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arXiv:1511.03149 · Physical Review A

How measurement reversal could erroneously suggest the capability to discriminate the preparation basis of a quantum ensemble

Sandeep K. Goyal, Rajeev Singh, Sibasish Ghosh A

Published Jan 19, 2016• 0 citations
arXivHTMLPDF

Abstract

Mixed states of a quantum system, represented by density operators, can be decomposed as a statistical mixture of pure states in a number of ways where each decomposition can be viewed as a different preparation recipe. However the fact that the density matrix contains full information about the ensemble makes it impossible to estimate the preparation basis for the quantum system. Here we present a measurement scheme to (seemingly) improve the performance of unsharp measurements. We argue that in some situations this scheme is capable of providing statistics from a single copy of the quantum system, thus making it possible to perform state tomography from a single copy. One of the by-products of the scheme is a way to distinguish between different preparation methods used to prepare the state of the quantum system. However, our numerical simulations disagree with our intuitive predictions. We show that a counterintuitive property of a biased classical random walk is responsible for the proposed mechanism not working.

DOI
10.1103/physreva.93.012114

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Mixed states of a quantum system, represented by density operators, can be decomposed as a statistical mixture of pure states in a number of ways where each decomposition can be viewed as a different preparation recipe. However the fact that the density matrix contains full information about the ensemble makes it impossible to estimate the preparation basis for the quantum system. Here we present a measurement scheme to (seemingly) improve the performance of unsharp measurements. We argue that in some situations this scheme is capable of providing statistics from a single copy of the quantum system, thus making it possible to perform state tomography from a single copy. One of the by-products of the scheme is a way to distinguish between different preparation methods used to prepare the state of the quantum system. However, our numerical simulations disagree with our intuitive predictions. We show that a counterintuitive property of a biased classical random walk is responsible for the proposed mechanism not working.
Abstract A central tenet of quantum theory is that ensembles sharing the same density matrix are indistinguishable under measurements on individual systems. We show that this indistinguishability can break down when additional structure is present across multiple copies. We investigate ensembles of qubits prepared in either the computational (ZZZ) or Hadamard (XXX) basis under a same-basis constraint. All qubits within a batch are prepared in the same basis, but with randomly distributed outcomes such that the reduced state of each individual qubit is maximally mixed. We introduce an experimentally realizable protocol that exploits multi-copy statistical structure to distinguish these ensembles. The graph-state protocol identifies the two basis through stabilizer measurements and higher-order statistical features. Our protocol has been implemented on the \texttt{ibm\_fez} superconducting quantum processor and demonstrates robust performance in the presence of realistic noise, achieving near-unity classification accuracy as the number of samples increases. While single-copy measurements are uninformative, aggregating outcomes across multiple instances produces a clear and scalable separation between basis families. Our result identifies a regime in which quantum information is not accessible at the level of individual systems but emerges through structured multi-copy measurements. This highlights the role of higher-order statistical correlations in quantum information processing and provides a practical route for exploiting such structure on near-term quantum devices.
We explore the possibility of using ``weak'' measurements to carry out quantum state tomography via numerical simulations. Given a fixed number of copies of identically prepared states of a qubit, we perform state tomography using weak as well as projective measurements. Due to the collapse of the state after measurement, we cannot reuse the state after a projective measurement. If the coupling strength between the quantum system and the measurement device is made weaker, the disturbance caused to the state can be lowered. This then allows us to reuse the same member of the ensemble for further measurements and thus extract more information from the system. However, this happens at the cost of getting imprecise information from the first measurement. We implement this scheme for a single qubit and show that under certain circumstances, it can outperform the projective measurement-based tomography scheme. This opens up the possibility of new ways of extracting information from quantum ensembles. We study the efficacy of this scheme for different coupling strengths and different ensemble sizes.
Tomography allows one to estimate the density matrix describing the state an ensemble of quantum systems are prepared in (for example, polarization tomography determines the polarization state of a beam of identically prepared photons). In general, it is not possible to uniquely decompose the density matrix into its pure state components. Agarwal et al. proposed a protocol which, for a mixture composed of any two pure states of a qubit (with arbitrary probabilities), allows an observer to infer not only the density matrix but the identity of those specific pure states and their weights - the additional requirement being that the qubits arrive in pairs, where both qubits in each pair are in the same state. We experimentally demonstrate this learning-from-pairs concept using photons in the polarization degree of freedom. We use tomography to measure a sequence of single photons and make use of their time-of-arrival information to 'pair up' the photons after the measurement. From here we are able to infer the photons' polarization states and their respective probabilities, and we demonstrate this for various different choices of polarization states and ratios. Finally, we investigate our ability to discriminate between two equal mixtures of distinct pairs of orthogonal polarization states. We find that on the order of approx. 10e4 photons is typically enough to achieve tomography fidelities of approximately 0.9999. This is sufficient to discriminate between two different preparations of the same mixed state, differing by angles of less than 5 degrees between the pure states used in the two preparations.
Random ensembles of pure states have proven to be extremely important in various aspects of quantum physics such as benchmarking the performance of quantum circuits, testing for quantum advantage, studying many-body thermalization, and the black hole information paradox. Although generating a truly random quantum ensemble is experimentally challenging, approximate realizations are equally valuable and are known to emerge naturally in a variety of physical models, including Rydberg setups. These are referred to as approximate quantum state designs, and verifying their degree of randomness can be a measurement-intensive task, similar to performing full quantum state tomography on many-body systems. In this theoretical work, we present a measurement scheme and analysis techniques to validate the degree of randomness of a quantum ensemble generated by a simulated experimental setup. This is achieved by translating the information residing in the complex many-body state into a succinct representation of classical data using projective measurements in randomly chosen bases, which is then processed using methods of statistical inference such as maximum-likelihood estimation and neural networks, benchmarked against the predictions of shadow tomography. Our scheme only requires individually addressed single-qubit operations to be performed in order to be employed, making it applicable for a range of physical platforms.
Suppose you receive a sequence of qubits where each qubit is guaranteed to be in one of two pure states, but you do not know what those states are. Your task is to determine the states. This can be viewed as a kind of quantum state learning -- or quantum state estimation. A problem is that, without more information, all that can be determined is the density matrix of the sequence and, in general, density matrices can be decomposed into pure states in many different ways. To solve the problem, additional information, either classical or quantum, is required. We show that if an additional copy of each qubit is supplied -- that is, one receives pairs of qubits, both in the same state, rather than single qubits -- the task can be accomplished. This is possible because the mixed two-qubit state has a unique decomposition into pure product states. For illustration, we simulate numerically the symmetric, informationally complete measurement of a sequence of qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred from the data with high accuracy. Finally, we propose an experiment that employs a product measurement and can be realized with existing technology, and we demonstrate how the data tell us the states and their probabilities. We find that it is enough to detect a few thousand qubit pairs.
Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many-body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as ${\ensuremath{\sum}}_{\mathbit{a},\mathbit{b}}{|\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathbit{a}\mathbit{b}}|}^{2}\ensuremath{\sim}{5}^{N}/{N}_{\mathrm{tot}}$ for full tomography, and ${\mathrm{\ensuremath{\Delta}}\ensuremath{\mu}}^{2}\ensuremath{\sim}{7}^{N}/{N}_{\mathrm{tot}}^{2}$ for purity estimation, for NNN qubits and Ntot{N}_{\mathrm{tot}}Ntot​ measurements performed.