Distinguishing computable mixtures of quantum states

Ignacio H. López Grande, Gabriel Senno, Gonzalo de la Torre, Miguel A. Larotonda, Ariel Bendersky, Santiago Figueira, and Antonio Acín

Phys. Rev. A 97, 052306 – Published 7 May 2018

Abstract

In this article we extend results from our previous work [Bendersky et al., Phys. Rev. Lett. 116, 230402 (2016)] by providing a protocol to distinguish in finite time and with arbitrarily high success probability any algorithmic mixture of pure states from the maximally mixed state. Moreover, we include an experimental realization, using a modified quantum key distribution setup, where two different random sequences of pure states are prepared; these sequences are indistinguishable according to quantum mechanics, but they become distinguishable when randomness is replaced with pseudorandomness within the experimental preparation process.

Received 10 August 2017
Revised 26 March 2018

DOI:https://doi.org/10.1103/PhysRevA.97.052306

Physics Subject Headings (PhySH)

Quantum foundations Quantum information processing

Quantum Information, Science & Technology

Authors & Affiliations

Ignacio H. López Grande¹, Gabriel Senno², Gonzalo de la Torre², Miguel A. Larotonda¹, Ariel Bendersky^3,4, Santiago Figueira^3,4, and Antonio Acín^2,5

¹DEILAP, CITEDEF-CONICET, Villa Martelli, Buenos Aires, Argentina
²ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
³Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Computación, 1428 Buenos Aires, Argentina
⁴CONICET-Universidad de Buenos Aires, Instituto de Investigación en Ciencias de la Computación (ICC), 1428 Buenos Aires, Argentina
⁵ICREA–Institucio Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

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Issue

Vol. 97, Iss. 5 — May 2018

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Images

Figure 1
Schematic description of the protocol given in [5] allowing a player Bob to tell if he is being given pseudorandom eigenstates of the $σ_{x}$ basis or of the $σ_{z}$ basis.
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Figure 2
Complete setup for implementing the transmission and search protocol: qubits encoded in polarized faint pulses are produced by infrared LEDs. Light is coupled into and decoupled from multimode fibers to obtain uniform beams for the four sources. The polarization state preparation is achieved by passing through a PBS (for H and V states) and an extra half-wave plate for the D and A paths. A nonpolarizing beam splitter cube couples the optical paths into an only exit light path. At the receiver's side a BS passively and randomly selects the detection basis for each incoming pulse. The outputs are coupled into multimode optical fibers, where different delays are imposed to make a polarization to time-bin transformation into a common output fiber. Finally a photon counter module and a temporal mask demultiplexer are used for detection.
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Figure 3
Experimentally estimated error rate (the error that Bob does make a wrong claim) obtained with the noise tolerant protocol (circles over red lines), compared with the theoretical bounds for the noiseless (lower line in blue) and noise tolerant (upper line in green) algorithms. The cyan line with squares is the computational simulation of the experimental data taking into account the average QBER.
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Figure 4
Green curve (squares) shows the total wrong claim probability (error rate) and the contribution to this quantity coming from the programs of length 1 (blue dots), together with the effective tolerance $q_{eff}$ (red triangles) for each value of $k$ over the 3100 experiment repetitions. The tolerance value $q$ was set to 0.15. Almost every error comes from $ℓ = 1$ programs and the probability of error occurrences coming from programs with $ℓ > 1$ vanishes as $k$ increases. The sudden increases of $q_{eff}$ in $k = 7$ and $k = 14$ appear due to the discrete nature of the maximum number of errors allowed on an accepted bit string (maximum Hamming distance). Also for each value of $k$ where the effective tolerance probability for $ℓ = 1$ increases, the error probability also increases. The vertical dashed lines delimit the regions where the maximum number of bit flips $N_{err}$ allowed (for $ℓ = 1$ ) is constant.
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Figure 5
Plotted data correspond to the wrong claim probability obtained by doing simulations of the experiment with an increasing set of programs (from $2^{1}$ to $2^{11}$ programs). For each value of $ℓ_{\max}$ we run a simulation of the experiment with $5 \times 10^{4}$ repetitions. The $P_{err}$ value stabilizes as the number of programs grows.
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