It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement, (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement, (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.

Jan 1, 20101 citation

Contributions to quantum probability

T. A. Fritz

Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.

Jan 1, 20100 citations

State distinguishability under weak measurement and postselection: A unified system and device perspective

Philipp Stammer

We quantify the disturbance of a quantum state undergoing a sequence of observations and particularly focus on a weak measurement followed by postselection and compare these results to the projective counterpart. Taking into account the distinguishability of both the system and the device, we obtain the exact trade-off between the system state disturbance and the change in the device pointer state. We show that for particular postselection procedures the coupling strength between the system and the device can be significantly reduced without losing measurement sensitivity, which is directly transferred to a reduced state disturbance of the system. We observe that a weak measurement alone does not provide this advantage, and only in combination with postselection is a significant improvement in terms of increased measurement sensitivity and reduced state disturbance found. We further show that under realistic experimental conditions this state disturbance is small, whereas the exact postselection probability is considerably larger than the approximate value given by the overlap of the initial and final states when neglecting the system state disturbance.We quantify the disturbance of a quantum state undergoing a sequence of observations and particularly focus on a weak measurement followed by postselection and compare these results to the projective counterpart. Taking into account the distinguishability of both the system and the device, we obtain the exact trade-off between the system state disturbance and the change in the device pointer state. We show that for particular postselection procedures the coupling strength between the system and the device can be significantly reduced without losing measurement sensitivity, which is directly transferred to a reduced state disturbance of the system. We observe that a weak measurement alone does not provide this advantage, and only in combination with postselection is a significant improvement in terms of increased measurement sensitivity and reduced state disturbance found. We further show that under realistic experimental conditions this state disturbance is small, whereas the exact postselection probability is considerably larger than the approximate value given by the overlap of the initial and final states when neglecting the system state disturbance.

arXiv:2011.04490Physical Review ADec 10, 20201 citation

Probability distributions of continuous measurement results for two noncommuting variables and postselected quantum evolution

A. Franquet, Yuli V. Nazarov

We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle ${\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1$ in the distribution of measurement outcomes ${\mathcal{O}}_{1,2}$ in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle ${\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1$ in the distribution of measurement outcomes ${\mathcal{O}}_{1,2}$ in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.

arXiv:1708.05662Physical Review ADec 6, 20191 citation

Probability distributions of continuous measurement results for conditioned quantum evolution

A. Franquet, Yuli V. Nazarov

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.

arXiv:1608.06858Physical review. B./Physical review. BFeb 21, 20177 citations

Quantum random walk and tight-binding model subject to projective measurements at random times

Debraj Das, Shamik Gupta

Abstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resuAbstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resu

arXiv:2105.00338Journal of Statistical Mechanics Theory and ExperimentMar 1, 202218 citations

{\mathcal{O}}_{1}^{2}+{\mathcal{O}}_{2}^{2}=1

in the distribution of measurement outcomes

{\mathcal{O}}_{1,2}

in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small.