Quantum limits to dynamical evolution

For a pure state with mean energy E and energy spread ΔE, the minimum time to reach fidelity ε is bounded below by T_ε(E,ΔE).

“Assuming (without loss of generality) zero ground-state energy eigenvalue, it is possible to prove that the minimum time at which this happens is bounded by the quantity”

• AssumptionThe authors state this as an assumption: The Hamiltonian H has zero ground-state energy, and all subsystem Hamiltonians H_k are redefined to have zero ground states without loss of generality.

The only separable pure states of the form (21) that can reach the quantum speed limit (3) for some ε are those in which all the energy spread ΔE is carried by a single subsystem k′; the other subsystems are in eigenstates of their Hamiltonians.

“In conclusion, the only states | Ψ 〉 of the form (21) that can reach the quantum speed limit (3) for some value of ǫ are those in which all the energy spread ∆ E is carried by the single subsystem k ′ . The other subsystems are in eigenstates of their Hamiltonians H k .”

• AssumptionThe authors state this as an assumption: The Hamiltonian H has zero ground-state energy, and all subsystem Hamiltonians H_k are redefined to have zero ground states without loss of generality.
• AssumptionThe authors state this as an assumption: The composite system consists of M non-interacting subsystems with Hamiltonian H = ∑_k H_k.

For a suitable choice of ξ, the entangled state |Ψ_ξ〉 achieves the quantum speed limit bound.

“However, unlike the separable state | Ψ s 〉 discussed before, for a suitable choice of the parameter ξ , | Ψ ξ 〉 achieves the quantum speed limit bound, as can be shown by comparison with the state | Ω ξ 〉 of Eq. (A7) in App. A 1.”

The first time t for which P(t) = ε for the state |Ω_ξ〉 is given by Eq. (A8).

“Solving the dynamical evolution of the state | Ω ξ 〉 , one can show that the first time t for which P ( t ) = ǫ is given by”

In each experimental run, only one of the subsystems evolves.

“Since the above derivation applies for any expansion p n , one can say that in each experimental run only one of the subsystems evolves.”

Given E and ΔE, the quantum speed limit defines a forbidden evolution regime where the probability P(t) is not allowed to enter for 0 ≤ t ≤ T_0(E,ΔE).

“Given E and ∆ E , the quantum speed limit defines a forbidden evolution regime where the probability P ( t ) is not allowed to enter. In fact, for 0 ⩽ t ⩽ T 0 ( E, ∆ E ), Eq. (3) implies”

The functions α(ε) and β^2(ε) are strictly subadditive.

“The functions α ( ǫ ) and β 2 ( ǫ ) are strictly subadditive, i.e. for ǫ k ∈ [0 , 1],”

The first derivative of P(t) is limited only by the energy spread ΔE.

“For the same reason, the bound (A12) on the first derivative of P ( t ) is not modified by the presence of the Margolus-Levitin type contribution of Eq. (6): | dP ( t ) /dt | is limited only by the energy spread ∆ E .”

If the system is in the Margolus-Levitin regime, then the single subsystem k′ carries also all the mean energy E, and the other subsystems are in their ground states.

“Moreover, if the system is in the Margolus-Levitin regime, then k ′ carries also all the mean energy E , the other subsystems being in their ground states.”

• AssumptionThe authors state this as an assumption: The Hamiltonian H has zero ground-state energy, and all subsystem Hamiltonians H_k are redefined to have zero ground states without loss of generality.

The ratio R(ε) between t_ε and T_ε(E,ΔE) for homogeneous separable states satisfies R(ε) ≥ 1.

“From the subadditivity properties (A16) and (A17), it follows that R ( ǫ ) ⩾ 1.”

Minimizing over ξ the right term of Eq. (A8) gives an upper bound for α(ε).

“Minimizing over ξ the right term of Eq. (A8) gives the following upper bound for α ( ǫ ), i.e.”

The only separable mixed states that reach the quantum speed limit bound (7) are those for which, in each statistical realization n of the mixture, a single subsystem evolves to an orthogonal configuration at its own maximum speed limit time (which coincides with T(E,ΔE) of the whole system), and all other subsystems do not evolve.

“This means that the only separable states ̺ that reach the bound (7) are those for which for any statistical realization n of the mixture (36), a single subsystem evolves to an orthogonal configuration at its own maximum speed limit time (which coincides with T ( E, ∆ E ) of the whole system). All the other subsystems do not evolve.”

• AssumptionThe authors state this as an assumption: The composite system consists of M non-interacting subsystems with Hamiltonian H = ∑_k H_k.

For ε = 0, R(ε) is always greater than √M.

“In particular, for ǫ = 0, R ( ǫ ) is always greater than √ M , as discussed in [6].”

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The quantum speed limit bound T_ε(E,ΔE) extends to mixed states: it gives the lower bound to the time for a state ̺ with energy E and spread ΔE to evolve to a configuration ̺(t) with fidelity F(̺,̺(t)) = ε.

“We will show that the notion of quantum speed limit bound (3) can be extended to the density matrices in the sense that T ǫ ( E, ∆ E ) gives the lower bound to the time it takes for a state ̺ with energy E and spread ∆ E to evolve to a configuration ̺ ( t ) such that”

The bound (A13) is achievable, for example by the state |Ω_{ξ=1/√2}〉.

“Notice that the bound (A13) is achievable since, for example, the state | Ω ξ =1 / √ 2 〉 of Eq. (A7) reaches it.”

The functions α(ε^2) and β^2(ε^2) are strictly convex.

“The functions α ( ǫ 2 ) and β 2 ( ǫ 2 ) are strictly convex, i.e. for ǫ n ∈ [0 , 1],”

The entangled state |Ψ_ξ〉 represents a homogeneous configuration where each subsystem has energy ξ^2 E_0 and spread ξ^2 √(1−ξ^2) E_0.

“The state | Ψ ξ 〉 represents an homogeneous configuration where each subsystem has energy ξ 2 E 0 and spread ξ 2 √ 1 -ξ 2 E 0 .”

There exist counterexamples showing that interpreting T_ε(E,ΔE) as the minimum time to reach Tr[̺′̺] = ε Tr[̺^2] does not work when ε = 0.

“However, it is possible to find counterexamples that show that this definition does not work when ǫ = 0. For example consider the mixture”

For homogeneous separable states with E_k = E/M and ΔE_k = ΔE/√M, the minimum time t_ε for the global state to rotate by ε is bounded below by T_{ε^{1/M}}(E/M, ΔE/√M).

“Applying the quantum speed limit (3) to each subsystem, one obtains that t ǫ ⩾ T ǫ 1 /M ( E/M, ∆ E/ √ M ).”

• AssumptionThe authors state this as an assumption: The composite system consists of M non-interacting subsystems with Hamiltonian H = ∑_k H_k.

For ε = 0, the condition F(̺,̺(t)) = 0 is equivalent to Tr[̺(t)̺] = 0.

“Notice, finally, that for ǫ = 0 we reobtain all the results that were given in [6], since in this case the condition (7) is equivalent to the condition Tr[ ̺ ( t ) ̺ ] = 0 that was employed there [8].”

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The smallest time t for which P(t) = ε is bounded by β(ε)πħ/(2ΔE).

“This means that the smallest time t for which P ( t ) = ǫ is bounded by the quantity β ( ǫ ) π ℏ / (2∆ E ) with β ( ǫ ) defined in (5).”

The fastest mixed states that reach the quantum speed limit are mixtures composed of pure states that all achieve the quantum speed limit bound for the same ε at the same time.

“This shows that the fastest states ̺ are mixture composed by pure states | φ n 〉 that all achieve the quantum speed limit bound for the same ǫ at the same time.”

Margolus and Levitin pointed out that characteristic times of a system can be related to the average energy E of the initial state.

“More recently, however, Margolus and Levitin have pointed out that one can relate the characteristic times of the system also to the average energy E of the initial state [5].”

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In previous work, entanglement was shown to be a fundamental resource to reach the quantum speed limit when the energy resources E or ΔE are distributed among subsystems.

“In this context we showed that entanglement is a fundamental resource to reach the quantum speed limit when the energy resources ( E or ∆ E ) are distributed among the subsystems.”

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Entanglement is a fundamental resource to achieve speedup in the dynamical evolution in a composite system.

“Proving that homogeneous separable states cannot exhibit speedup while at least one homogeneous entangled case that exhibits speedup exists, we have shown that entanglement is a fundamental resource in this context.”

Quantum correlations among subsystems allow the state of the system to rotate in Hilbert space faster if the energy resources are not devoted to a single subsystem.

“Quantum correlations among subsystems allow the state of the system to rotate in Hilbert space faster if the energy resources are not devoted to a single subsystem.”

The fidelity F satisfies the purification property stated in prior work.

“The fidelity F has been shown [7, 9] to satisfy the property”

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