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Multi-Qubit Systems, Entanglement, and Quantum Weirdness

Alice Flarend, Bob Hilborn

Published Apr 15, 2022• 0 citations
Publisher

Abstract

Abstract Quantum state entanglement is a crucial ingredient in many quantum algorithms. This chapter introduces the description of multiqubit quantum systems and then explains what it means to say the state of that system is entangled. If the quantum state of the system is an entangled state, none of the constituents of the system are described by their own individual states. For a two-qubit entangled state, measurements on one of the qubits apparently affects the measurement results of the other even though the two qubits may be far apart. Entanglement is explained using both joint and conditional probabilities of measurement results.

DOI
10.1093/oso/9780192857972.003.0009

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•Mar 27, 2000•0 citations
Abstract Quantum state entanglement is a crucial ingredient in many quantum algorithms. This chapter introduces the description of multiqubit quantum systems and then explains what it means to say the state of that system is entangled. If the quantum state of the system is an entangled state, none of the constituents of the system are described by their own individual states. For a two-qubit entangled state, measurements on one of the qubits apparently affects the measurement results of the other even though the two qubits may be far apart. Entanglement is explained using both joint and conditional probabilities of measurement results.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.
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.
Abstract The importance of quantum entanglement in the development of quantum cryptography, quantum communications, and other areas of science related to modern quantum technologies cannot be overestimated. While the measures of entanglement for a two-level system in a pure state are well-known, the mathematical description of quantum entanglement for multi-level systems is still an open question. The paper proposes a formula that allows us to calculate the numerical value of the quantum entanglement of systems in pure states based on their state vector. The value of quantum entanglement lies in the range [0, 1], and for Greenberg–Horne–Zeilinger states, this value is equal to one.
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbitrary state of two qubits, pure or mixed, quantifies a tradeoff between the amounts of entanglement, separable uncertainty, and single particle properties that are encoded in the quantum state. The separable uncertainty is a natural measure of our ignorance about the properties possessed by individual subsystems, and may be used to completely characterize the relationship between entanglement and mixedness in two-qubit systems. The two-qubit complementarity relation yields a useful geometric picture in which the root mean square values of local subsystem properties act like coordinates in the space of density matrices, and suggests possible insights into the problem of interpreting quantum mechanics.
Consider a two-qubit maximally entangled state that gets corrupted when only one of the qubits is subjected to local interaction with the environment via amplitude damping channel. We show there are situations where this corrupted state gets better by allowing the other qubit also to interact with the environment. Moreover if the corrupted state is non teleporting, then it can always be made teleporting by the same prescription i.e. by allowing the second qubit to interact with the environment. Quantum entanglement [1] is a property of composite systems by virtue of which the local subsystems exhibit certain correlations among themselves that cannot be explained in classical terms. Often termed as quantum nonlocality, the subject had been under extensive study and debate since 1935 when Einstein, Podolsky and Rosen questioned the completeness of quantum mechanics [2]. Later, in 1964 in his pioneering work John Bell [3] showed that the correlations exhibited by the local subsystems that may be far apart are inherently quantum mechanical and any model admitting the description of local hidden variables (or one may prefer to say local realism) fails to explain such correlations.