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, and Raymond Laflamme

Phys. Rev. Lett. 116, 230501 – Published 7 June 2016

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.

Received 24 February 2016

DOI:https://doi.org/10.1103/PhysRevLett.116.230501

Physics Subject Headings (PhySH)

Entanglement detection Entanglement measures Quantum tomography

Nuclear magnetic resonance

General PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Dawei Lu¹, Tao Xin^1,2, Nengkun Yu^1,3,4, Zhengfeng Ji^1,5, Jianxin Chen⁶, Guilu Long², Jonathan Baugh¹, Xinhua Peng⁷, Bei Zeng^1,4,8,*, and Raymond Laflamme^1,8,9

¹Institute for Quantum Computing, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
²State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
³Centre for Quantum Computation and Intelligent Systems, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia
⁴Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada
⁵State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
⁶Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
⁷Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230036, China
⁸Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
⁹Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

^*zengb@uoguelph.ca

Verifying the Quantumness of Bipartite Correlations

Claudio Carmeli, Teiko Heinosaari, Antti Karlsson, Jussi Schultz, and Alessandro Toigo

Phys. Rev. Lett. 116, 230403 (2016)

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Vol. 116, Iss. 23 — 10 June 2016

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Images

Figure 1
Geometry of separable and entangled states. The top pink oval represents the set of all states, denoted by $A$ . (a) A set (blue) inside $A$ which is not cylinderlike. Hence, no projection onto any plane exists that can separate the set with the rest states. (b) A set (blue) that is an intersection of a generalized cylinder with $A$ (i.e., cylinderlike). The projection onto the plane that is orthogonal to the boundary lines of the cylinder separates this set with the rest of the states. The bottom ovals are the images of the top sets onto a plane, which clearly show a separation of the images of the blue set from the pink set in (b), but an overlap of images in (a).
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Figure 2
(a) Schematic circuit for implementing the filter-based entanglement distillation proposal for an unknown two-qubit state $ρ_{A B}^{0}$ . $F_{A, B}^{i}$ is the $i$ th local filter applied on $A$ or $B$ , designed from the previous reduced density matrix. The gray dots mean a single-qubit tomography is implemented. (b) Simulated variation of concurrence and fidelity as the increased number $1 \leq m \leq 5$ of applied filters. The simulated state is chosen as Eq. (5) with $λ = 0.2$ . When $m \leq 4$ , the reconstructed state is not unique due to the lack of constraints, so both the concurrence (blue squares) and fidelity (red circles) have some distributions. When $m = 5$ , the input state can be uniquely determined, and the concurrence and fidelity converges to a single point. The dashed blue and red curves show the envelopes of the variations of concurrence and fidelity along with $m$ , respectively.
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Figure 3
(a) Molecular structure and Hamiltonian parameters of ${}^{13}C$ -labeled transcrotonic acid. (b) NMR sequence to realize the filter-based protocol of entanglement distillation. In particular, this sequence displays how to realize the first two filters $F_{A}^{0}$ and $F_{B}^{1}$ in terms of NMR pulses. First, the four-qubit system is prepared to the PPS by the spatial average technique. Then the system qubits $A$ and $B$ are initialized to $| ϕ_{B} ⟩$ , $| ϕ_{1} ⟩$ , and $| ϕ_{2} ⟩$ via three independent experiments illustrated in the lower-right inset, respectively. The right part of the sequence is used to implement the filters. $V_{A}$ , $V_{B}$ , $U_{A}$ , $U_{B}$ , $θ_{1}$ , and $θ_{2}$ [18] all depend on the measured single-qubit state in the last step.
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Figure 4
Concurrence of $ρ_{A B}^{0}$ , $ρ_{0}^{e}$ , and $ρ_{f}^{e}$ as a function of the Bell-state weight $λ$ . The brown curve shows the concurrence computed by the theoretical state $ρ_{A B}^{0}$ ; the blue squares represent the concurrence of $ρ_{0}^{e}$ , the state obtained from two-qubit FST right after the input state preparation; the red circles represent the concurrence of $ρ_{f}^{e}$ , the state reproduced from the single-qubit information. The colored band-region accounts for an artificial noise of the strength 5.79% which is roughly our experimental error source.
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