Sequential Quantum Measurements Research Articles

Secure and robust randomness with sequential quantum measurements

Quantum correlations between measurements of separated observers are crucial for applications like randomness generation and key distribution. Although device-independent security can be certified with minimal assumptions, current protocols have limited performance. Here, we exploit sequential measurements, defined with a precise temporal order, to enhance performance by reusing quantum states. We provide a geometric perspective and a general mathematical framework, analytically proving a Tsirelson-like boundary for sequential quantum correlations, which represents a trade-off in nonlocality shared by sequential users. This boundary is advantageous for secure quantum randomness generation, certifying maximum bits per state with one remote and two sequential parties, even if one sequential user shares no nonlocality. Our simple qubit protocol reaches this boundary, and numerical analysis shows improved robustness under realistic noise. A photonic implementation confirms feasibility and robustness. This study advances the understanding of sequential quantum correlations and offers insights for efficient device-independent protocols.

Objectivity of classical quantum stochastic processes

We investigate what can be concluded about a quantum system when sequential quantum measurements of its observable – a prominent example of the so-called quantum stochastic process – fulfill the Kolmogorov consistency condition and thus appear to an observer as a sampling of a classical trajectory. We identify a set of physical conditions imposed on the system dynamics, that when satisfied, lead to the aforementioned trajectory interpretation of the measurement results. We then show that when another quantum system is coupled to the observable, the operator representing it can be replaced by external noise. Crucially, the realizations of this surrogate (classical) stochastic process follow the same trajectories as those measured by the observer. Therefore, it can be said that the trajectory interpretation suggested by the Kolmogorov consistent measurements also applies in contexts other than sequential measurements.

Causal potency of consciousness in the physical world

The evolution of the human mind through natural selection mandates that our conscious experiences are causally potent in order to leave a tangible impact on the surrounding physical world. Any attempt to construct a functional theory of the conscious mind within the framework of classical physics, however, inevitably leads to causally impotent conscious experiences in direct contradiction to evolution theory. Here, we derive several rigorous theorems that identify the origin of the latter impasse in the mathematical properties of ordinary differential equations employed in combination with the alleged functional production of the mind by the brain. Then, we demonstrate that a mind–brain theory consistent with causally potent conscious experiences is provided by modern quantum physics, in which the unobservable conscious mind is reductively identified with the quantum state of the brain and the observable brain is constructed by the physical measurement of quantum brain observables. The resulting quantum stochastic dynamics obtained from sequential quantum measurements of the brain is governed by stochastic differential equations, which permit genuine free will exercised through sequential conscious choices of future courses of action. Thus, quantum reductionism provides a solid theoretical foundation for the causal potency of consciousness, free will and cultural transmission.

Multiplexed Photon Number Measurement

When a two-level system -- a qubit -- is used as a probe of a larger system, it naturally leads to answering a single yes-no question about the system state. Here we propose a method where a single qubit is able to extract, not a single, but many bits of information about the photon number of a microwave resonator using continuous measurement. We realize a proof-of-principle experiment by recording the fluorescence emitted by a superconducting qubit reflecting a frequency comb, thus implementing multiplexed photon counting where the information about each Fock state -- from 0 to 8 -- is simultaneously encoded in independent measurement channels. Direct Wigner tomography of the quantum state of the resonator evidences the back-action of the measurement as well as the optimal information extraction parameters. Our experiment unleashes the full potential of quantum meters by replacing a sequential quantum measurements with simultaneous and continuous measurements separated in the frequency domain.

Entanglement-preserving limit cycles from sequential quantum measurements and feedback

Entanglement generation and preservation is a key task in quantum information processing, and a variety of protocols exist to entangle remote qubits via measurement of their spontaneous emission. We here propose feedback methods, based on monitoring the fluorescence of two qubits and using only local pi-pulses for control, to increase the yield and/or lifetime of entangled two-qubit states. Specifically, we describe a protocol based on photodetection of spontaneous emission (i.e. using quantum jump trajectories) which allows for entanglement preservation via measurement undoing, creating a limit cycle around a Bell states. We then demonstrate that a similar modification can be made to a recent feedback scheme based on homodyne measurement (i.e. using diffusive quantum trajectories), [L. S. Martin and K. B. Whaley, arXiv:1912.00067] in order to increase the lifetime of the entanglement it creates. Our schemes are most effective for high measurement efficiencies, and the impact of less-than-ideal measurement efficiency is quantified. The method we describe here combines proven techniques in a novel way, complementing existing protocols, and offering a pathway towards generating and protecting entangled states so that they may be used in various applications on demand.

Measurement-Induced Boolean Dynamics from Open Quantum Networks

In this paper, we study the induced probabilistic Boolean dynamics for dynamical quantum networks subject to sequential quantum measurements. In this part of the paper, we focus on closed networks of quits whose states evolutions are described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo random jumps and the corresponding measurement outcomes can be described by a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean networks can be explicitly represented through realification of the master equation. Next, when the open quantum dynamics is relaxing in the sense that it possesses a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of the relaxing property. Finally, we show that for quantum consensus networks as a type of non-relaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph.

Error-Disturbance Trade-off in Sequential Quantum Measurements.

We derive a state-dependent error-disturbance trade-off based on a statistical distance in the sequential measurements of a pair of noncommutative observables and experimentally verify the relation with a photonic qubit system. We anticipate that this Letter may further stimulate the study on the quantum uncertainty principle and related applications in quantum measurements.

Phase transitions in sequential weak measurements

Quantum measurements and phase transitions are seemingly uncorrelated topics, but here we show that phase transitions occur in sequential quantum measurements. We find that the probability distribution of the measurement results of a sequence of quantum measurements on a two-level system (e.g. a qubit) is equivalent to the Boltzmann distribution of a classical lattice spin model. So the measurement results present phase transitions similar to those in the lattice spin model. In sequential commuting positive-operator valued measurements, the probability distribution is mapped to a long-range Ising model in the weak-measurement regime, and a projective measurement emerges from a sequence of weak measurement when the strength or the number of measurements becomes above certain critical values, which correspond to a second-order ferromagnetic phase transition of the lattice spin model. These findings not only provide new insights on sequential quantum measurements, but may also have potential applications in quantum technologies.

On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements

In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions (of measurement results) on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility ($A-A$) and separated reproducibility($A-B-A$). The first one is reproducibility with probability 1 of a result of measurement of some observable $A$ measured twice, one $A$ measurement after the other. The second one, $A-B-A$, is reproducibility with probability 1 of a result of $A$ measurement when another quantum observable $B$ is measured between two $A$'s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, for an important class of quantum observables given by positive operator valued measures (POVMs), this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables $A$ and $B.$ However, the additional constraint in the form of separated reproducibility of the $B-A-B$ type makes this coexistence impossible. Mathematically, this paper is about the operator algebra for effects composing POVMs. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements.

Universality of sequential quantum measurements

We show that any jointly measurable pair of quantum observables can be obtained in a sequential measurement scheme, even if the second observable will be decided after the first measurement. This means that it is possible to perform a measurement of any quantum observable in a way that does not disturb the subsequent measurements more than is dictated by joint measurability. Only measurements with a specific structure have this universality feature. As a supplementing result, we provide a characterization of all possible joint measurements obtained from a sequential measurement lacking universality.

Extracting information from a qubit by multiple observers: toward a theory of sequential state discrimination.

We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e., misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the state, and we maximize this probability. The probability that Charlie's measurement succeeds depends on how much information about the state Alice sent is left in the qubit after Bob's measurement, and this information can be quantified by the overlap between the two possible states in which Bob's measurement leaves the qubit. This Letter is a first step toward developing a theory of nondestructive sequential quantum measurements, which could be useful in quantum communication schemes.

Testing sequential quantum measurements: how can maximal knowledge be extracted?

The extraction of information from a quantum system unavoidably implies a modification of the measured system itself. In this framework partial measurements can be carried out in order to extract only a portion of the information encoded in a quantum system, at the cost of inducing a limited amount of disturbance. Here we analyze experimentally the dynamics of sequential partial measurements carried out on a quantum system, focusing on the trade-off between the maximal information extractable and the disturbance. In particular we implement two sequential measurements observing that, by exploiting an adaptive strategy, is possible to find an optimal trade-off between the two quantities.

Coherent versus incoherent sequential quantum measurements

We compare a trade-off between knowledge and decoherence for the incoherent and coherent partial sequential compatible measurements on single-qubit systems. The individual partial measurement nondestructively monitors basis states of the system by single-qubit meter. For the same decoherence caused by this unbiased measurement, the individual coherent measurement gives more knowledge than the incoherent one. For identical sequential coherent measurements, knowledge accumulated not additively increases more slowly than for the incoherent measurements. The overall knowledge can be accumulated using an adaptive measurement strategy on the meters if the single-qubit coherence of meters is kept. On the other hand, preservation of the mutual qubit coherence between the meters necessary for the collective measurement strategy is not required. A loss of single-qubit coherence degrades the coherent measurements back to the incoherent ones. Since the decoherence caused by the measurement process is a quadratic function of knowledge extracted by the individual measurement, Zeno-like behavior can be observed for repetitive weak compatible measurements. This unconditional universal effect does not depend on any dynamics of the qubit and it is a direct consequence of optimally controlled sequential evolution of quantum information.

Sequential Product and Jordan Product of Quantum Effects

The quantum effects for a physical system can be described by the set \(\mathcal{E(H)}\) of positive operators on a complex Hilbert space \(\mathcal{H}\) that are bounded above by the identity operator I. We denote the set of sharp effects by \({\mathcal{P(H) }}\). For \(A,B\in\mathcal{E(H)}\), the operation of sequential product \(A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}\) was proposed as a model for sequential quantum measurements. Denote by \(A\ast B=\frac{AB+BA}{2}\) the Jordan product of \(A,B\in\mathcal{E(H)}\). The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on A∗B imply that A and B commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if A∗B satisfies certain associative laws, then AB=BA. Moreover, \(A\ast B\in{\mathcal{P(H) }}\) if and only if \(A\circ B\in{\mathcal{P(H)}}\).

Generalized infimum and sequential product of quantum effects

The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. For A, B∊E(H), the operation of sequential product A∘B=A1∕2BA1∕2 was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,” J. Math. Phys. 42, 5212 (2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum A⨅B are studied.

Sequential product of quantum effects

Unsharp quantum measurements can be modelled by means of the class E ( H ) \mathcal {E}(\mathcal {H}) of positive contractions on a Hilbert space H \mathcal {H} , in brief, quantum effects. For A , B ∈ E ( H ) A,B\in \mathcal {E}(\mathcal {H}) the operation of sequential product A ∘ B = A 1 / 2 B A 1 / 2 A\circ B=A^{1/2}BA^{1/2} was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption A ∘ B ≥ B A\circ B\geq B implies A B = B A = B AB=BA=B . Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.

Sequential quantum measurements

A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.

Accessible information in combined and sequential quantum measurementson a binary-state signal

The accessible information obtained in combined and sequential quantum measurements is calculated for a binary pure quantum state signal. It is shown that the accessible information obtained in the combined measurement is equal to that obtained in the sequential measurement. The result indicates that the conjecture by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] is not valid, at least for a binary quantum state signal. Furthermore, the result is compared with that obtained by Brody and Meister [Phys. Rev. Lett. 76, 1 (1996)].