Final State Projection: Foundations & Applications

• Final state projection is a method that imposes terminal constraints on quantum evolution by selecting outcomes compatible with a specified final configuration.
• It spans various domains including black-hole evaporation, quantum measurement, relativistic scattering, and trajectory optimization, each utilizing tailored projection operators.
• Its applications elucidate how conditioning quantum dynamics can reconcile unitarity, entanglement, and information recovery across diverse physical systems.

Searching arXiv for the papers on arXiv and closely related work on final state projection. Tool unavailable in this interface; proceeding using the supplied arXiv records and ids. Final state projection denotes a family of constructions in which a terminal condition selects those components of an evolution that are compatible with a specified final configuration. In the black-hole information literature, it denotes a fixed pure-state boundary condition at the spacelike singularity, implemented by postselection and yielding a teleportation-like map from infalling matter to outgoing Hawking radiation (Lloyd et al., 2013). In quantum measurement theory, it denotes the transformation of an initial superposition into an eigenstate conditioned on the measurement outcome (Kakuyanagi et al., 2010). In relativistic scattering, control theory, and quantum computing, the same phrase refers to Fock-space operators that select semi-inclusive final states, nonlinear projection operators that return candidate curves to the manifold of trajectories with prescribed terminal state, and ancilla-postselected filters that isolate symmetry sectors and energies (Dickinson et al., 2017, Notarnicola et al., 2017, Stetcu et al., 2022).

1. General formal structure

A common formal pattern is the imposition of a constraint on admissible outcomes, followed by renormalization or conditioning. In post-selection language, if the allowed histories form a linear subspace and $P$ is the projector onto that subspace, the update takes the form

$$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$

with the normalization factor $N(\rho)=\mathrm{Tr}(P\rho P)$ depending on the input state; this dependence is the source of nonlinearity in black-hole final-state models (Devin, 2014). In scattering theory, the same operational role is played by an effect operator $E$, with outcome probability

$$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$

so that a “final state projection” is an operator in Fock space selecting the measured outgoing sector while summing over unobserved quanta (Dickinson et al., 2017).

In standard quantum measurement theory, state projection is the transformation of an initial superposition

$|\psi\rangle = c_g|g\rangle + c_e|e\rangle$

into one of the eigenstates of the measured observable with probabilities $|c_g|^2, |c_e|^2$, conditioned on the measurement outcome (Kakuyanagi et al., 2010). A recurring source of confusion is that these uses are not identical. In the Horowitz–Maldacena setting, the projection is a fixed boundary condition at a singularity; in superconducting-qubit readout and in QED detection, it is standard Born-rule conditionalization; in control and optimization, it is a projection operator on trajectory space rather than on a Hilbert-space state.

2. Horowitz–Maldacena final-state projection and black-hole evaporation

In the Horowitz–Maldacena proposal, the black-hole Hilbert space is decomposed as

$$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$

where $\mathcal{H}_M$ describes infalling matter, $\mathcal{H}_{\rm in}$ the infalling Hawking modes behind the horizon, and $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$0 the outgoing Hawking radiation. The horizon is modeled by an Unruh state that is maximally entangled across $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$1,

$$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$2

while the singularity carries a fixed maximally entangled final state of $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$3 and $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$4,

$$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$5

For an initial matter state $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$6, the postselected contraction yields an outgoing state

$$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$7

so the effective S-matrix is unitary when the Unruh state and final state are appropriately matched (Lloyd et al., 2013).

The same analysis shows that exact unitarity requires fine-tuning. Entanglement across the horizon must be full rank and appropriately matched to the final state, and interactions involving $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$8 must be controlled so that the net circuit remains equivalent to pure teleportation. Yet the model does not require exact tuning to be nearly unitary: for a generic final-state boundary condition, deviations from unitarity are of order $$\rho \mapsto \frac{P \rho P}{\mathrm{Tr}(P \rho P)} , \qquad P = |\Psi_f\rangle \langle \Psi_f| ,$$9, and the fidelity of recovering the initial state from the radiation differs from $N(\rho)=\mathrm{Tr}(P\rho P)$0 by terms of order $N(\rho)=\mathrm{Tr}(P\rho P)$1 (Lloyd et al., 2013).

The principal conceptual significance of the model is that it relaxes several standard quantum-information constraints. On a spacelike slice, both the original matter $N(\rho)=\mathrm{Tr}(P\rho P)$2 and the outgoing radiation $N(\rho)=\mathrm{Tr}(P\rho P)$3 contain the same quantum information, which is cloning in the naively global Hilbert-space description. In the AMPS setup for an old black hole, a Hawking mode $N(\rho)=\mathrm{Tr}(P\rho P)$4 can be maximally entangled both with its interior partner $N(\rho)=\mathrm{Tr}(P\rho P)$5 and with the early radiation $N(\rho)=\mathrm{Tr}(P\rho P)$6, yielding polygamous entanglement. In this way the model reconciles unitarity of the outside S-matrix, smoothness of the horizon, and approximate validity of local EFT outside, at the price of giving up entanglement monogamy and strict linearity in the underlying dynamics. The same framework also permits chronology-violating circuit structures, though the computational complexity of decoding the Hawking radiation may render the resulting causality violation unobservable (Lloyd et al., 2013).

3. Variants, diagnostics, and phenomenology in black-hole settings

A more explicitly post-selected formulation treats the singularity as a spacelike boundary where all quantum fields take unique boundary values, equivalently a zero-entropy singularity whose internal degrees of freedom are constrained to lie in a single pure state $N(\rho)=\mathrm{Tr}(P\rho P)$7. In that language, the final-state projection is

$N(\rho)=\mathrm{Tr}(P\rho P)$8

and the state-dependent normalization factor is central. If $N(\rho)=\mathrm{Tr}(P\rho P)$9, no branch satisfies the boundary condition; approximate post-selection replaces exact paradoxes by histories of tiny prior probability, a mechanism described as paradox censorship. The same framework relaxes monogamy of entanglement, is explicitly nonlinear, and has been used to motivate speculative phenomenology such as suppression of black-hole formation, complementary incoming radiation, and “un-emissions” (Devin, 2014).

A distinct development studies final-state projection in two-dimensional conformal field theory. There one fixes an initial state $E$0, a postselected final state $E$1 at time $E$2, and defines at intermediate time $E$3 the pair

$E$4

For a subsystem $E$5, the reduced transition matrix is

$E$6

and the pseudo-entropy is

$E$7

The real part of pseudo-entropy was used to estimate the amount of quantum entanglement averaged over histories between the initial and final states, and it was found to display a Page curve-like behavior (Akal et al., 2021). This suggests that final-state projection can reproduce a rise–peak–fall structure for radiation-like entanglement diagnostics without invoking the standard entropy of a single evolving state.

A third line of work replaces literal post-selection by an emergent attractor. The black-hole Hilbert space is split as

$E$8

and unitarity at complete evaporation is argued to require a unique interior final state. Instead of imposing that state as a boundary condition, the proposal is that a UV theory with infinitely many fields drives all black-hole interiors toward an effectively unique final state by a mechanism analogous to dissipative quantum mechanics. The final state then plays the role of a final-state projector in Horowitz–Maldacena-type models, but as an emergent attractor rather than an imposed post-selection (Ho, 2021). This sharpens a central controversy: whether final-state uniqueness is fundamental boundary data or a consequence of unitary UV dynamics.

4. Measurement-induced final-state projection in superconducting qubits

In superconducting-circuit experiments using a Josephson bifurcation amplifier (JBA), final state projection denotes the state of the qubit after a readout pulse rather than a boundary condition at a singularity. The JBA is a nonlinear resonator with two classically stable oscillation states, low-amplitude and high-amplitude oscillations; because the qubit is magnetically coupled to the SQUID, the bifurcation threshold depends on the qubit state. The measurement pulse is the front part of the full readout pulse, with rise $E$9 ns followed by a $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$0 ns plateau, and its height $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$1 controls the measurement strength (Kakuyanagi et al., 2010).

The experimental protocol places the projection pulse between two qubit control pulses. If no projection occurs, the qubit remains coherent and

$$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$2

If full projection into the energy basis occurs after the first pulse, then

$$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$3

Intermediate cases are modeled by

$$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$4

where $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$5 is a projection indicator: $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$6 corresponds to fully coherent evolution, $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$7 to full projection, and intermediate $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$8 to partial projection or partial coherence (Kakuyanagi et al., 2010).

The central empirical result is that projection and visibility are not the same. For $$P = \mathrm{Tr}(E\,\rho_f) = \mathrm{Tr}(E\,U_{fi}\,\rho_i\,U_{fi}^\dagger) ,$$9–1.03, the JBA readout has nonzero visibility and $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$0, so the qubit has been projected into an energy eigenstate in an informative regime. For $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$1, the visibility returns to zero because both $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$2 and $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$3 drive the JBA to essentially the same final state, but $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$4 remains $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$5: the qubit has still been projected into an eigenstate even though the readout carries no usable information. For $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$6, the state is over-disturbed and dependence on the first control pulse disappears completely. The experiment therefore separates back-action from information gain and shows that final state projection can persist when asymptotic detector distinguishability vanishes (Kakuyanagi et al., 2010).

5. Final-state projection in relativistic scattering and entangled radiation states

In scattering theory, final-state projection is formulated directly in Fock space. For a bosonic field, the number operator counting quanta in a momentum-space region $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$7 is

$|\psi\rangle = c_g|g\rangle + c_e|e\rangle$8

and the projector onto zero quanta in $|\psi\rangle = c_g|g\rangle + c_e|e\rangle$9 is

$|c_g|^2, |c_e|^2$0

Its action is exact: it leaves a Fock state unchanged if there are zero quanta in $|c_g|^2, |c_e|^2$1, and annihilates it otherwise. More generally,

$|c_g|^2, |c_e|^2$2

projects onto states with exactly $|c_g|^2, |c_e|^2$3 particles in $|c_g|^2, |c_e|^2$4, inclusive over everything outside $|c_g|^2, |c_e|^2$5. These operators encode semi-inclusive measurements, acceptance regions, and veto conditions such as non-emission in specified subregions of momentum space (Dickinson et al., 2017).

A different but related use appears in Vavilov–Cherenkov radiation and in the equivalent-photon emission problem. There the evolved final state

$|c_g|^2, |c_e|^2$6

is first constructed as a unitary QED two-particle state before any detection or post-selection. In twisted-mode expansions, the final state is a coherent superposition of twisted electron and twisted photon modes with total angular momentum conservation enforced by Kronecker deltas such as $|c_g|^2, |c_e|^2$7 for a plane-wave initial electron and $|c_g|^2, |c_e|^2$8 for a twisted initial electron. A detector sensitive to a twisted electron with definite projection of total angular momentum therefore projects the photon onto a twisted state with definite total angular momentum projection fixed by that conservation law; the converse statement for photon detection is equally valid (Chaikovskaia et al., 2023).

This relativistic-QED example is notable because the evolution is strictly unitary and the final-state projection is nothing beyond the standard Born rule conditionalization on measurement outcomes. The phrase “final state projection” therefore covers both nonlinear post-selected dynamics in black-hole models and entirely standard measurement projections on entangled final states in QED. A plausible implication is that the shared terminology should be read structurally, not ontologically: what is common is the selection of an outgoing sector, not a universal mechanism.

6. Projection operators in trajectory optimization and quantum state preparation

In nonlinear optimal control, final-state projection is a projection operator acting on curves rather than a projector on a quantum state. The problem is

$|c_g|^2, |c_e|^2$9

subject to

$$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$0

Standard PRONTO provides a nonlinear projection operator $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$1 onto the manifold of trajectories satisfying the dynamics and initial condition. The final-state constrained extension constructs a constrained operator $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$2 that maps arbitrary state–input curves into the manifold $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$3 of trajectories satisfying both the dynamics and the hard equality $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$4. The algorithm alternates between solving a quadratic approximation of the nonlinear problem to obtain a descent direction satisfying the terminal constraint to first order, and projecting the candidate update back into $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$5. Its defining property is recursive feasibility: every intermediate iterate exactly satisfies the final-state constraint (Notarnicola et al., 2017).

On quantum computers, the “Quantum Projection Filter” implements final-state projection by ancilla postselection. With system operator $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$6 and ancilla Pauli $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$7, each filtering step applies

$$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$8

so that after measuring the ancilla and postselecting $$\mathcal{H}_M\otimes \mathcal{H}_{\rm in}\otimes \mathcal{H}_{\rm out},$$9, the system is mapped by

$\mathcal{H}_M$0

After $\mathcal{H}_M$1 successful steps, the filter is

$\mathcal{H}_M$2

which approximates a projector onto a desired symmetry sector or energy eigenspace. The method first isolates individual quantum numbers and then uses time evolution to isolate the energy; in its simplest form it requires only one additional auxiliary qubit. The total time evolved for an accurate solution is proportional to the ratio of the spectrum range of the trial state to the gap to the lowest excited state, and the accuracy increases exponentially with the time evolved. The success rate is dominated by the square overlap of the original state to the desired state (Stetcu et al., 2022).

Across these algorithmic settings, final-state projection no longer denotes a modification of quantum mechanics. It denotes an operator-level mechanism for returning an iterate, curve, or wavefunction to a terminally constrained manifold or eigenspace. This suggests a useful taxonomy: in black-hole physics the term names a boundary condition with deep implications for unitarity, entanglement, and causality; in laboratory quantum systems, scattering theory, control, and quantum algorithms it names concrete projection procedures whose role is to condition, filter, or enforce terminal feasibility.