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arXiv:quant-ph/9907104

Entanglement and the linearity of quantum mechanics

G. Alber

Published Jul 30, 1999• 2 citations
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Abstract

Optimal universal entanglement processes are discussed which entangle two quantum systems in an optimal way for all possible initial states. It is demonstrated that the linear character of quantum theory which enforces the peaceful coexistence of quantum mechanics and relativity imposes severe restrictions on the structure of the resulting optimally entangled states. Depending on the dimension of the one-particle Hilbert space such a universal process generates either a pure Bell state or mixed entangled states. In the limit of very large dimensions of the one-particle Hilbert space the von-Neumann entropy of the optimally entangled state differs from the one of the maximally mixed two-particle state by one bit only.

DOI
10.48550/arxiv.quant-ph/9907104

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Optimal universal entanglement processes are discussed which entangle two quantum systems in an optimal way for all possible initial states. It is demonstrated that the linear character of quantum theory which enforces the peaceful coexistence of quantum mechanics and relativity imposes severe restrictions on the structure of the resulting optimally entangled states. Depending on the dimension of the one-particle Hilbert space such a universal process generates either a pure Bell state or mixed entangled states. In the limit of very large dimensions of the one-particle Hilbert space the von-Neumann entropy of the optimally entangled state differs from the one of the maximally mixed two-particle state by one bit only.
Universal two-particle entanglement processes are analyzed in arbitrary dimensional Hilbert spaces. On the basis of this analysis the class of possible optimal universal entanglement processes is determined whose resulting output states do not contain any separable states. It is shown that these processes form a one-parameter family. For all Hilbert space dimensions larger than two the resulting optimally entangled output states are mixtures of anti-symmetric states which are freely entangled and which also preserve information about input states. Within this one-parameter family there is only one process by which all information about any input state is destroyed completely.
Within the class of all possible universal (covariant) two-particle quantum processes in arbitrary dimensional Hilbert spaces those universal quantum processes are determined whose output states optimize the recently proposed entanglement measure of Vidal and Werner. It is demonstrated that these optimal entanglement processes belong to a one-parameter family of universal entanglement processes whose output states do not contain any separable components. It is shown that these optimal universal entanglement processes generate antisymmetric output states and, with the single exception of qubit systems, they preserve information about the initial input state.
We show that the entanglement evolution of an open quantum system is the same for the vast majority of initial pure states, in the limit of large Hilbert space dimensions. The analytical results are illustrated by a numerical study of the evolution of negativity for many-qubit systems undergoing a local decoherence process. In the limit of many subsystems, the arising mixed state entanglement between one subsystem and the rest grows more robust, whereas for two equally sized partitions it becomes more fragile.
Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy. Along the way, we determine the states that, for a given value of entropy, achieve maximal violation of Bell's inequality.
We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of this issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type (p,q) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space. The phenomenon of entanglement is of central importance in the interpretation of quantum mechanics. Historically, entanglement was the focus of the famous Einstein-Podolsky-Rosen (EPR) paper [1], which suggested that standard quantum mechanics is an incomplete theory of physical reality. The central argument of the EPR paper was that more information about incompatible variables such as momentum and position could in principle be deduced about an entangled two particle quantum state than quantum mechanics permits, effectively giving information about each particle
A fundamental starting point in quantum information theory is the consideration of the von Neumann entropy and its generalization to relative quantum entropies. A particular feature of quantum relative entropies is their relation via their Hessian to monotonic Riemannian metrics on the dense set of invertible mixed quantum states (Lesniewski and Ruskai 1999). These metrics are also known as quantum Fisher information metrics and provide a direct link to quantum estimation theory (Helstrom 1969). Quantum Fisher information metrics which are extendable to pure states coincide all with the Fubini Study metric of the projective Hilbert space of complex rays. This theses outlines possible advantages of an inverse approach to quantum information theory, by starting with the Fubini Study metric rather then with the von Neumann entropy. This is done in a first step by associating to the Fubini Study metric a covariant and a contra-variant structure on the punctured Hilbert space as being available in the geometric formulation of quantum mechanics. While the contra-variant structure leads to a quantum version of the Cramér-Rao inequality for general 1-dimensional submanifolds of pure states, the covariant structure provides alternative entanglement monotones by identifying an inner product on the pullback tensor fields on local unitary group orbits of quantum states. It is shown in the case of two qubits that these monotones yield a more efficient estimation of entanglement than standard measures from the literature as those associated with the linearization of the von Neumann entropy.