Scrambling and quantum teleportation

Scrambling is a concept introduced from information loss problem arising in black hole. In this paper we discuss the effect of scrambling from a perspective of pure quantum information theory regardless of the information loss problem. We introduce 7-qubit quantum circuit for a quantum teleportation. It is shown that the teleportation can be perfect if a maximal scrambling unitary is used. From this fact we conjecture that “the quantity of scrambling is proportional to the fidelity of teleportation”. In order to confirm the conjecture, we introduce θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-dependent partially scrambling unitary, which reduces to no scrambling and maximal scrambling at θ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta = 0$$\end{document} and θ=π/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta = \pi / 2$$\end{document}, respectively. Then, we compute the average fidelity analytically, and numerically by making use of qiskit (version 0.36.2) and 7-qubit real quantum computer ibm_\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\_$$\end{document}oslo. Finally, we show that our conjecture can be true or false depending on the choice of qubits for Bell measurement.


I. INTRODUCTION
Nowadays, quantum information theories (QIT) [1] is one of the subjects, which attract much attention recently.This seems to be mainly due to the rapid development of quantum technology such as realization of quantum cryptography [2,3] and quantum computer [4,5].
In QIT quantum entanglement [1,6,7] plays an important role as a physical resource in the various types of quantum information processing (QIP).It is used in many QIP such as in quantum teleportation [8,9], superdense coding [10], quantum cloning [11], quantum cryptography [2,12], quantum metrology [13], and quantum computers [4,14,15].In particular, quantum computing attracted a lot of attention recently after IBM and Google independently realized quantum computers.It is debatable whether "quantum supremacy" is achieved or not in the quantum computation.
Quantum Gravity (QG) is a field of physics that seeks to describe gravity according to the principles of quantum mechanics.Experimental access to QG, however, is challenging at present since it requires the ability to measure miniscule physical effects.Recent rapid development of quantum computer, however, may allow different possibility to test QG indirectly.Using quantum simulators and quantum computers we may be able to probe QG in the laboratory [16][17][18][19].
Already there were several papers investigating QG toward this direction.In Ref. [20] matrix quantum mechanics is simulated by adopting the quantum-classical hybrid algorithm called VQE [21].In particular, the authors of Ref. [20] computed the low-energy spectra of bosonic and supersymmetric matrix models and compare them to the results of Monte Carlo simulations.In Ref. [22] quantum teleportation with scrambling unitary was implemented on a fully-connected trapped-ion quantum computer [23].This is based on the Hayden-Preskill protocol [24][25][26].The intuition behind their approach is to reinterpret the black hole's information loss problem via the quantum teleportation.In Ref. [27] wormhole-inspired teleportation was simulated by making use of Quantinuum's trapped-ion System Model H1-1 and five IBM superconducting quantum processing units.This is indirect approach to verify the ER=EPR conjecture [28,29], which assumes that the quantum channel generated by entangled quantum state is nothing but the wormhole.It was shown that the teleportation signals reach 80% of theoretical predictions.
In this paper we study the teleportation scheme with a scrambling unitary from a view- point of pure QIT.The scrambling [25,26,[30][31][32] is a concept introduced from information loss problem [33,34] in black hole physics.Although there is more rigorous definition [30,31], roughly speaking, "scrambling" means the delocalization of quantum information.In other words, when the quantum information of the subsystem is completely mixed with remaining systems, we use the terminology "scrambling"1 .
The quantum circuit for this scheme is different from usual quantum teleportation as shown in Fig, 1. Fig. 1a is well-known 3-qubit quantum circuit for usual quantum teleportation.Alice has first two qubits and Bob has last one.The vertical line means the maximally entangled state |β 0 = 1 √ 2 (|00 + |11 ).The task is to teleport the unknown state |ψ = α|0 + β|1 to Bob.It is easy to show that the quantum state |Ψ in Fig. 1a is Thus, the task is completed by applying X and/or Z to Bob's qubit appropriately, where X, Y , and Z are the Pauli operators.
Fig. 1b is a 7-qubit quantum circuit for teleportation with scrambling.First qubit, i.e. 0 th -qubit, in Fig. 1b represents the Alice's secret qubit.In this paper we assume if U is partially scrambling unitary, the fidelity of teleportation is lowered from one even though Daniel performs the optimal quantum measurement.This means that the quantity of scrambling of U is probably proportional to the fidelity for the teleportation.The purpose of the paper is to examine this conjecture.In order to explore this problem we introduce U (θ), where θ = 0 and θ = π/2 correspond to the no scrambling and maximally scrambling.
Since there is no measure which quantify the scrambling, we cannot say how much quantum information is scrambled by U (θ).But from the parametrization in θ, we guess that the quantity of scrambling of U (θ) is proportional to θ.Then, we will compute the θ-dependence of the fidelities between |ψ A and ρ B analytically.In order to examine the noise effect we also compute the fidelities numerically by making use of qiskit (version 0.36.2) and 7-qubit real quantum computer ibm oslo.From the analytical and numerical results we conclude that our conjecture "the quantity of scrambling is proportional to the fidelity of teleportation" can be true or false depending on the Daniel's choice of qubits for Bell measurement.
The paper is organized as follows.In next section we examine the quantum teleportation with maximally scrambling U .If Daniel chooses Bell measurement in one of {2, 3}, {1, 4} or {0, 5} qubits and notifies the outcomes to Bob, it is shown that the perfect teleportation is possible.In section III we examine the teleportation again with U (θ), which is no scrambling at θ = 0 and maximally scrambling at θ = π/2.If Daniel takes Bell measurement of either {2, 3} or {1, 4} qubits, it is shown that the fidelities are the exactly the same.The θdependence of average fidelity is monotonically increasing function with respect to θ, which supports the conjecture.If, however, Daniel takes Bell measurement of {0, 5} qubits, it is shown that the average fidelity is not monotonic.In section IV the numerical calculation for the fidelities is discussed.Comparing the analytically computed fidelities with the numerical ones, it is shown that qiskit and ibm oslo yields errors less that 1%.Therefore, the effect of noise is negligible in the calculation of fidelities.Thus, if we need to discuss a similar issue in the future with large number of qubits, we can adopt the numerical approach without producing much error.In section V a brief conclusion is given.In appendix A the partial scrambling property of U (θ) is more clearly verified.The numerical results are summarized in appendix B and appendix C.
In this section we choose U in a form: This unitary operator can be experimentally implemented up to the global phase by a quantum circuit in Fig. 2. It is straightforward to show where X, Y , Z and I are the three Pauli operators and the Identity operator.Eq. (2.2) verifies the maximal scrambling property of U by showing that it delocalizes all singlet-qubit into three-qubit operators.One can show that the quantum state |Ψ in Fig. 1b is From Eq. (2.3) it is easy to show that the teleportation process is completed if Daniel performs a Bell measurement in one of {2, 3}, {1, 4}, or {0, 5} qubits and notifies the measurement outcomes to Bob.The Bell measurement can be easily implemented by using a quantum circuit of Fig. 3.This circuit transforms the Bell states into the computation basis as FIG. 4: (Color online) Quantum circuit for implementing the unitary in Eq. (3.1).
In the previous section we showed that perfect quantum teleportation is possible if the maximal scrambling unitary (2.1) is used.In order to understand the role of scrambling property in the teleportation process more clearly, we consider in this section the teleportation with partial scrambling unitary.For this purpose we choose U as a θ-dependent unitary in the form: where  It is easy to show 3 i=0 P i = 1.After measurement, the Bob's 6 th -qubit state should be derived by taking a partial trace over remaining qubits.Therefore, Bob's state can be gen-erally mixed state.In order to examine how well the quantum teleportation is accomplished, we will compute the fidelity F(ρ, σ) = Tr ρ 1/2 σρ 1/2 between Alice's secret state and Bob's last-qubit state.If F = 1, this means a perfect teleportation.measurement outcome definition Bloch vector s of Bob's 6 th -qubit state  In Table I  Then, it is straightforward to compute the fidelities F 2 j = F 2 (ρ A , σ j,B ), where ρ A = |ψ A ψ|, whose explicit expressions are in the form: If α is real and β = √ 1 − α 2 e iφ , Eq. (3.6) becomes The θ-dependence of F 2 j is plotted in Fig. 5a when α = 1/ √ 3 and φ = 0.This figure shows that F 2 j do not reach to 1 at θ = π/2 except j = 0.In fact, this can be expected from Eq. (2.3).
In order to increase the fidelities at θ = π/2 we define Then, F 2 j ≡ F 2 (ρ A , σ j,B ) becomes The θ-dependence of F 2 j is plotted in Fig. 5b when α = 1/ √ 3 and φ = 0.As expected, this figure shows that all F 2 j approach to 1 at θ = π/2, which indicates the perfect teleportation in the maximal scrambling unitary (2.1).In Fig. 6a we plot the θ-dependence of the average fidelity defined when α = 1/ √ 3 and φ = 0.It approaches to 0.5 and 1 when θ = 0 (no scrambling) and θ = π/2 (maximal scrambling).The monotonic increasing behavior of F 2 avg supports the conjecture "the quantity of scrambling is proportional to the fidelity of quantum teleportation".In Fig. 6b we plot the φ-dependence of F  In this subsection we assume that Daniel takes {0, 5} qubits as a Bell measurement.From Eq. (3.3) it is straightforward to show that the probability Q j for each outcome is

measurement outcome definition
Bloch vector s of Bob's final state Table II: Bob's 6 th -qubit state.The quantities x 1 ,x 2 , x 3 ,, y 1 , y 3 , z 1 and z 3 are explicitly given in Eq. (3.12).
After taking a partial trace over remaining qubits, Bob's 6 th -qubit state, ρ j,B , for each measurement outcome is summarized in Table II, where In order to increase the fidelities at θ = π/2 we define It is interesting to note that ρ 0,B and ρ j,B (j = 1, 2, 3) reduce to ρ A = |ψ A ψ| at θ = 0 and π/2.Therefore, the fidelities F 2 0 ≡ F 2 (ρ A , ρ 0,B ) and F 2 j ≡ F 2 (ρ A , ρ j,B ) should be one at both θ = 0 and π/2.The explicit expressions of those fidelities are

IV. NUMERICAL SIMULATION
In order to examine the noise effect we compute the fidelities numerically in this section by making use of the qiskit and 7-qubit real quantum computer ibm oslo, and compare them with the theoretical results.First, we assume that Alice's secret state is |ψ A =

1 3 |0 + 2 3 1 3
|1 .In order to compute the fidelities numerically we prepare a quantum circuit of Fig.8a.In this figure θ = 1.5 is chosen and we assume that Daniel chooses {2, 3} qubits for Bell measurement.In the circuit the gates of purple color represent U and U * presented in Eq. (3.1).The numerical experiment is repeated 10 3 times and we compute their average value.Next, we take |ψA = |0 + 2 3 e iφ |1 .In this case we should prepare a quantum The 1 th -and 2 th -qubits are Charlie's qubits.Thus, unitary operator U scrambles the quantum information of Alice's and Charlie's qubits.The 3 th -and 4 th -qubits denote Daniel's qubits.Finally, the 5 th -and 6 th -qubits are Bob's ancillary qubits2.The vertical lines in Fig.1bmeans |β 0 too.Of course, U * is a complex conjugate of unitary U .Here, we assume that Daniel can access to all parties.Therefore, Daniel can select the unitary operator U and quantum measurement freely.Then, the question is as follows: is it possible to teleport Alice's qubit |ψ A to Bob's 6 th qubit if Daniel selects U

Table I :
Bob's 6 th -qubit state for each measurement outcome.The quantities a, a ± , b 1 , b 2 , c, d 1 and d 2 are explicitly given in Eq. (3.5).
Bob's 6 th -qubit state is summarized for each measurement outcome, where As expected, this figure exhibits oscillatory behavior.In Fig 6 the red crossing and blue dot are numerical results computed by qiskit and ibm oslo.This will be discussed in next section.