Noise-Resistant Quantum Teleportation, Ansibles, and the No-Projector Theorem
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Noise-Resistant Quantum Teleportation, Ansibles, and the No-Projector Theorem Samuel R. Hedemann The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA (Dated: November 9, 2018) A method is presented for achieving entanglement-free teleportation of a quantum state subject to any quantum noise. We apply this as a light-speed noise-resistant communicator, but also treat the possibility of a quantum ansible, a device for effectively superluminal communication and quantum broadcasting. The results suggest a \no-projector theorem" analogous to the no-cloning theorem. We then show how to build a pseudo-ansible for connection-free light-speed communication. PACS numbers: 03.67.Hk, 03.67.Pp, 03.65.Ta, 03.65.Ud I. INTRODUCTION II. UQEC APPLIED TO NOISE-RESISTANT QUANTUM TELEPORTATION Since the birth of special relativity in 1905 [1,2], many have tried to break the classical light-speed limit to Unassisted quantum error correction (UQEC) in its achieve superluminal communication. The most popular full form can protect any quantum state from any noise attempts use entanglement and quantum cloning [3,4], channel, as shown in App. A. However, for conceptual but such methods have been proven impossible [5{8]. simplicity we begin by reviewing a special case called Here, we propose a new communication method that limited-direct UQEC, which only protects a limited fam- pairs two big ideas: entanglement-free quantum telepor- ily of states from all noise. We then discuss how to realize tation to send information as a state, and quantum error the process and examine the crucial role of projectors. correction (QEC) to protect the state during teleporta- tion. We call this noise-resistant quantum teleportation (NRQT). In one realization, it is light-speed limited and A. Overview of Unassisted Quantum Error requires a physical connection. However, the theory also Correction suggests that if true projectors are possible, then infor- mation could be sent faster than light, without violating The process of limited-direct UQEC starts by only con- special relativity. We can call any device that achieves sidering as input families of states with the form this an ansible, a term coined by science-fiction author n Ursula K. Le Guin for the idea proposed by earlier au- X thors like Isaac Asimov, inspired by Einstein. This sug- ρ ≡ ρ(θ) ≡ λj(θ)jejihejj; (1) gests a \no-projector theorem," but also indicates that a j=1 connection-free light-speed pseudo-ansible is possible. where fjejig is a set of n orthonormal states that are the defining constant of the family ρ(θ), and where λj ≡ P y Pn Alice ρ ! j Lj E(·) R(·) Lj ! ρ Bob λj(θ) 2 [0; 1] are eigenvalues where j=1 λj = 1, so the π family's free parameters are θ ≡ fθ1; : : : ; θn−1g 2 [0; 2 ]. FIG. 1: (color online) Unassisted QEC (UQEC) combined Given input ρ(θ), we can get limited-direct UQEC as with teleportation to achieve NRQT. E is any error channel, L and Ly are synchronized encoding/decoding operations, n n j j X X X y y y y and R is a recovery channel, all in the space of ρ. Alice can ρ(θ) = LjRqEkLjρ(θ)LjEkRqLj; (2) send any message to Bob, regardless of noise. j=1 q=1 k where we define encoding and recovery operators as The protection against quantum noise comes from a new form of QEC that is capable of correcting all errors, Lj ≡ j ihejj and Rq ≡ j ihφqj; (3) arXiv:1605.09233v1 [physics.gen-ph] 25 May 2016 without help from ancillas, called unassisted quantum er- ror correction (UQEC), depicted in Fig. 1. UQEC both where j i is the reference state, which is any pure state, protects against noise and enables the teleportation, but fjφqig is any orthonormal complete basis, fjejig is the is impractical for quantum computation (see App. A). constant set of orthonormal eigenstates defining ρ(θ), To present ideas organically, Sec. II reviews UQEC and and fEkg are Kraus operators of noise channel E where P y its role in NRQT, showing how the existence of projectors kEkEk = I. Operators like those in (3) can be decom- leads to effective superluminality. Then, Sec. III shows posed as a unitary operation and rank-1 projector, as how to build an ansible if true projectors could be re- shown in Sec. II D 3. See App. A for a derivation of (2). alized, motivating a \no-projector theorem" in Sec. IV. Next, to see how (2) enables us to protect any family Finally, Sec. V shows how to build a pseudo-ansible for of states ρ(θ) in the form of (1) from any quantum noise connection-free light-speed communication. channel, we will look at a single-qubit example. 2 B. Single-Qubit Example of UQEC where C is an \error-dependent" scalar, defined as n X X 2 Here, we do a step-by-step example showing that all C ≡ j(Ek)q;1j : (12) possible input states of a certain family can be recovered q=1 k from all possible error channels acting on one qubit (n = 2). For simplicity, let fjuig be computational basis kets Appendix A 1 proves that C = 1 for all channels, but here generically labeled starting on 1 as fj1i;:::; jnig. we keep it. Now we see that the matrix to the right of C First, define the state family of constant eigenstates by in (11) only contains information about ρ, and putting (4) into (11) and normalizing (unnecessarily) gives picking fjejig ≡ fj1i; j2ig, so that 2 2 2 1 X X X y y y y X 2 2 tr(Cρ) LjRqEkLjρLjEkRqLj = ρ, (13) ρ ≡ ρ(θ) ≡ λjjejihejj = c j1ih1j + s j2ih2j; (4) θ θ j=1 q=1 k j=1 which is identical to the input state, as desired. Notice where cθ ≡ cos(θ) and sθ ≡ sin(θ). Then, choosing j i= that no ancilla systems are needed, and this same method 1 0 j1i and fjφqig ≡ fj1i; j2ig, where j1i ≡ 0 and j2i ≡ 1 , works for all n-level systems, even with nonlocal noise. (3) gives both the encoding and recovery operators as 1 0 0 1 C. Numerical Demonstration of UQEC L = R = and L = R = ; (5) 1 1 0 0 2 2 0 0 Here, we look at a numerical test to see that UQEC where Sec. II D 3 explains their physical implementation. works, but see App. A for a proof that it always works. The first step in the protection procedure is to ran- Figure 2 subjects an arbitrary qutrit ρ to an arbitrary domly choose and apply one of the Lj to ρ, keeping in P y y noise channel E(σ) ≡ k EkσEk. This qutrit is treated mind that later, we will also have to apply its adjoint Lj. A physical process for this is given in Sec. II D 2, but for 2π (a) φ now, the possible results of the encoding step are 1:0 3π 0:5 2 1 0 0:0 π L ρLy = λ ; (6) 1 j j j ρ 2 π 0 0 2 3 3 1 2 0 where the coefficients λj ≡ hejjρjeji are given by 2π (b) 1:0 φ 3π 0:5 λ = ρ = c2 and λ = ρ = s2; (7) 2 1 1;1 θ 2 2;2 θ 0:0 π 1 ρE 2 π where ρa;b ≡ hajρjbi. Then one of the Ek acts, producing 2 3 3 1 2 0 j(E ) j2 (E ) (E )∗ 2π y y k 1;1 k 1;1 k 2;1 (c) 1:0 φ EkLjρLjEk = λj ∗ 2 ; 3π (Ek)2;1(Ek)1;1 j(Ek)2;1j 0:5 2 (8) 0:0 π where again, (E ) ≡ hajE jbi. Next we randomly ap- 1 k a;b k ρR 2 π 2 ply one of the recovery operators, producing 3 3 1 2 0 y y y 2 1 0 FIG. 2: (color online) Limited-direct UQEC on an arbitrary RqEkLjρLjEkRq = λjj(Ek)q;1j : (9) 0 0 qutrit in an arbitrary noise channel. These plots depict density-matrix representations of the states. For each matrix y Then, applying the matched decoder Lj = jejih1j yields element, labeled by row indices on the bottom left and column indices on the bottom right, the height of the box gives the LyR E L ρLyEyRyL = λ j(E ) j2je ihe j: (10) magnitude, the color represents the phase-angle magnitude, j q k j j k q j j k q;1 j j and an asterisk on top indicates a negative phase angle. (a) shows input ρ, (b) shows damaged state ρE resulting from no Now, since we don't know which Ek happened, or protection, and (c) shows recovered state ρ resulting from which R was applied, or which pair fL ;Lyg was used, R q j j full protection. Note that ρR = ρ exactly. then before we look at the result, the potentially unnor- malized state is the sum over all possible results, as as part of a constant family ρ(θ1,θ2) by using its eigen- 2 2 2 states to construct Lj. Figure 2 then compares ρ to X X X y y y y X LjRqEkLjρLjEkRqLj = C λjjejihejj; the error-damaged state ρE ≡ E(ρ), and to the recov- Pn Pn P y y y y j=1 q=1 k j=1 ered state ρR ≡ j=1 q=1 k LjRqEkLjρLjEkRqLj, (11) which includes the effects of E. 3 D. How to Realize the Quantum Operations for Noise-Resistance Fk y nF Here we discuss the fine points of the UQEC procedure, ρ FkρF P y k F(ρ)= FkρFk focusing on practical methods of implementation, to see k=1 how it applies to noise-resistant quantum teleportation fF1;:::;FnFg (NRQT), and to identify the problem with projectors.