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arXiv:1511.00581 · Physical Review Letters

Tomography is Necessary for Universal Entanglement Detection with Single-Copy Observables

Dawei Lu, Tao Xin, Nengkun Yu, Zhengfeng Ji, Jianxin Chen, Guilu Long, Jonathan Baugh, Xinhua Peng, Bei Zeng, Raymond Laflamme

Published Jun 7, 2016• 53 citations
arXivHTMLPDF

Abstract

Entanglement, one of the central mysteries of quantum mechanics, plays an essential role in numerous tasks of quantum information science. A natural question of both theoretical and experimental importance is whether universal entanglement detection can be accomplished without full state tomography. In this Letter, we prove a no-go theorem that rules out this possibility for nonadaptive schemes that employ single-copy measurements only. We also examine a previously implemented experiment [H. Park et al., Phys. Rev. Lett. 105, 230404 (2010)], which claimed to detect entanglement of two-qubit states via adaptive single-copy measurements without full state tomography. In contrast, our simulation and experiment both support the opposite conclusion that the protocol, indeed, leads to full state tomography, which supplements our no-go theorem. These results reveal a fundamental limit of single-copy measurements in entanglement detection and provide a general framework of the detection of other interesting properties of quantum states, such as the positivity of partial transpose and the k-symmetric extendibility.

DOI
10.1103/physrevlett.116.230501

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Entanglement, one of the central mysteries of quantum mechanics, plays an essential role in numerous tasks of quantum information science. A natural question of both theoretical and experimental importance is whether universal entanglement detection can be accomplished without full state tomography. In this Letter, we prove a no-go theorem that rules out this possibility for nonadaptive schemes that employ single-copy measurements only. We also examine a previously implemented experiment [H. Park et al., Phys. Rev. Lett. 105, 230404 (2010)], which claimed to detect entanglement of two-qubit states via adaptive single-copy measurements without full state tomography. In contrast, our simulation and experiment both support the opposite conclusion that the protocol, indeed, leads to full state tomography, which supplements our no-go theorem. These results reveal a fundamental limit of single-copy measurements in entanglement detection and provide a general framework of the detection of other interesting properties of quantum states, such as the positivity of partial transpose and the k-symmetric extendibility.
The entanglement, one of the central mysteries of the quantum mechanics, plays a significant role in a variety of applications of the quantum information theory. A natural question in theoretical and experimental importance is whether it is possible to detect a universal entanglement without full diagnostics. The diagnostics relies on a set of quantum trajectories and their records from measurements. This model reflects the probability that each of the measurements may be damaged from interference and decoherence, and may also be associated with recording of continuous signals for an end-time period. The goal is then to retrieve the quantum state such as it had been in the beginning of this measurement process. The proposed solution relies on explicit expression of the probability function through effective matrices contained in the quantum approximation and solutions of ad-joint quantum filters. In this article, we prove а no-go theorem, which outlines this possibility for non-adaptive schemes, which use only single-copy measurements. We also examine in detail а previously conducted experiment, for which it is claimed that detects the entanglement of two-qubit states through adaptive single-copy measurements without full diagnostics. With the conduct of the experiment and the analysis of the data, we demonstrate that the information collected is really sufficient to reconstruct the state. These results reveal a fundamental limit of the single-copy measurements upon the entanglement detection, and provide a common framework for learning the detection of other interesting properties of the quantum states, such as the positivity of partial transposition and the k-symmetric-extendibility.
Certifying multipartite entanglement is a fundamental task. Since n-qubit state is parameterized by 4n- 1 real numbers, it is interesting to design a measurement setup that detects multipartite entanglement with as little effort as possible, and at a minimum without fully revealing the whole information of the state, the so-called “tomography”. In this paper, we study the relationship between multipartite entanglement certification and tomography, with the constraint that only single-copy measurements are allowed. We show that by using nonadaptive single-copy measurements, universal entanglement detection, among all states, can not be accomplished without full state tomography. Moreover, we show that almost all multipartite correlations, including the genuine entanglement and the entanglement depth, require full state tomography to detect in this measurement setting. We also observe that universal entanglement detection, among pure states, can be accomplished using much fewer measurements than full state tomography even using only local measurements.
Experimental detection of entanglement of an arbitrary state of a given bipartite system is crucial for exploring many areas of quantum information processing. But such a detection should be made in a device-independent way if the preparation process of the state is considered to be faithful, in order to avoid detection of a separable state as an entangled one. The recently developed scheme of detecting bipartite entanglement in a measurement-device-independent way [Phys. Rev. Lett. 110, 060405 (2013)] does require information about the state. Here, by using Auguisiak et al.'s universal entanglement witness scheme for two-qubit states [Phys. Rev. A 77, 030301 (2008)], we provide a universal entanglement detection scheme for two-qubit states in a measurement-device-independent way. We also provide a set of universal witness operators for detecting NPT-ness (negative under partial transpose) of two-qudit states in a measurement-device-independent way. We conjecture that no such universal entanglement witness scheme exists for PPT (positive under partial transpose) entangled states. We also analyze the robustness of some of the experimental schemes---for detecting entanglement in a measurement-device-independent way---under the influence of noise in the inputs (from the referee) as well as in the measurement operator.
Quantum state tomography is a fundamental problem in quantum computing. Given n copies of an unknown N -qubit state ρ∈ℂ d × d , d =2 N , the goal is to learn the state up to an accuracy ε in trace distance, say with at least constant probability 0.99. We are interested in the copy complexity, the minimum number of copies of ρ needed to fulfill the task. As current quantum devices are physically limited, Pauli measurements have attracted significant attention due to their ease of implementation. However, a large gap exists in the literature for tomography with Pauli measurements. The best-known upper bound is O ( N · 12 N /ε 2 ), and no non-trivial lower bound is known besides the general single-copy lower bound of Ω(8 N /ε 2 ), achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of O (10 N /ε 2 ). To complement it, we also obtain a lower bound of Ω(9.118 N /ε 2 ), which holds even with adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements, while prior lower-bound techniques for tomography only work with measurement-independent constructions. Moreover, we connect the copy complexity lower bound of tomography to the eigenvalues of the measurement information channel, which governs the measurement’s capacity to distinguish between states. To demonstrate the generality of the new framework, we obtain tight bounds for adaptive quantum state tomography with k -outcome measurements, where we recover existing results and establish new ones.
Abstract One of the main challenges of quantum information is the reliable verification of quantum entanglement. The conventional detection schemes require repeated measurement on a large number of identically prepared systems. This is hard to achieve in practice when dealing with large-scale entangled quantum systems. In this letter we formulate verification as a decision procedure, i.e., entanglement is seen as the ability of quantum system to answer certain “yes-no questions”. We show that for a variety of large quantum states even a single copy suffices to detect entanglement with a high probability by using local measurements. For example, a single copy of a 16-qubit k-producible state or one copy of 24-qubit linear cluster state suffices to verify entanglement with more than 95% confidence. Our method is applicable to many important classes of states, such as cluster states or ground states of local Hamiltonians in general.
Entanglement witness is an effective method to detect entanglement in unknown states without doing full tomography. One of the most widespread schemes of witnessing entanglement is measuring its fidelity with respect to a pure entangled state. Recently, a large class of states whose entanglement can not be detected with the fidelity witness has been discovered in Phys.Rev.Lett \textbf{124},200502(2020). They are called unfaithful states. In this paper we propose a new way to detect entanglement by calculating the lower bound of entanglement using measurement results. Numerical simulation shows our method can detect entanglement in unfaithful states with a small number of measurements. Moreover, we generalize our scheme to multipartite states and show that it can tolerate higher noise than previous entanglement witness operators with same number of measurement settings.