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Random Witnesses and the Classical Character of Macroscopic Objects

Itamar Pitowsky

Published Apr 8, 2004• 0 citations
OpenAlex

Abstract

An observable $W$ that distinguishes all the unentangled states from some entangled states is called a witness. We consider witnesses on $n$ qbits, and use the following normalization: A witness $W$ satisfies $| tr(W\\rho)| \\leq 1$ for all separable states $\\rho $, while $| W| >1$, with the norm being the maximum among the absolute values of the eigenvalues of $W$. Although there are $n$-qbit witnesses whose norm is exponential in $n$, we conjecture that for a large majority of $n$-qbit witnesses $| W| \\leq O(\\sqrt{n\\log n})$. 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''.

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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\\rho 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''.
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 $\ensuremath{\mid}\mathrm{tr}(W\ensuremath{\rho})\ensuremath{\mid}\ensuremath{\leqslant}1$ for all separable states $\ensuremath{\rho}$, then the efficiency of WWW 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 nnn qubits for which $\ensuremath{\parallel}W\ensuremath{\parallel}$ is exponential in nnn. However, we conjecture that for a large majority of nnn-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)].
Entanglement witnesses are of special interest for distinguishing between separable and entangled states. For three-qubit states, a family of optimal entanglement witnesses has been introduced by Jafarizadeh et al. [Eur. Phys. J. D 50 (2008) 107]. We give an alternative demonstration for some elements of this family and show that they are able to detect bound entangled states. To illustrate this fact, we consider two families of bound entangled three-qubit states, introduced by Kay [Phys. Rev. A 83 (2011) 020303] and Kye [J. Phys. A, Math. Theor. 48 (2015) 235303], and show that these witnesses detect the entanglement of all states of these two families.
Quantum Mechanics Quantum entanglement produces correlations between particles that cannot be described classically, and it is a resource sought out in many applications to leverage the quantum advantage over classical operation. However, determining whether particles are entangled is a complex process that can require a set of measurements (the witness) to establish the extent of the correlations between them. Shanhandeh et al. describe a protocol that should simplify the job of the witness somewhat. Using already known information about the systems, they demonstrate that entangled and unentangled states can be distinguished using fewer measurements. Extendable to larger systems, the simplified but enhanced witness should prove useful in quantum devices for which verifying entanglement is required. Phys. Rev. Lett. 118 , 110502 (2017).
In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking optimality of entanglement witnesses developed already in [M. Lewenstein et al., Phys. Rev. A 62, 052310 (2000)], however, hardly exploited so far in the literature.
We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus Tr Ωρ≥0\textrm{Tr}\,Ωρ\geq 0TrΩρ≥0 when ΩΩΩ is a witness and ρρρ a pure product state, ρ=ψψ†ρ=ψψ^{\dagger}ρ=ψψ† with ψ=ϕ⊗χψ=ϕ\otimesχψ=ϕ⊗χ. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value f(ϕ,χ)=Tr Ωρf(ϕ,χ)=\mathrm{Tr}\,Ωρf(ϕ,χ)=TrΩρ as a function of ϕϕϕ and χχχ is a nonnegative biquadratic form. Every zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) imposes real-linear constraints on fff and ΩΩΩ. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of f(ϕ,χ)f(ϕ,χ)f(ϕ,χ) is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions lead to an extremal witness which is usually quadratic with the minimum number of zeros. We discuss some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the structural physical approximation (SPA) of an optimal witness. We discuss how to treat the entanglement witnesses on a complex Hilbert space as witnesses on a real Hilbert space.
We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.