Unavoidable disturbance caused by a quantum measurement implies that the realizable subsequent measurements are getting limited after one performs some measurement. The obvious general limitation that one cannot circumvent by sequential or any other method is that the actually implemented measurements must be jointly measurable. In this work we show that any jointly measurable pair of observables can be obtained in a sequential measurement scheme, even if the second observable would be decided after the first measurement. This universality feature holds only for measurement schemes with a specific structure. As a supplementing result, we provide a characterization of all possible joint measurements obtained from a sequential measurement lacking universality.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
Measurement-based quantum computing (MBQC) is a model of quantum computation where quantum information is coherently processed by means of projective measurements on highly entangled states. Following the introduction of MBQC, cluster states have been studied extensively both from the theoretical and experimental point of view. Indeed, the study of MBQC was catalysed by the realisation that cluster states are universal for MBQC with (X,Y)-plane and Z measurements. Here we examine the question of whether the requirement for Z measurements can be dropped while maintaining universality. We answer…
We present optimal and minimal measurements on identical copies of an unknown state of a qubit when the quality of measuring strategies is quantified with the gain of information (Kullback of probability distributions). We also show that the maximal gain of information occurs, among isotropic priors, when the state is known to be pure. Universality of optimal measurements follows from our results: using the fidelity or the gain of information, two different figures of merits, leads to exactly the same conclusions. We finally investigate the optimal capacity of $N$ copies of an unknown state as…
The graph state formalism offers strong connections between quantum information processing and graph theory. Exploring these connections, first we show that any graph is a pivot-minor of a planar graph, and even a pivot minor of a triangular grid. Then, we prove that the application of measurements in the (X,Z)-plane over graph states represented by triangular grids is a universal measurement-based model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.
We investigate for which resource states an efficient classical simulation of measurement based quantum computation is possible. We show that the Schmidt--rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt--rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement based quantum computation is possible for all states with logarithmically bounded Schmidt--rank width (with respect to the system size). For graph states …
Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires $X$ and $Z$-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only $X$ and $Z$-bas…
We reason about possible future development of quantum game theory and its impact on information processing and the emerging information society. Two of the authors have recently proposed a quantum description of financial market in terms of quantum game theory. These "new games" cannot by themselves create extraordinary profits or multiplication of goods, but they may cause the dynamism of transaction which would result in more effective markets and capital flow into hands of the most efficient traders. We focus upon the problem of universality of measurement in quantum market games. Quantum-…
We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction [quant-ph/0108020]. Then, we prove the universality of simple discrete sets of 2-qubit measurements, again using a scheme simplifying the initial construction [quant-ph/0111122]. Finally, we show how to use a single 4-qubit measurement to achieve universal quantum computation, and outline a proof for the universality of almost all maximally entangling 4-qubit measuremen…
The relation between projective measurements and generalized quantum measurements is a fundamental problem in quantum physics, and clarifying this issue is also important to quantum technologies. While it has been intuitively known that projective measurements can be constructed from sequential generalized or weak measurements, there is still lack of a proof of this hypothesis in general cases. Here we prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit. Specifically, a selectiv…
YZ-plane measurement-based quantum computation: Universality and Parity Architecture implementation
Parity Quantum Computing GmbH（Parity量子计算有限公司）；Parity Quantum Computing Germany GmbH（Parity量子计算德国有限公司）；Institute for Theoretical Physics, University of Innsbruck（因斯布鲁克大学理论物理研究所）
State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bi-partite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal seque…
We present two new quantum union bounds for sequential projective measurements. These bounds estimate the disturbance accumulation and probability of outcomes when the measurements are performed sequentially. These results are based on a trigonometric representation of quantum states and should have wide application in quantum information theory for information processing tasks such as communication and state discrimination, and perhaps even in the analysis of quantum algorithms.
Sequential weak measurements of non-commuting observables is not only fundamentally interesting in quantum measurement but also shown potential in various applications. The previous reported methods, however, can only realize limited sequential weak measurements experimentally. In this Letter, we propose the realization of sequential measurements of arbitrary observables and experimentally demonstrate for the first time the measurement of sequential weak values of three non-commuting Pauli observables by using genuine single photons. The results presented here will not only improve our underst…
We investigate an optimization problem of finding quantum sequential measurements, which forms a wide class of state discrimination problems with the restriction that only sequential measurements are allowed. Sequential measurements from Alice to Bob on a bipartite system are considered. Using the fact that the optimization problem can be formulated as a problem with only Alice's measurement and is convex programming, we derive its dual problem and necessary and sufficient conditions for an optimal solution. In the problem we address, the output of Alice's measurement can be infinite or contin…
We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without post-selection yield the corresponding correlations of the Wigner function, offering direct quantum tomography through the moments of the canonical variables. Spin-half sequential measurements are, on the contrary, constrained kinematically, they are equivalent with single WMs. In sequential WMs with post-selection, a new anomaly occurs, different from the weak value anomaly of single WMs. In particular, the spread of polarization $\hatσ$, as measu…
We sketch applications of the so-called J-equation to quantum information theory concerning fundamental properties of the von Neumann entropy. The J-equation has recently be proposed as a sort of progenitor of the various versions of the Jarzynski equation. It has been derived within a general framework of sequential measurements that is slightly generalised here.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
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.
We present a new paradigm for capturing the complementarity of two observables. It is based on the entanglement created by the interaction between the system observed and the two measurement devices used to measure the observables sequentially. Our main result is a lower bound on this entanglement and resembles well-known entropic uncertainty relations. Besides its fundamental interest, this result directly bounds the effectiveness of sequential bipartite operations---corresponding to the measurement interactions---for entanglement generation. We further discuss the intimate connection of our …
Unsharp measurements are increasingly important for foundational insights in quantum theory and quantum information applications. Here, we report an experimental implementation of unsharp qubit measurements in a sequential communication protocol, based on a quantum random access code. The protocol involves three parties; the first party prepares a qubit system, the second party performs operations which return both a classical and quantum outcome, and the latter is measured by the third party. We demonstrate a nearly-optimal sequential quantum random access code that outperforms both the best …
The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation.
This work analyzes correlations arising from quantum systems subject to sequential projective measurements to certify that the system in question has a quantum dimension greater than some $d$. We refine previous known methods and show that dimension greater than two can be certified in scenarios which are considerably simpler than the ones presented before and, for the first time in this sequential projective scenario, we certify quantum systems with dimension strictly greater than three. We also perform a systematic numerical analysis in terms of robustness and conclude that performing random…
The extraction of information from a quantum system unavoidably implies a modification of the measured system itself. It has been demonstrated recently that 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 consider two different regimes of measureme…
A sequential measurement of two non-commuting quantum observables results in a joint probability distribution for all output combinations that can be explained in terms of an initial joint quasi-probability of the non-commuting observables, modified by the resolution errors and back-action of the initial measurement. Here, we show that the error statistics of a sequential measurement of photon polarization performed at different measurement strengths can be described consistently by an imaginary correlation between the statistics of resolution and back-action. The experimental setup was design…
Quantum correlation is a fundamental property which distinguishes quantum systems from classical ones, and it is also a fragile resource under projective measurement. Recently, it has been shown that a subsystem in entangled pairs can share nonlocality with multiple observers in sequence. Here we present a new steering scenario where both subsystems are accessible by multiple observers. And it is found that the two qubits in singlet state can be simultaneously steered by two sequential observers, respectively.
The sequential quantum random access code (QRAC) allows two or more decoders to obtain a desired message with higher success probability than the best classical bounds by appropriately modulating the measurement sharpness. Here, we propose an entanglement-assisted sequential QRAC protocol which can enable device-independent tasks. By relaxing the equal sharpness and mutually unbiased measurement limits, we widen the sharpness modulation region from a one-dimensional interval to a two-dimensional triangle. Then, we demonstrate our scheme experimentally and get more than 27 standard deviations a…
We address the problem of whether parties who cannot communicate but share nonsignaling quantum correlations between the outcomes of sharp measurements can distinguish, just from the value of a correlation observable, whether their outcomes were produced by sequential compatible measurements on single systems or by measurements on spatially separated subsystems. We show that there are quantum correlations between the outcomes of sequential measurements which cannot be attained with spatially separated systems. We present examples of correlations between spatially separated systems whose quantu…
We utilize a discrete (sequential) measurement protocol to investigate quantum process tomography of a single two-level quantum system, with an unknown initial state, undergoing Rabi oscillations. The ignorance of the dynamical parameters is encoded into a continuous-variable classical system which is coupled to the two-level quantum system via a generalized Hamiltonian. This combined estimate of the quantum state and dynamical parameters is updated by using the information obtained from sequential measurements on the quantum system and, after a sufficient waiting period, faithful state monito…
Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to includ…
A general scenario for an $N$-sequential conclusive state discrimination introduced recently in Loubenets and Namkung [arXiv:2102.04747] can provide a multipartite quantum communication realizable in the presence of a noise. In the present article, we propose a new experimental scheme for the implementation of a sequential conclusive discrimination between binary coherent states via indirect measurements within the Jaynes-Cummings interaction model. We find that if the mean photon number is less than 1.6, then, for our two-sequential state discrimination scheme, the optimal success probability…
Unsharp measurements play an increasingly important role in quantum information theory. In this paper, we study a three-party prepare-transform-measure experiment with unsharp measurements based on $ 3 \rightarrow 1 $ sequential random access codes (RACs). We derive optimal trade-off between the two correlation witnesses in $ 3 \rightarrow 1 $ sequential quantum random access codes (QRACs), and use the result to complete the self-testing of quantum preparations, instruments and measurements for three sequential parties. We also give the upper and lower bounds of the sharpness parameter to comp…
Entanglement is a fundamental resource for quantum information science. However, bipartite entanglement is destroyed when one particle is observed via projective (sharp) measurements, as it is typically the case in most experiments. Here we experimentally show that, if instead of sharp measurements, one performs many sequential unsharp measurements on one particle which are suitably chosen depending on the previous outcomes, then entanglement is preserved and it is possible to reveal quantum correlations through measurements on the second particle at any step of the sequence. Specifically, we …
We study the efficiency of estimation procedures where the temperature of an external bath is indirectly recovered by monitoring the transformations induced on a probing system that is put in thermal contact with the bath. In particular we compare the performances of sequential measurement schemes where the probe is initialized only once and measured repeatedly during its interaction with the bath, with those of measure & re-prepare approaches where instead, after each interaction-and-measurement stage, the probe is reinitialized into the same fiduciary state. From our analysis it is revealed …
Universal quantum computation can be achieved by simply performing single-qubit measurements on a highly entangled resource state, such as cluster states. Cai, Miyake, Dür, and Briegel recently constructed a ground state of a two-dimensional quantum magnet by combining multiple Affleck-Kennedy-Lieb-Tasaki quasichains of mixed spin-3/2 and spin-1/2 entities and by mapping pairs of neighboring spin-1/2 particles to individual spin-3/2 particles [Phys. Rev. A 82, 052309 (2010)]. They showed that this state enables universal quantum computation by single-spin measurements. Here, we give an alterna…
The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as 1/<N>, where <N> is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited, to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts. Our result gives the first completely general, constraint-free and non-asymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including mu…
Non-locality and quantum measurement are two fundamental topics in quantum theory and theirinterplay attracts intensive focus since the discovery of Bell theorem. Non-locality sharing amongmultiple observers is predicted and experimentally observed. However, only one-sided sequentialcase, i.e., one Alice and multiple Bobs is widely discussed and little is known about two-sided case.Here, we theoretically and experimentally explore the non-locality sharing in two-sided sequentialmeasurements case in which one entangled pair is distributed to multiple Alices and Bobs. Weexperimentally observed d…
Non-locality sharing for a three-qubit system via multilateral sequential measurements was deeply discussed. Different from 2-qubit case, it is shown that non-locality sharing between $\mathrm{Alice_{1}-Bob_{1}-Charlie_{1}}$ and $\mathrm{Alice_{2}-Bob_{2}-Charlie_{2}}$ in 3-qubit system can be observed, where two Mermin-Ardehali-Belinskii-Klyshko (MABK) inequalities can be violated simultaneously. What's more, a complete non-locality sharing with 8 MABK inequalities violations simultaneously can be also observed. Compared with 2-qubit case, the nonlocal sharing in a three qubit system shows mo…
A sequence of moments encode the corresponding probability distribution. Probing if quantum joint probability distribution can be retrieved from the associated set of moments -- realized in the sequential measurement of a dichotomic observable at different time intervals -- reveals a negative answer i.e., the joint probabilities of sequential measurements do not agree with the ones obtained by inverting the moments. This is indeed a reflection of the non-existence of a bonafide grand joint probability distribution, consistent with all the physical marginal probability distributions. Here we ex…
We study the transmission of classical information in quantum channels. We present a decoding procedure that is very simple but still achieves the channel capacity. It is used to give an alternative straightforward proof that the classical capacity is given by the regularized Holevo bound. This procedure uses only projective measurements and is based on successive "yes"/"no" tests only.
A basic assumption behind the inequalities used for testing noncontextual hidden variable models is that the observables measured on the same individual system are perfectly compatible. However, compatibility is not perfect in actual experiments using sequential measurements. We discuss the resulting "compatibility loophole" and present several methods to rule out certain hidden variable models which obey a kind of extended noncontextuality. Finally, we present a detailed analysis of experimental imperfections in a recent trapped ion experiment and apply our analysis to that case.
We model a typical Bell-test experimental situation by considering that Alice and Bob perform incompatible measurements in a sequential way, with mixed orders of execution. After emphasizing that order effects will generally produce a violation of the marginal laws, we derive an upper limit for the observed correlations. More precisely, when Alice's and Bob's measurements are compatible, the marginal laws are obeyed and Tsirelson's bound limits the quantum correlations in the Bell-CHSH inequality to $2\sqrt{2}$. On the other hand, when Alice and Bob perform incompatible mixed sequential measur…
A possible alternative to the standard model of measurement-based quantum computation (MBQC) is offered by the sequential model of MBQC -- a particular class of quantum computation via ancillae. Although these two models are equivalent under ideal conditions, their relative resilience to noise in practical conditions is not yet known. We analyze this relationship for various noise models in the ancilla preparation and in the entangling-gate implementation. The comparison of the two models is performed utilizing both the gate infidelity and the diamond distance as figures of merit. Our results …
Lund University（隆德大学）；University of Jyväskylä（于韦斯屈莱大学）；Technical University of Denmark（丹麦技术大学）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to sequential measurement processes is the aim of this paper. Given only a simple set of postulates, a probability measure is derived utilising the concept of Liouville space and the most general permissible quantum channel arises in the same manner. Super-Liouville space is constructed and a Bayesian interpretation of this object is provided. An important applica…
Recently, it has been shown that at most two observers (Bobs) can sequentially demonstrate bipartite nonlocality with a spatially separated single observer (Alice) invoking a scenario where an entangled system of two spin-$\frac{1}{2}$ particles are shared between a single Alice in one wing and several Bobs on the other wing, who act sequentially and independently of each other [Phys. Rev. Lett. 114, 250401 (2015)]. This has been probed through the quantum violations of CHSH inequality, i. e., when each observer performs two dichotomic measurements. In the present study we investigate how many…
We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider…
