We consider the simultaneous and continuous measurement of qubit observables $σ_z$ and $σ_z\cosφ+ σ_x\sinφ$, focusing on the temporal correlations of the two output signals. Using quantum Bayesian theory, we derive analytical expressions for the correlators, which we find to be in very good agreement with experimentally measured output signals. We further discuss how the correlators can be applied to parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data.
We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two non-commuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with…
We consider multi-time correlators for output signals from linear detectors, continuously measuring several qubit observables at the same time. Using the quantum Bayesian formalism, we show that for unital (symmetric) evolution in the absence of phase backaction, an $N$-time correlator can be expressed as a product of two-time correlators when $N$ is even. For odd $N$, there is a similar factorization, which also includes a single-time average. Theoretical predictions agree well with experimental results for two detectors, which simultaneously measure non-commuting qubit observables.
Considering the simultaneous measurement of non-commuting observables, we define a geometric measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. The rationality of our geometric measure is demonstrated and the application of it is presented. The connection between our measure and Heisenberg's uncertainty principle is discussed as well. Our work deepens the understanding of quantum non-commuting measurement.
Simultaneous measurement of several noncommuting observables is modeled by using semigroups of completely positive maps on an algebra with a non-trivial center. The resulting piecewise-deterministic dynamics leads to chaos and to nonlinear iterated function systems (quantum fractals) on complex projective spaces.
We analyze the operation of a four-qubit Bacon-Shor code with simultaneous continuous measurement of non-commuting gauge operators. The error syndrome in this case is monitored via time-averaged cross-correlators of the output signals. We find the logical error rate for several models of decoherence, and also find the termination rate for this quantum error detecting code. The code operation is comparable to that based on projective measurements when the collapse timescale due to continuous measurements is an order of magnitude less than the time period between the projective measurements. An …
We analyze the physical meaning of a possible experiment aimed at the simultaneous measurement of two non-commuting spin components. We demonstrate that switching of a strong spin-orbit interaction, e.g., in a solid state or a cold-atom system, for a short period of time simulates a simultaneous von Neumann measurement of spin components, $σ_x$ and $σ_y$. By mapping spin dynamics onto coordinate-space motion, such a measurement determines time averages of $σ_x$ and $σ_y$ which, unlike their instantaneous values, may be evaluated simultaneously to an arbitrary accuracy.
It is shown that the full unknown state of a spin-1/2 system, S, which, within Born's statistical interpretation, is meant as the state of an ensamble of identically prepared systems, can be determined with a simultaneous measurement with the help of an "assistant" system A whose initial state is known. The idea is to let S and A interact with each other in a known way during a proper interaction time, to measure simultaneously two observables, one of S and one of A and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated expe…
In quantum mechanics, measurements cause wavefunction collapse that yields precise outcomes, for non-commuting observables such as position and momentum Heisenberg's uncertainty principle limits the intrinsic precision of a state. Although theoretical work has demonstrated the possibility to perform simultaneous non-commuting measurements and has revealed the limits on measurement outcomes, only recently has the dynamics of the quantum state been discussed. To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables to a …
We consider the effect of phase backaction on the correlator $\langle I(t)\, I(t+τ)\rangle$ for the output signal $I(t)$ from continuous measurement of a qubit. We demonstrate that the interplay between informational and phase backactions in the presence of Rabi oscillations can lead to the correlator becoming larger than 1, even though $|\langle I\rangle|\leq 1$. The correlators can be calculated using the generalized "collapse recipe" which we validate using the quantum Bayesian formalism. The recipe can be further generalized to the case of multi-time correlators and arbitrary number of det…
The fact that not all quantum observables are jointly measurable is one of the major differences between quantum and classical theory. In the former, non-commuting observables can only be simultaneously measured with limited precision. We report on an experimental implementation of such a simultaneous measurement of two non-commuting observables based on the framework of weak von Neumann measurements. We use a photonic setup where the polarisation degree of freedom acts as the system and the two components of the transversal position correspond to the pointers of the measurement apparatuses. I…
The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin-$\frac{1}{2}$ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be non-ideal, as (i) the joint statistics do not c…
In contrast to a projective quantum measurement in which the system is projected onto an eigenstate of the measured operator, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A full description of such measurement should describe the measurement protocol and provide an explicit form of the measurement operator that transform the quantum state to its post measurement form. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as p…
Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators
University of Exeter（埃克塞特大学）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Combinatorics & Optimization, University of Waterloo（滑铁卢大学组合优化系）
Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitio…
Einstein, Podolski and Rosen (EPR) have shown that any wave function (subject to the Schrödinger equation) can describe the physical reality completely, and any two observables associated to two non-commuting operators can have simultaneous reality. In contrast, quantum theory claims that the wave function can capture the physical reality completely, and the physical quantities associated to two non-commuting operators cannot have simultaneous reality. The above contradiction is known as the EPR paradox. Here, we unambiguously expose that there is a hidden assumption made by EPR, which gives r…
Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as $N^4$ for typical problems of interest--a daunting growth rate that poses a serious li…
In the conventional formulation, it is broadly accepted that simultaneous measurability and commutativity of observables are equivalent. However, several objections have been claimed that there are cases in which even nowhere commuting observables can be measured simultaneously. Here, we outline a new theory of simultaneous measurements based on a state-dependent formulation, in which nowhere commuting observables are shown to have simultaneous measurements in some states, so that the known objections to the conventional theory are theoretically justified. We also discuss new results on the re…
In constructing his theorem, Bell assumed that correlation functions among non-commuting variables are the same as those among commuting variables. However, in quantum mechanics, multiple data values exist simultaneously for commuting operations while for non-commuting operations data are conditional on prior outcomes, or may be predicted as alternative outcomes of the non-commuting operations. Given these qualitative differences, there is no reason why correlation functions among non-commuting variables should be the same as those among commuting variables, as assumed by Bell. When data for c…
The problem of inferring the outcome of a simultaneous measurement of two non-commuting observables is addressed. We show that for certain pairs with dense spectra, precise inferences of the measurement outcomes are possible in pre-and post-selected ensembles, and if the selections involve entangled states with some other system. We show that the problem is related to the problem of assigning weak values to a continuous family of operators, and give explicit examples where this problem is solvable. Some foundational implications are briefly discussed.
Heisenberg uncertainty relation in quantum mechanics sets the limit on the measurement precision of non-commuting observables, which prevents us from measuring them accurately at the same time. In some applications, however, the information are embedded in two or more non-commuting observables. On the other hand, quantum entanglement allows us to infer through Einstein-Podolsky-Rosen correlations two conjugate observables with precision better than what is allowed by Heisenberg uncertainty relation. With the help of the newly developed SU(1,1) interferometer, we implement a scheme to measure j…
If noncontextuality is defined as the robustness of a system's response to a measurement against other simultaneous measurements, then the Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality. Namely, for the argument to be effective, (i) each operator must be uniquely associated with a measurement and (ii) commuting operators must represent simultaneous measurements. However, in all Kochen-Specker arguments discussed in the literature either (i) or (ii) is not met. Arguments meeting (i) contain at least one subset of mutually commuting operators which do not re…
The recently established universal uncertainty principle revealed that two nowhere commuting observables can be measured simultaneously in some state, whereas they have no joint probability distribution in any state. Thus, one measuring apparatus can simultaneously measure two observables that have no simultaneous reality. In order to reconcile this discrepancy, an approach based on quantum logic is proposed to establish the relation between quantum reality and measurement. We provide a language speaking of values of observables independent of measurement based on quantum logic and we construc…
Quantum mechanics predicts the joint probability distribution of the outcomes of simultaneous measurements of commuting observables, but, in the state of the art, has lacked the operational definition of simultaneous measurements. The question is answered as to when the consecutive applications of measuring apparatuses give a simultaneous measurement of their observables. For this purpose, all the possible state reductions caused by measurements of an observable is also characterized by their operations.
Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq δ$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq ε$? A theorem of H. Lin shows that this is uniformly true, in that for every $ε>0$ there exists a $δ>0$, independent of the size $N$ of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how $δ$ depends on $ε$. We give uniform bounds relating $δ$ and $ε$. We provide tighter bounds in the case of block tridiagonal and tridiagonal matric…
Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective …
Variational Quantum Eigensolver (VQE) is a promising algorithm for near-term quantum machines. It can be used to estimate the ground state energy of a molecule by performing separate measurements of $O(N^4)$ terms. Several recent papers observed that this scaling may be reducible to $O(N^3)$ by partitioning the terms into linear-sized commuting families that can be measured simultaneously. We confirm these empirical observations by studying the MIN-COMMUTING-PARTITION problem at the level of the fermionic Hamiltonian and its encoding into qubits. Moreover, we provide a fast, pre-computable pro…
We analyze the continuous measurement of two non-commuting observables for a qubit, and investigate whether the simultaneously observed noisy signals are consistent with the evolution of an equivalent classical system. Following the approach outlined by Leggett and Garg, we show that the readouts violate macrorealistic inequalities for arbitrarily short temporal correlations. Moreover, the derived inequalities are manifestly violated even in the absence of Hamiltonian evolution, unlike for Leggett-Garg inequalities that use a single continuous measurement. Such a violation should indicate the …
We present a new approach for obtaining quantum quasi-probability distributions, $P(α,β)$, for two arbitrary operators, $\mathbf{a}$ and $\mathbf{b}$, where $α$ and $β$ are the corresponding c-variables. We show that the quantum expectation value of an arbitrary operator can always be expressed as a phase space integral over $α$ and $β$, where the integrand is a product of two terms: One dependent only on the quantum state, and the other only on the operator. In this formulation, the concepts of quasi-probability and correspondence rule arise naturally in that simultaneously with the derivatio…
For any pair of bounded observables $A$ and $B$ with pure point spectra, we construct an associated "joint observable" which gives rise to a notion of a joint (projective) measurement of $A$ and $B$, and which conforms to the intuition that one can measure non-commuting observables simultaneously, provided one is willing to give up arbitrary precision. As an application, we show how our notion of a joint observable naturally allows for a construction of a "functional calculus," so that for any pair of observables $A$ and $B$ as above, and any (Borel measurable) function $f: \mathbb{R}^2 \right…
We address the statistics of a simultaneous CWLM of two non-commuting variables on a few-state quantum system subject to a conditioned evolution. Both conditioned quantum measurement and that of two non-commuting variables differ drastically for either classical or quantum projective measurement, and we explore the peculiarities brought by the combination of the two. We put forward a proper formalism for the evaluation of the distributions of measurement outcomes. We compute and discuss the statistics in idealized and experimentally relevant setups. We demonstrate the visibility and manifestat…
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caus…
Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state. In general, the possible paths will depend on the future and past boundary conditions, as well as the system's dynamics. We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously, each with probability one, depending on which measurement was performed. This experiment involves the measurement of observables that do not correspond to Hermitian operators. Our main result is a scheme for measuring these operators. The sc…
By using finite resolution measurements it is possible to simultaneously obtain noisy information on two non-commuting polarization components of a single photon. This method can be applied to a pair of entangled photons with polarization statistics that violate Bell's inequalities. The theoretically predicted results show that the non-classical nature of entanglement arises from negative joint probabilities for the non-commuting polarization components. These negative probabilities allow a "disentanglement" of the statistics, providing new insights into the non-classical properties of quantum…
We show how simultaneous, back-action evading tracking of non-commuting observables can be achieved in a widely-used sensing technology, atomic interferometry. Using high-dynamic-range dynamically-decoupled quantum non-demolition (QND) measurements on a precessing atomic spin ensemble, we track the collective spin angle and amplitude with negligible effects from back action, giving steady-state tracking sensitivity 2.9 dB beyond the standard quantum limit and 7.0 dB beyond Poisson statistics.
The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical n…
We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commutative functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories - symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of no…
Given the state of a quantum system, one can calculate the expectation value of any observable of the system. However, the inverse problem of determining the state by performing different measurements is not a trivial task. In various experimental setups it is reasonably straightforward to reconstruct the state of a quantum system employing linear tomographic technique. In this way the elements of the density matrix can be linearly related to a set of measured quantities. But since different observables of a quantum system are not commuting with each other, one often has to perform series of s…
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to er…
The concept of experimental accuracy is investigated in the context of the unbiased joint measurement processes defined by Arthurs and Kelly. A distinction is made between the errors of retrodiction and prediction. Four error-disturbance relationships are derived, analogous to the single error-disturbance relationship derived by Braginsky and Khalili in the context of single measurements of position only. A retrodictive and a predictive error-error relationship are also derived. The connection between these relationships and the extended Uncertainty Principle of Arthurs and Kelly is discussed.…
It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we give an efficient algorithm to explicitly compute a transfer matrix which realizes the simultaneous block diagonalization of unitary matrices whose decomposition in irreducible blocks (common invariant subspaces) is known from elsewhere. Our main motivation lies in particle physics, where the resulting transfer matrix must be known explicitly in order to uneq…
We establish a quantitative relation between Hardy's paradox and the breaking of uncertainty principle in the sense of measurement-disturbance relations in the conditional measurement of non-commuting operators. The analysis of the inconsistency of local realism with entanglement by Hardy is simplified if this breaking of measurement-disturbance relations is taken into account, and a much simplified experimental test of local realism is illustrated in the framework of Hardy's thought experiment. The essence of Hardy's model is identified as a combination of two conditional measurements, which …
Although quantum metrology allows us to make precision measurement beyond the standard quantum limit, it mostly works on the measurement of only one observable due to Heisenberg uncertainty relation on the measurement precision of non-commuting observables for one system. In this paper, we study the schemes of joint measurement of multiple observables which do not commute with each other by using the quantum entanglement between two systems. We focus on analyzing the performance of newly developed SU(1,1) interferometer on fulfilling the task of joint measurement. The results show that the inf…
We discuss coupling of violation of Bell's inequality and non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions of non-Kolmogorovness. There can be found other sufficient conditions of non-Kolmogorovness and, hence, violation of Bell's inequality. We find one important source of non-Kolmogorovness by analyzing axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting opera…
The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or compatible operators simultaneously. Unfortunately, the current hardware permits measuring only a much more limited subset of operators that share a common tensor product eigen-basis. We introduce unitary transformations that transform any fully commuting group of operators to a group that can be measured on current hardware. These unitary operations can be en…
Joint, or simultaneous, measurements of non-commuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin 1/2 particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.
The weak value, the average result of a weak measurement, has proven useful for probing quantum and classical systems. Examples include the amplification of small signals, investigating quantum paradoxes, and elucidating fundamental quantum phenomena such as geometric phase. A key characteristic of the weak value is that it can be complex, in contrast to a standard expectation value. However, typically only either the real or imaginary component of the weak value is determined in a given experimental setup. Weak measurements can be used to, in a sense, simultaneously measure non-commuting obse…
To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as $N^4$, where $N$ is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of th…
