Partial Trace

The partial trace operation performs a measurement or a selective collapse on the specified register(s). It is the key means of extracting classical information in quantum algorithms.

Overview

Partial trace operators overview

Operator	Operation	Return value
PartialTrace	Randomly measure the specified register(s)	(measured_values, probability)
PartialTraceSelect	Collapse to specified values	probability
PartialTraceSelectRange	Collapse to within a specified range	probability

Physically, a partial trace “measures away” a register from the quantum state — after the measurement that register collapses to a definite value, while the remaining registers enter the corresponding conditional state according to the measurement outcome.

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PartialTrace (random measurement)

Operation: Performs a random measurement on the specified register(s) (sampling from the probability distribution), returning the measurement outcome and the corresponding probability.

Parameters: Register identifier(s) (a name string, a list of names, an ID, or a list of IDs).

Returns: (measured_values, probability)

• measured_values — the list of register values obtained from the measurement

• probability — the probability of this measurement outcome

import pysparq as ps

ps.System.clear()
ps.System.add_register("addr", ps.UnsignedInteger, 3)
ps.System.add_register("data", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("addr", 3)(state)

# Randomly measure the addr register
values, prob = ps.PartialTrace("addr")(state)
print(f"Measurement result: {values}, probability: {prob:.4f}")
# Example: Measurement result: [3], probability: 0.1250

# After the measurement addr has collapsed; only the corresponding basis state remains in state

When measuring multiple registers, the returned value list is ordered according to the parameter order:

values, prob = ps.PartialTrace(["addr", "data"])(state)

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PartialTraceSelect (selective collapse)

Operation: Collapses the specified register(s) to the given values instead of measuring randomly.

Parameters: A register-to-value mapping (dictionary or lists).

Returns: probability — the probability of this collapse outcome.

ps.System.clear()
ps.System.add_register("a", ps.UnsignedInteger, 2)
ps.System.add_register("b", ps.UnsignedInteger, 2)

state = ps.SparseState()
ps.Hadamard_Int("a", 2)(state)

# Collapse a to the value 1
prob = ps.PartialTraceSelect({"a": 1})(state)
print(f"Collapse probability: {prob:.4f}")
# Collapse probability: 0.2500

# A list form also works
prob = ps.PartialTraceSelect(["a"], [1])(state)

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PartialTraceSelectRange (range collapse)

Operation: Collapses the specified register to within a given value range, keeping the basis states that satisfy the condition and renormalizing.

Parameters: Register identifier and value range (min, max).

Returns: probability — the total probability within the range.

ps.System.clear()
ps.System.add_register("x", ps.UnsignedInteger, 4)

state = ps.SparseState()
ps.Hadamard_Int("x", 4)(state)

# Keep only the basis states with x ∈ [2, 5]
prob = ps.PartialTraceSelectRange("x", (2, 5))(state)
print(f"Probability within range: {prob:.4f}")
# Probability within range: 0.2500 (4 out of 16 basis states)