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arXiv:1212.3503 · Physical Review A

Uniqueness of quantum states compatible with given measurement results

Jianxin Chen, Hillary Dawkins, Zhengfeng Ji, Nathaniel Johnston, David W. Kribs, Frederic W. Shultz, Bei Zeng

Published Jul 11, 2013• 63 citations
arXivHTMLPDF

Abstract

We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a $d$-dimensional Hilbert space, there exists a set of $4d\ensuremath{-}5$ observables that uniquely determines any pure state. We show that for case (2), $5d\ensuremath{-}7$ observables suffice to uniquely determine any pure state. Thus, there is a gap between the results for (1) and (2), and we give some examples to illustrate this. Unique determination of a pure state by its reduced density matrices (RDMs), a special case of determination by observables, is also discussed. We improve the best-known bound on local dimensions in which almost all pure states are uniquely determined by their RDMs for case (2). We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low-rank quantum states.

DOI
10.1103/physreva.88.012109

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We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a ddd-dimensional Hilbert space, there exists a set of 4d\ensuremath−54d\ensuremath{-}54d\ensuremath−5 observables that uniquely determines any pure state. We show that for case (2), 5d\ensuremath−75d\ensuremath{-}75d\ensuremath−7 observables suffice to uniquely determine any pure state. Thus, there is a gap between the results for (1) and (2), and we give some examples to illustrate this. Unique determination of a pure state by its reduced density matrices (RDMs), a special case of determination by observables, is also discussed. We improve the best-known bound on local dimensions in which almost all pure states are uniquely determined by their RDMs for case (2). We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low-rank quantum states.
Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However, it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work, we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally, we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results demonstrate the advantages and possible pitfalls of quantum state tomography with local measurements.
In quantum systems, entanglement corresponds to nonclassical correlation of nonlocal observables. Thus, entanglement (or, to the contrary, separability) of a given quantum state is not uniquely determined by properties of the state, but may depend on the choice of the factorization of the algebra of observables. In the present work, we expose and systematize some recently reported results about the possibility to represent a single quantum state as either entangled or separable. We will distinguish in particular the cases of pure and mixed states. For pure states, it has been shown that observables can always be constructed such that any state has any amount of entanglement possible. For mixed states, the situation is more complex and only partial results are known: while it is always possible to choose a factorization such that a state appears separable, a general criterion to determine whether a state can be represented as entangled is not known. These results will be illustrated by several examples, the phenomenon of quantum teleportation, and the geometry of the states of two qubits.
We study the discrimination of NNN mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of NNN states in a ddd-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating NNN equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in ddd dimensions. In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.
Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring O(d2)\mathcal{O}({d}^{2})O(d2) measurements for a general ddd-dimensional state. This complexity can be substantially reduced to O(d)\mathcal{O}(d)O(d) in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among aall states (UDA). The UDP determination inherently involves nonconvexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the mugmented Lagrangian ethod (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parametrizing quantum states and employing ALM to handle the constrained nonconvex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems, generalized Greenberger-Horne-Zeilinger states and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (a) UDA, (b) UDP but not UDA, and (c) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.
Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.
We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value rrr. In this scenario two notions of informationally complete measurements emerge: rank-rrr-complete measurements and rank-rrr strictly-complete measurements. Whereas in the first notion, a rank-rrr state is uniquely identified from within the set of rank-rrr states, in the second notion the same state is uniquely identified from within the set of all physical states, of any rank. We argue, therefore, that strictly-complete measurements are compatible with convex optimization, and we prove that they allow robust quantum-state estimation in the presence of experimental noise. We also show that rank-rrr strictly-complete measurements are as efficient as rank-rrr-complete measurements. We construct examples of strictly-complete measurements and give a complete description of their structure in the context of matrix completion. Moreover, we numerically show that a few random bases form such measurements. We demonstrate the efficiency-robustness property for different strictly-complete measurements with numerical experiments. We thus conclude that only strictly-complete measurements are useful for practical tomography.