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Measure and remove quantum systems with project_traceout!

This tutorial shows how to use project_traceout! for discrete and continuous measurements. You will measure qubits and qumodes. You will also check the state that remains after each measurement.

project_traceout! does two operations. First, it projects the measured system onto a measurement outcome. This projection changes any system that is entangled with it. Then, it removes the measured system from the register slot.

For a discrete measurement, the function returns a one-based index into the measurement basis. The index is not an eigenvalue. For a homodyne measurement, the function returns continuous quadrature data.

The first few examples use the default QuantumOpticsRepr(). The homodyne example selects GabsRepr explicitly.

Measure a qubit with a Pauli operator

Start with a Bell state. Its two qubits have the same result in the $Z$ basis. Pass Z to select its eigenbasis, (Z1, Z2).

using QuantumSavory

qubits = Register(2)
bell = (Z1 ⊗ Z1 + Z2 ⊗ Z2) / sqrt(2)
initialize!(qubits[1:2], bell)

first_outcome = project_traceout!(qubits[1], Z)
partner_state = (Z1, Z2)[first_outcome]
partner_fidelity = observable(qubits[2], SProjector(partner_state))
second_outcome = project_traceout!(qubits[2], Z)

(
    measurements_match = first_outcome == second_outcome,
    partner_matches_first = isapprox(partner_fidelity, 1; atol = 1e-12),
    both_slots_are_empty = !isassigned(qubits, 1) && !isassigned(qubits, 2),
)
(measurements_match = true, partner_matches_first = true, both_slots_are_empty = true)

Each outcome is either 1 or 2. The two values are always equal because the Bell state has equal $Z$-basis results. The projector observable checks that the second qubit is in the state selected by the first outcome. Both slots are empty after the second measurement.

You can use X or Y in the same way. Each operator selects its own pair of eigenstates.

For CliffordRepr, symbolic X, Y, and Z are the currently supported measurement bases. The explicit basis-vector form in the next section is available with QuantumOpticsRepr and QuantumMCRepr.

Give the qubit basis explicitly

An explicit tuple or vector sets both the basis and the order of the outcomes. Measure a new Bell pair in the $X$ basis.

qubits = Register(2)
initialize!(qubits[1:2], bell)

x_basis = (X1, X2)
first_outcome = project_traceout!(qubits[1], x_basis)
partner_state = x_basis[first_outcome]
partner_fidelity = observable(qubits[2], SProjector(partner_state))
second_outcome = project_traceout!(qubits[2], x_basis)

(
    measurements_match = first_outcome == second_outcome,
    partner_matches_first = isapprox(partner_fidelity, 1; atol = 1e-12),
)
(measurements_match = true, partner_matches_first = true)

The Bell pair is also correlated in the $X$ basis. If the first outcome is 1, the selected state is X1. If it is 2, the selected state is X2. The projector observable and the second measurement both confirm the correlation without requiring either value.

Use this form when you need direct control of the basis order. The basis states must form a complete orthonormal basis for the measured system.

Measure photon number in a qumode

A qumode has a Fock basis of photon-number states. This example uses the default QuantumOpticsRepr(), which has Fock states F0, F1, and FockState(2). The prepared pair has either zero photons in both modes or one photon in both modes.

modes = Register(fill(Qumode(), 2))
mode_pair = (F0 ⊗ F0 + F1 ⊗ F1) / sqrt(2)
initialize!(modes[1:2], mode_pair)

fock_basis = (F0, F1, FockState(2))
first_outcome = project_traceout!(modes[1], fock_basis)
partner_state = fock_basis[first_outcome]
partner_photon_number = real(observable(modes[2], N))
partner_fidelity = observable(modes[2], SProjector(partner_state))
second_outcome = project_traceout!(modes[2], fock_basis)

(
    populated_outcome = first_outcome in (1, 2),
    measurements_match = first_outcome == second_outcome,
    photon_number_matches = isapprox(
        partner_photon_number,
        first_outcome - 1;
        atol = 1e-12,
    ),
    partner_matches_first = isapprox(partner_fidelity, 1; atol = 1e-12),
)
(populated_outcome = true, measurements_match = true, photon_number_matches = true, partner_matches_first = true)

The first outcome can select F0 or F1. It cannot select FockState(2) because that state has zero amplitude in the prepared pair. The photon-number observable is zero after an F0 result and one after an F1 result. The two mode measurements have the same basis index.

Pass the Fock states explicitly for this measurement. N is available as an observable, but it is not currently available as the basis argument of project_traceout!.

Measure a continuous quadrature

A homodyne measurement returns a continuous value. Use 0.0 as the angle for the $x$ quadrature. Use pi / 2 for the $p$ quadrature. This operation is available for qumodes that use GabsRepr.

The measured coherent state and the remaining vacuum state are independent in this small example. The homodyne result is random, but the remaining mode must still have zero photons.

using Gabs

gabs_repr = GabsRepr(QuadBlockBasis)
modes = Register(fill(Qumode(), 2), fill(gabs_repr, 2))
initialize!(modes[1:2], CoherentState(0.3 + 0.2im) ⊗ F0)

result = project_traceout!(
    modes[1],
    HomodyneMeasurement([0.0]; squeeze = 1e-12),
)

# GabsRepr does not yet support `observable`. Convert a copy for this check.
remaining_state = copy(QuantumSavory.stateof(modes[2]).state[])
check_state = express(remaining_state, QuantumOpticsRepr())

(
    result_has_x_and_p = length(result) == 2,
    measured_x_is_finite = isfinite(result[1]),
    measured_slot_is_empty = !isassigned(modes, 1),
    remaining_mode_is_vacuum = isapprox(
        real(observable(check_state, [1], N)),
        0;
        atol = 1e-12,
    ),
)
(result_has_x_and_p = true, measured_x_is_finite = true, measured_slot_is_empty = true, remaining_mode_is_vacuum = true)

For one mode, the result contains $x$ and $p$ phase-space values. Index 1 is the measured $x$ value in this example. For a $p$ measurement, use angle pi / 2 and read index 2. The other value is the conjugate quadrature in the finite-squeezing approximation.

The sampled $x$ value can change between runs. The example checks only that the result has the expected form and contains a finite measured value. The final observable confirms that the independent second mode remains in the vacuum state. The conversion is only a check on a copy of the state. It does not change the register.

What to carry forward

  • Pass X, Y, or Z for a qubit Pauli measurement.
  • With QuantumOpticsRepr or QuantumMCRepr, pass an ordered tuple or vector for an explicit discrete basis.
  • Use an explicit Fock basis for a discrete qumode measurement with these representations.
  • Pass HomodyneMeasurement to GabsRepr for a continuous qumode measurement.
  • Use the returned discrete index to select the matching basis state.
  • Remember that the measured slot is empty after every successful call.

For exact signatures, see the Register Interface. For backend limits, see Backend Simulators.