a curve in state space, with the measurement ending when one of the end points is reach…
The weak value of a variable O is a description of an effective interaction with that variable in the limit of weak coupling. It is particularly important for a pre- and post-selected quantum system.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in t…
Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements…
Weak values and measurements have been proposed as means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur with highly suppressed probabilities. This raises three vital questions in weak-measurement-based metrology, namely, (Q1) Does post-selection enhance the measurement precision? (Q2) Does weak measurement offer better precision than strong measurement? (Q3) Is it possible to beat the standard quantum limit or to achieve the Heisenberg limit with weak measure…
The measurement process in quantum mechanics is usually described by the von Neumann projection postulate, which forms a basic constituent of the laws of quantum mechanics. Since this postulate requires the outside observer of the system, it is hard to apply quantum mechanics to the whole Universe. Therefore we propose that the quantum measurement process is actually a physical process associated with the ubiquitous mechanism of spontaneous symmetry breaking. Based on this proposal, we construct a quantum measurement model in which the von Neumann projection is described as the dynamical pro-c…
Pancharatnam's geometric phase is associated with the phase of a complex-valued weak value arising in a certain type of weak measurement in pre- and post-selected quantum ensembles. This makes it possible to test the nontransitive nature of the relative phase in quantum mechanics, in the weak measurement scenario.
Resource-Theoretic Quantifiers of Weak and Strong Symmetry Breaking: Strong Entanglement Asymmetry and Beyond
Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Department of Physics, Kyushu University（九州大学物理系）；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理研跨学科理论与数学科学中心(iTHEMS)）；Institut des Hautes Études Scientifiques (IHES)（高等科学研究所(IHES)）；Department of Informatics, Faculty of Information Science and Electrical Engineering, Kyushu University（九州大学信息科学与电气工程学院信息学系）
Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.
The two-dimensional cluster state, a universal resource for measurement-based quantum computation, is also the gapped ground state of a short-ranged Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian. We prove that, provided the perturbation is sufficiently small and respects a certain symmetry, the perturbed ground state remains a universal resource. We do this by characterising the operation of an adaptive measurement protocol throughout a suitable symmetry-protected quantum phase, relying on generic properties of the phase rather than any analytic control over the…
Topological phase transition induced by spatially-tuned single-site measurement is investigated in a measurement-only circuit, in which three different types of projective measurement operator. Specific spatial setting and combination of commutation relations among three measurement operators generate such a transition. In practice, symmetry protected topological (SPT) phase is recovered on even sublattice by eliminating a projective measurement disturbing the SPT via applying another spatially-tuned projective measurement on odd sublattice. We further investigate the critical properties of th…
The weak measurements based amplification of ultra-small phase was proposed in our previous work. Due to the technical imperfections, the ability of amplification is usually limited in practice. Here we introduce the concept of cascaded weak measurements amplification so that the ability of weak measurements amplification can be further improved. A formula of determining the cascaded number of weak measurements amplification is derived. The method presented here could be found useful in high precision measurement e.g., gravitational waves detection.
Measurements in classical and quantum physics are described in fundamentally different ways. Nevertheless, one can formally define similar measurement procedures with respect to the disturbance they cause. Obviously, strong measurements, both classical and quantum, are invasive -- they disturb the measured system. We show that it is possible to define general weak measurements, which are noninvasive: the disturbance becomes negligible as the measurement strength goes to zero. Classical intuition suggests that noninvasive measurements should be time symmetric (if the system dynamics is reversib…
We study various optimality criteria for quantum observables. Observables are represented as covariant positive operator valued measures and we consider the case when the symmetry group is compact. Phase observables are examined as an example.
For an invertible (bounded) linear operator Q acting in a Hilbert space ${\cal H}$, we consider the consequences of the QT-symmetry of a non-Hermitian Hamiltonian $H:{\cal H}\to{\cal H}$ where T is the time-reversal operator. If H is symmetric in the sense that ${\cal T} H^\dagger {\cal T}=H$, then QT-symmetry is equivalent to Q^{-1}-weak-pseudo-Hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non-PT-symmetric Hamiltonians that share the spectral properties of PT-symmetric Hamiltonians.
Why does spontaneous symmetry breaking occur? Why is a state breaking symmetry realized? We explore an idea that measurement selects such a state even if a system is given in a state respecting the symmetry of the system. We point out that the spectrum of the relevant observable is important, and simply apply the projection postulate for quantum measurement. We first show that this approach correctly describes the well-known interference of Bose-Einstein condensates. We then examine a fermionic system and prove that superconducting states with a definite relative phase are selected by the meas…
