Quantum Speed Limit for Physical Processes

It is possible to extend the concept of minimal evolution time between two distinguishable states to nonunitary processes, using an attainable lower bound connected to the quantum Fisher information for time estimation.

“Here we show that it is possible to extend this concept to nonunitary processes, using an attainable lower bound that is connected to the quantum Fisher information for time estimation.”

For a dephasing channel, the bound yields very good lower bounds for the minimal evolution time between two nonorthogonal states.

“For a dephasing channel, it yields very good lower bounds for the minimal evolution time between two nonorthogonal states.”

For the above process, the distance of the evolved state from the initial state can reach at most √<ΔZ^2> π/4.

“Notice that, for the above process, the distance of the evolved state from the initial state can reach at most √ 〈 ∆ ˆ Z 2 〉 π/ 4.”

Eq. (14) captures the fact that the evolved and initial states never become orthogonal for r under a critical value rcrit ≃ 2.6.

“Another feature of (14) is that it captures the fact that the evolved and initial states never become orthogonal for r under a critical value r crit ≃ 2 . 6 [47].”

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For the amplitude-damping channel, the distance of the evolved state from the initial state can reach at most <σ+σ-> π/2.

“Notice that, for the above process, the distance of the evolved state from the initial state can reach at most 〈 ˆ σ + ˆ σ -〉 π/ 2.”

The more tractable bound based on purification procedure leads to attainable upper bounds for the quantum Fisher information.

“For situations when the general bound is too hard to evaluate, we have introduced a more tractable bound, based on a purification procedure that leads to attainable upper bounds for the quantum Fisher information.”

For N-qubit dephasing, the bound on minimum evolution time is given by Eq. (18).

“where α ( t ), β ( t ), and δ ( t ) are optimization variables. We get then [47]”

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The bound (14) stays close to the exact result up to the first minimum of the latter.

“This bound is compared to an exact calculation of D in Fig. 1, which shows that it stays close to the exact result up to the first minimum of the latter.”

For separable states, the lower bound goes from a τ ∼ 1/√N dependence for γ√N ≪ ω0 to ∼1/N for γ√N ≫ ω0.

“For separable states, on the other hand, the lower bound goes from a τ ∼ 1 / √ N dependence for γ √ N ≪ ω 0 to ∼ 1 /N for γ √ N ≫ ω 0 , as shown in Fig. 2.”

• AssumptionRests on an assumption: For a separable state, q ≤ 1/N (equality if symmetrical on the N qubits).

Eq. (14) consistently guarantees the eigenstates of Z not to evolve.

“Eq. (14) consistently guarantees the eigenstates of ˆ Z not to evolve.”

Eq. (15) saturates for pure initial states with <ΔZ^2>=1.

“which saturates for pure initial states with 〈 ∆ ˆ Z 2 〉 = 1.”

For any upper bound CQ(t) to FQ(t), one can obtain an implicit lower bound to the evolution time τ, given by Eq. (6).

“Then, for any upper bound C Q ( t ) to F Q ( t ), one can obtain an implicit lower bound to the evolution time τ , given by”

A general lower bound on the time τ necessary for a quantum system evolving under some physical process to reach a final state at distance D from its initial state is given by Eq. (3), valid for arbitrary physical processes and any metric dependent on the Bures fidelity.

“One gets then a general implicit lower bound on the evolution time τ , valid for arbitrary physical processes and any metric dependent on the Bures fidelity (see Supplemental Material [47]):”

This transition to faster behavior can be corroborated via direct calculations on symmetric, separable states.

“This transition to faster behavior can be corroborated via direct calculations on symmetric, separable states [47].”

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• AssumptionRests on an assumption: For a separable state, q ≤ 1/N (equality if symmetrical on the N qubits).

For the amplitude-damping channel, τ is bounded by Eq. (10).

“Setting ˆ u E ( t ) as an identity operator and inserting the variance of ˆ H S,E ( t ), obtained from the unitary operator (9) via definition (5), into (6), it is straightforward to show that τ is bounded by”

The bound in Eq. (3) is invariant under rescaling D' = kD.

“Notice that the square root on the l.h.s. of the above inequality is proportional to the curvature of D ( F B ) at F B = 1 and that this bound is invariant by a rescaling D ′ = kD .”

For Markovian dephasing, the minimum of CQ(t) over all Hermitian 2x2 operators hE(t), written in terms of the constant variance of Z, is given by Eq. (13).

“In order to find the best possible bound on τ within our approach, we now minimize C Q ( t ) over the whole set of Hermitian 2 × 2 operators ˆ h E ( t ). The minimum, written in terms of the (constant) variance of ˆ Z , is [47]”

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For a time-independent Hamiltonian, inequality (4) leads to the Mandelstam-Tamm bound: τ ≥ (πħ/2)/√<ΔH^2>.

“For a time-independent Hamiltonian ˆ H , with ˆ U ( t ) = e -i ˆ Ht/ ℏ , then ˆ H ( t ) = ˆ H , and for D = π/ 2 (orthogonal states), inequality (4) leads to the Mandelstam-Tamm bound: τ ≥ ( π ℏ / 2) / √ 〈 ∆ ˆ H 2 〉 .”

This geodesic connection remains valid for any monotonically decreasing P(t) with P(0)=1.

“This remains valid for any monotonically decreasing P ( t ), with P (0) = 1, since, under this condition, changing the form of Θ( t ) in (9) corresponds to a mere rescaling of the time parameter, which does not change the path in state space followed by a given initial state.”

For γ→0, inequality (14) yields the Mandelstam-Tamm bound, since the process becomes unitary.

“In the extreme cases γ → 0 and ω 0 → 0, inequality (14) yields simple analytical expressions for the bound on the minimal time. For the former, the Mandelstam-Tamm bound [2, 6] is recovered, since the process becomes unitary;”

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The bound is attained if and only if the evolution occurs on a geodesic.

“The above discussion makes it clear that this bound is attained if and only if the evolution occurs on a geodesic, which is, incidentally, the same condition for attainability of the Mandelstam-Tamm bound for unitary processes with time-independent Hamiltonians [14].”

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The quantum Fisher information of S+E is an upper bound to the quantum Fisher information of system S.

“The quantum Fisher information of S + E is an upper bound to the quantum Fisher information of system S , since, from the point of view of parameter-estimation theory, S + E does not contain less information about the parameter t than S alone.”

Under the condition γ√N ≫ ω0, the evolution speeds of separable and entangled states scale in the same way with respect to the number of qubits.

“Under this condition, the evolution speeds of separable and entangled states scale in the same way with respect to the number of qubits.”

• AssumptionRests on an assumption: For a separable state, q ≤ 1/N (equality if symmetrical on the N qubits).

In this situation, pure Markovian dephasing (ω0→0) links a pure state to a fully mixed state through a geodesic path in state space.

“In this situation, pure Markovian dephasing ( ω 0 → 0) links a pure state to a fully mixed state through a geodesic path in state space.”

The general result allows the estimation of the impact of the environment on the speed of quantum computation and information processing.

“Our general result allows the estimation of the impact of the environment on the speed of quantum computation and information processing.”

The values q=1 and <ΔZ^2>=1, achievable for GHZ states, yield a lower bound valid for any initial state.

“The values q = 1 and 〈 ∆ ˆ Z 2 〉 = 1, achievable for GHZ states [ | 0 . . . 0 〉 + e iφ | 1 . . . 1 〉 ] / √ 2, yield a lower bound valid for any initial state.”

It is always possible to fulfill CQ(t)=FQ(t) by minimizing CQ(t) over all operators uE(t), for given U_S,E(t).

“In fact, as it was shown in [50, 51], it is always possible to fulfill this condition by minimizing C Q ( t ) over all operators ˆ u E ( t ), for given ˆ U S,E ( t ).”

• cites [50] B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, Phys. Rev. Lett. 109 , 190404 (2012). sourceNot checked yet

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• cites [51] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Phys. 7 , 406 (2011); Braz. J. Phys. 41 , 229, (2011). sourceNot checked yet

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The bound based on CQ(t) can only be tight when CQ(t)=FQ(t), in which case it reduces to bound (4).

“However, it can only be tight when C Q ( t ) = F Q ( t ), in which case it reduces to bound (4).”

The saturation of bound (10) implies that the amplitude-damping channel connects the states |1> and |0> along a geodesic path, which includes mixed states.

“Furthermore, the fact that the bound is saturated implies that the amplitude-damping channel connects the states | 1 〉 and | 0 〉 along a geodesic path, which includes mixed states.”

For N-qubit dephasing, the evolution speed-up due to entanglement of its subsystems, previously demonstrated for unitary evolution, is shown to hold, in the nonunitary case, also for separable states.

“For N -qubit dephasing, the evolution speed-up due to entanglement of its subsystems, previously demonstrated for unitary evolution, is shown to hold, in the nonunitary case, also for separable states.”

• AssumptionRests on an assumption: For a separable state, q ≤ 1/N (equality if symmetrical on the N qubits).

This result invalidates claims that there is no general bound valid for all possible (unitary and nonunitary) quantum evolutions.

“This result invalidates claims that there is no general bound valid for all possible (unitary and nonunitary) quantum evolutions [40].”

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Eq. (4) also recovers the known implicit bounds on τ for time-dependent Hamiltonians.

“Eq. (4) also recovers the known implicit bounds on τ for time-dependent Hamiltonians [9, 36].”

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For unitary evolution, the quantum Fisher information is FQ(t)=4<ΔH^2(t)>/ħ^2.

“In this case, F Q ( t ) = 4 〈 ∆ ˆ H 2 ( t ) 〉 / ℏ 2 [49], where 〈 ∆ ˆ H 2 ( t ) 〉 is the variance in the initial state of a Hermitian operator ˆ H ( t ) defined as”

• cites [49] S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Phys. Rev. Lett. 98 , 090401 (2007). sourceNot checked yet

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The general bound depends on the quantum Fisher information of the system, rather than the initial variance of the Hamiltonian of the system alone.

“It is important to note that our general bound depends on the quantum Fisher information of the system, rather than the initial variance of the Hamiltonian of the system alone.”