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arXiv:quant-ph/0404051 · Physical Review A

Macroscopic objects in quantum mechanics: A combinatorial approach

Itamar Pitowsky

Published Aug 16, 2004• 14 citations
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Abstract

Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles $n$ grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness $W$ distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness $W$ to satisfy $\ensuremath{\mid}\mathrm{tr}(W\ensuremath{\rho})\ensuremath{\mid}\ensuremath{\leqslant}1$ for all separable states $\ensuremath{\rho}$, then the efficiency of $W$ depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm $\ensuremath{\parallel}W\ensuremath{\parallel}$. It is known that there are witnesses on the space of $n$ qubits for which $\ensuremath{\parallel}W\ensuremath{\parallel}$ is exponential in $n$. However, we conjecture that for a large majority of $n$-qubit witnesses $\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath{\leqslant}O(\sqrt{n\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}n})$. Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].

DOI
10.1103/physreva.70.022103

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Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles nnn grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness WWW distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness WWW to satisfy \ensuremath∣tr(W\ensuremathρ)\ensuremath∣\ensuremath⩽1\ensuremath{\mid}\mathrm{tr}(W\ensuremath{\rho})\ensuremath{\mid}\ensuremath{\leqslant}1\ensuremath∣tr(W\ensuremathρ)\ensuremath∣\ensuremath⩽1 for all separable states \ensuremathρ\ensuremath{\rho}\ensuremathρ, then the efficiency of WWW depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm \ensuremath∥W\ensuremath∥\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath∥W\ensuremath∥. It is known that there are witnesses on the space of nnn qubits for which \ensuremath∥W\ensuremath∥\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath∥W\ensuremath∥ is exponential in nnn. However, we conjecture that for a large majority of nnn-qubit witnesses \ensuremath∥W\ensuremath∥\ensuremath⩽O(nlogn)\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath{\leqslant}O(\sqrt{n\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}n})\ensuremath∥W\ensuremath∥\ensuremath⩽O(nlogn​). Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].
An observable WWW that distinguishes all the unentangled states from some entangled states is called a witness. We consider witnesses on nnn qbits, and use the following normalization: A witness WWW satisfies ∣tr(Wrho)∣leq1| tr(W\\rho)| \\leq 1∣tr(Wrho)∣leq1 for all separable states $\\rho $, while ∣W∣>1| W| >1∣W∣>1, with the norm being the maximum among the absolute values of the eigenvalues of WWW. Although there are nnn-qbit witnesses whose norm is exponential in nnn, we conjecture that for a large majority of nnn-qbit witnesses ∣W∣leqO(sqrtnlogn)| W| \\leq O(\\sqrt{n\\log n})∣W∣leqO(sqrtnlogn). We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf (Phys. Rev. A 64, 032112 (2001)). Assuming the conjecture is valid we argue that multiparticle entanglement can be detected only if a system has been carefully prepared in a very special state. Otherwise, multiparticle entanglement lies below the possibility of detection, even if it exists, and even if decoherence has been ``turned off''.
Let ${H}^{[N]}{=H}^{[{d}_{1}]}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{H}^{[{d}_{n}]}$ be a tensor product of Hilbert spaces and let ${\ensuremath{\tau}}_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state ${\ensuremath{\rho}}_{0}.$ Let ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ denote the closest separable state to ${\ensuremath{\rho}}_{0}$ along the line segment from I/NI/NI/N to ${\ensuremath{\rho}}_{0}$ where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W0{W}_{0}W0​ detecting the entanglement of ${\ensuremath{\rho}}_{0}$ can be constructed in terms of I, ${\ensuremath{\tau}}_{0},$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ If representations of ${\ensuremath{\tau}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ as convex combinations of separable projections are known, then the entanglement of ${\ensuremath{\rho}}_{0}$ can be detected by local measurements. G\"uhne et al. [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, ${\ensuremath{\tau}}_{0}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which ${\ensuremath{\tau}}_{0}\ensuremath{\ne}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a ``far face'' of the separable states relative to a bound entangled state ${\ensuremath{\rho}}_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density ${\ensuremath{\mu}}_{0}$ on the far face from ${\ensuremath{\rho}}_{0}.$ Up to a normalization this coincides with the witness obtained by G\"uhne et al. for the particular example analyzed there.
Macroscopic quantum phenomena refer to quantum features in objects of 'large' sizes, systems with many components or degrees of freedom, organized in some ways where they can be identified as macroscopic objects. This emerging field is ushered in by several categories of definitive experiments in superconductivity, electromechanical systems, Bose-Einstein condensates and others. Yet this new field which is rich in open issues at the foundation of quantum and statistical physics remains little explored theoretically (with the important exception of the work of A J Leggett [1], while touched upon or implied by several groups of authors represented in this conference. Our attitude differs in that we believe in the full validity of quantum mechanics stretching from the testable micro to meso scales, with no need for the introduction of new laws of physics.) This talk summarizes our thoughts in attempting a systematic investigation into some key foundational issues of quantum macroscopic phenomena, with the goal of ultimately revealing or building a viable theoretical framework. Three major themes discussed in three intended essays are the large N expansion [2], the correlation hierarchy [3] and quantum entanglement [4]. We give a sketch of the first two themes and then discuss several key issues in the consideration of macro and quantum, namely, a) recognition that there exist many levels of structure in a composite body and only by judicious choice of an appropriate set of collective variables can one give the best description of the dynamics of a specific level of structure. Capturing the quantum features of a macroscopic object is greatly facilitated by the existence and functioning of these collective variables; b) quantum entanglement, an exclusively quantum feature [5], is known to persist to high temperatures [6] and large scales [7] under certain conditions, and may actually decrease with increased connectivity in a quantum network [8]. We use entanglement as a measure of quantumness here and pick out these somewhat counter-intuitive examples to show that there are blind spots worthy of our attention and issues which we need to analyze closer. Our purpose is to try to remove the stigma that quantum only pertains to micro, in order to make way for deeper probes into the conditions whereby quantum features of macroscopic systems manifest.
The possibility to observe quantum features at the macroscopic level stands out as one of the most challenging yet fascinating open questions in modern quantum physics. The difficulties inherent in such a quest are manifold. First, the unavoidable interaction with the surrounding environment lies deeply at the roots of the loss of coherence and corruption of quantum correlations in a quantum state, being more severe as the size of the system being studied grows. Second, one faces the debated problem of achieving a measurement-precision that is sufficient to observe quantum effects at such macro-scales. Recent studies have shown that, depending on the Hamiltonian models, classicality may emerge from the coarse-graininess of measurements performed over a quantum system.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.
Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponentially many resources with regard to system size or qubit number, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to evaluate the detection capability of entanglement criteria theoretically. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double exponentially. Furthermore, if multicopy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems. Our results may shed light on why quantum phenomena are difficult to observe in large noisy systems.