Phase-preserving steady-state operations beyond Lindblad master equations

The gist: This work implements a class of phase-preserving non-unitary operations using a single time-independent Hamiltonian coupled to an infinite ancilla, proving these operations are not describable as maps from initial to steady state of a time-independent Lindblad equation.

Class of Non-Unitary Operations

The paper defines a class of quantum operations denoted by O [ρA] = UQQρAQU†Q + UP P ρAP U†P (1). This operation is defined for a general Hilbert space HA and includes examples such as the decay operation where P = 1⟩⟨1, UP = σx, and Q = 1, resulting in Odecay [ρA] = 1 0 / 0 0 (2). The class of operations in eq. (1) cannot be implemented as a map from initial to steady state of a time-independent Lindblad operator because this would require L[O[ρA]] = 0 for all ρA, which leads to a contradiction when considering specific conditions on UP and UQ (2).

Spectral Switching Mechanism

The operations are realized via coupling the system to an infinite-dimensional ancilla HB to reach a steady state in HA, utilizing a single time-independent Hamiltonian H = HA + Hint (3). The core idea is that the system Hamiltonian HA allows or suppresses the interaction between states mediated by Hint through spectral resonance (6). The energy difference µ = EP −EQ and normalized values µB = µ/B and CB = C/B are used to define a matrix hX (7) whose spectrum determines whether an operation is active or suppressed. The spectral criterion (8) provides the condition for hX to not have discrete energy levels, which acts as a programmable selector for measurement-and-feedforward operations (8).

Spectral Switching Theorem

The Spectral Switching Theorem guarantees that when the spectral criterion (8) is met, the time-evolved expectation values ⟨nj (t)⟩ have the limit lim t→+∞ ⟨nj (t)⟩ = 0 for any initial state ψx⟩ ∈ HX (9). This theorem is used to prove that the steady state of the operation O[ρA] is achieved, as it shows that ρA(t → ∞) = QψA⟩⟨ψAQ + UP PψA⟩⟨ψAP U†P (13). This construction generalizes to starting from a mixed state ρ(t = 0) = ρA,0 ⊗ 0⟩⟨0, yielding the same steady-state form (14).

Phase Preserving Steady States

The framework allows for the implementation of phase-preserving operations by generalizing eq. (1) using an extended ancilla Hilbert space HB with positive and negative indices to define states xψA0⟩ and yψA0⟩ (20). The action of the restricted Hamiltonian HX on these subspaces is described by a matrix similar to hX, allowing the application of the Spectral Switching Theorem (5). When the spectral criterion applies, both ⟨nX0(t → ∞)⟩ = 0 and ⟨nY0(t → ∞)⟩ = 0, resulting in a phase-preserving steady state where ρA(t → ∞) = UP PψA⟩⟨ψAP U†P + UQQψA⟩⟨ψAQU†Q (23).

Non-Lindbladian Proof and Robustness

The paper proves that the general operation eq. (1) cannot be implemented as a map from initial to steady state of a time-independent Lindblad operator by considering a subset of operations where UP and UQ are block diagonal in the P and Q subspaces (24). This requires uP = 1P up to an irrelevant global phase, showing that any operation with uP ≠ 1P or uQ ≠ 1Q is outside the class of Lindblad steady state operations (26, 27, 28). Furthermore, the robustness of phase preservation is demonstrated in a four-dimensional system example where coherence terms ρ(0,1)A and ρ(2,3)A remain close to their target values even in the presence of disorder (30).

Conclusion and Costs

The construction provides a minimal, time-independent Hamiltonian that implements measurement-and-feedforward operations without measurement or classical feedback (V). The key features are: no external timing is required, the operation is spectrally addressable via criterion (8), and the steady states include phase-preserving operations provably not arising from any time-independent Lindblad equation. However, there are two costs: the ancilla must be prepared in a definite initial state, and in a finite ancilla of size L, the steady state is only an intermediate-time plateau (V).

How it works

1. The operation O [ρA] = UQQρAQU†Q + UP P ρAP U†P is defined to be implemented without measurement by coupling the system to an infinite set of ancilla states (1).

2. A single time-independent Hamiltonian H = HA + Hint is used, where HA acts in the main subsystem and Hint implements the desired unitary on just a specific state i⟩ (6).

3. The dynamics are analyzed in a projected Hilbert space HX, whose Hamiltonian hX has an absolutely continuous spectrum in [-1, 1] determined by parameters µB and CB (7).

4. The Spectral Switching Theorem guarantees that for states within the spectral criterion (8), the time-evolved expectation values ⟨nj (t)⟩ decay to zero as t → +∞, leading to a steady state operation O[ρA] = QψA⟩⟨ψAQ + UP PψA⟩⟨ψAP U†P when the criterion is met (13).

5. The sharpness of the switch is quantified by the closed-form expression in eq. (8), which shows that as CB → 0, the steady state becomes sharper, with ⟨n0(t → ∞)⟩ = 0 for µB 2 (17).

Phase Preserving Steady States

The generalized implementation uses a Hamiltonian Hint that couples states in a chain-like way, leading to the decomposition of the restricted Hilbert space into HX + HY (5). By utilizing the Spectral Switching Theorem on this structure, the steady state is reached when both components decay to zero (5). The resulting steady state operation preserves relative phase information between specific pairs of states like 0⟩ and 1⟩ or 2⟩ and 3⟩ in a four-dimensional system example (30).

Non-Lindbladian Proof

The argument against Lindblad implementability relies on restricting the operations to those where UP and UQ are block diagonal in the P and Q subspaces (24). For these restricted cases, achieving a steady state requires uP = 1P and uQ = 1Q, meaning any operation with uP ≠ 1P or uQ ≠ 1Q is outside the Lindblad class (28).

Robustness of Phase Preservation

The robustness of phase preservation is shown by studying operations defined in a four-dimensional space where P = 0⟩⟨0+1⟩⟨1 and Q = 2⟩⟨2 + 3⟩⟨3, with UP = σx ⊕ 1Q and UQ involving σx and σz (6). Numerical analysis shows that the relative phases ρ(0,1)A and ρ(2,3)A remain close to their target values throughout the window in which the steady state holds, even when disorder is included (30).

Conclusion

This work introduces a minimal time-independent Hamiltonian that implements a class of measurement-and-feedforward operations as a map from an initial state to a local steady state (V). The construction is distinguished by being measurement-free, spectrally addressable via criterion (8), and achieving phase-preserving steady states that cannot arise from any time-independent Lindblad equation (V).

How it works

1. The operation O [ρA] = UQQρAQU†Q + UP P ρAP U†P is defined to be implemented without measurement by coupling the system to an infinite set of ancilla states (1).

2. A single time-independent Hamiltonian H = HA + Hint is used, where HA acts in the main subsystem and Hint implements the desired unitary on just a specific state i⟩ (6).

3. The dynamics are analyzed in a projected Hilbert space HX, whose Hamiltonian hX has an absolutely continuous spectrum in [-1, 1] determined by parameters µB and CB (7).

4. The Spectral Switching Theorem guarantees that for states within the spectral criterion (8), the time-evolved expectation values ⟨nj (t)⟩ have the limit lim t→+∞ ⟨nj (t)⟩ = 0 for any initial state ψx⟩ ∈ HX (9).

5. The sharpness of the switch is quantified by the closed-form expression in eq. (8), which shows that as CB → 0, the steady state becomes sharper, with ⟨n0(t → ∞)⟩ = 0 for µB 2 (17).

Phase Preserving Steady States

The generalized implementation uses a Hamiltonian Hint that couples states in a chain-like way, leading to the decomposition of the restricted Hilbert space into HX + HY (5).

1. textbfActive Autonomous Quantum Control via Spectral Switching: Implementing Non-Unitary Operations without Measurement or Classical Feedback (Eq. 1). Developing an AI system capable of executing state preparation or reset protocols (like qubit reset) autonomously by coupling a main system to an infinite ancilla, where the operation is controlled via a ‘spectral switch’ mechanism.

2. textbfRobust Phase Preservation Against Disorder: Achieving Autonomous Reset Fidelity in Noisy Environments. The improved system will maintain phase information during non-unitary operations even when subjected to moderate disorder, as demonstrated by the result that the phase preservation survives moderate disorder, over a time window set by the size of the ancilla.

3. textbfAdaptive Operation Switching: Implementing Measurement-and-Feedforward Operations with Tunable Control. The AI system will dynamically switch between executing a target unitary operation and suppressing it based on energy levels, as shown by the criterion (8) which acts as a switch that turns the operation on and off, allowing for precise control over the sharpness of this switch.

4. textbfBeyond Lindblad Mapping: Constructing Non-Lindbladian Steady States. The system can implement operations provably not describable as a mapping from initial to steady state of a time-independent Lindblad equation, providing access to quantum control strategies that bypass standard dissipative engineering limitations.

5. textbfProgrammable Quantum Dynamics via Hamiltonian Engineering: Implementing General Non-Unitary Maps (Eq. 1). The system can implement complex operations like O [ρA] = UQQρAQU†Q + UP P ρAP U†P by defining a single time-independent Hamiltonian, enabling the creation of a class of non-unitary operations that are not achievable through standard dissipative engineering.

We consider a class of nonunitary operations that is naturally implemented via a combination of projective measurement and unitary operators. We implement these operations without using measurements by coupling the system to an infinite set of ancilla states and time-evolving with a single time-independent Hamiltonian. The infinite ancilla enables the main system to reach a local steady state that preserves initial phase information. We prove that these steady-state operations are not describable as a mapping from an initial state to a steady state of a time-independent Lindblad master equation acting only on the main system. As an additional degree of control, we show that the spectral resonance between the system and the ancilla acts as a switch that turns the operation on and off, and we derive a closed-form expression that quantifies the sharpness of this switch. We further find numerically that the phase preservation survives moderate disorder, over a time window set by the size of the ancilla. This Hamiltonian framework therefore provides a route to autonomous quantum control using the tunability of a closed system.