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Tuesday, July 28, 2026

What Is a Quantum Observable?

The question, the possible outcomes, and the average estimated from many runs.

The question a quantum measurement asks

A quantum stateThe mathematical description used to predict the probabilities of different measurement outcomes for a quantum system. describes a system, but it does not automatically provide a readable answer. To extract information, we must decide what property to measure. That property is called an observableA measurable property represented mathematically by a Hermitian operator. One measurement returns one of its allowed outcomes; repeated measurements can be used to estimate its expectation value..1

The observable defines the question. One measurement gives one outcome. Repeated measurements estimate the expectation value.

An observable may represent energy, spin, particle number, magnetization, parityWhether a specified particle number, occupation number, or bit count is even or odd., or a correlation between qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once.. It is the mathematical object that defines the question being asked of the quantum state. Four ideas here are related without being interchangeable:

  • The quantum state describes the system.
  • The observable defines what is measured.
  • The measurement produces an outcome.
  • Repeated measurements can be used to estimate an expectation value.

Observables and Hermitian operators

In quantum mechanics, an observable is represented by a Hermitian operatorThe kind of mathematical object used to represent an observable, chosen because the outcomes it allows always come out as ordinary real numbers., usually written as A. A Hermitian operator satisfies:1

A  =  A†

Here A† is the conjugate transpose of A. The importance of that condition is physical, not merely mathematical. Hermitian operators have real eigenvaluesOne of the possible outcomes associated with measuring an observable. For a Hamiltonian, its eigenvalues are the system's allowed energies., and those eigenvalues represent the possible outcomes of an ideal projective measurementThe idealized textbook measurement: it returns one of the observable's allowed outcomes and leaves the system in a state associated with that outcome.. The Pauli-Z observable, for instance, has two eigenvalues: +1, associated with |0⟩, and −1, associated with |1⟩.

The observable does not itself force the state to collapseThe state update associated with a measurement outcome. In an ideal projective measurement, the state is projected into the eigenspace corresponding to that outcome.. It defines the possible outcomes and the states or eigenspacesThe set of states that all share the same measurement outcome for a given observable. associated with them. The physical measurementA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. process produces one of those outcomes and updates the quantum state accordingly. The observable therefore defines the landscape of possible answers. It does not determine which answer will appear in one particular run.

Outcome versus expectation value

A single measurementA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. gives one outcome. It does not usually reveal the full statistical behavior of the state. Consider the state:

|+⟩  =  (|0⟩ + |1⟩) / √2

When this state is measured using the Pauli-Z observableA measurable property represented mathematically by a Hermitian operator. One measurement returns one of its allowed outcomes; repeated measurements can be used to estimate its expectation value., the result is +1 with probability one-half and −1 with probability one-half. No individual run returns zero. The expectation valueThe probability-weighted average of an observable's possible outcomes for a particular quantum state. It is estimated from many repeated measurements., however, is:2

⟨A⟩ψ  =  ⟨ψ|A|ψ⟩  =  ∑a p(a) a

The symbol a represents a possible measurement outcome, while p(a) is the probability of obtaining it. The expectation value is their probability-weighted average. For the Pauli-Z measurement of |+⟩:

⟨Z⟩  =  ½(+1) + ½(−1)  =  0

This is the key distinction. The observable is Z. The possible outcomes are +1 and −1. One run returns one of those outcomes. The expectation value is 0.

A common mistake is to call the expectation value “the observable.”

The observable defines what is measured. The expectation value describes the average result associated with a particular state. Those are two different objects, and the gap between them is exactly the zero that no single run ever returns.

A hand-drawn diagram showing a Pauli-Z measurement of the state |+⟩. Each measurement returns either plus one or minus one, and the running average of many outcomes approaches a red line at zero labelled the expectation value.
Figure 1: A Pauli-Z measurement returns only +1 or −1. For the state |+⟩, the running average approaches zero even though no individual measurement returns zero.

How quantum hardware measures an observable

Quantum hardware commonly reports computational-basis bitstringsThe string of 0s and 1s returned by a computational-basis readout. It is raw measurement data that must be interpreted according to the operator and measurement basis. made of 0s and 1s. For a single Pauli-Z measurementA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. the mapping is straightforward: 0 becomes +1, and 1 becomes −1. For a multi-qubit Pauli termOne of the simple building blocks, labelled I, X, Y and Z, that a complicated observable is broken into so hardware can measure it a piece at a time., the full bitstring is converted into an eigenvalueOne of the possible outcomes associated with measuring an observable. For a Hamiltonian, its eigenvalues are the system's allowed energies. sample for that term. When an X-type or Y-type observableA measurable property represented mathematically by a Hermitian operator. One measurement returns one of its allowed outcomes; repeated measurements can be used to estimate its expectation value. must be measured, basis-rotation gates are applied before the final computational-basis readoutThe hardware process that converts a qubit measurement into classical data, usually a reported 0 or 1..3

The same circuit is run again and again. Each execution is one shot, and one shot is one sample.

The resulting samples are averaged to estimate the expectation valueThe probability-weighted average of an observable's possible outcomes for a particular quantum state. It is estimated from many repeated measurements.. This becomes especially important in variational quantum algorithmsA hybrid method: the quantum computer prepares a trial state and estimates its energy, and a classical optimizer adjusts the state to push that energy down.. After a molecular problem has been encoded onto qubitsThe basic unit of a quantum computer. Like a 'bit' in a normal computer, but instead of being only 0 or 1 it can be 0, 1, or a blend of both at once., its HamiltonianThe operator describing a system's total energy and how it changes over time. Writing one down is usually the first step in stating a quantum problem. can be written as a weighted sum of Pauli terms, and its energy expectation value is reconstructed from those parts:3

H  =  ∑i hi Pi       ⟨H⟩  =  ∑i hi ⟨Pi⟩

Each Pauli expectation value may require many shotsA single execution of a quantum circuit, ending in one readout. Estimating anything useful takes many of them., and terms that cannot be measured in the same basis require separate measurement settingsOne configuration of the readout, including any basis-rotation gates applied first. Terms needing different settings cannot be measured in the same run..3 Under ordinary independent sampling, statistical uncertainty decreases approximately as 1 / √N, where N is the number of shots.2 Halving the statistical uncertainty therefore requires roughly four times as many circuit executions.

A hand-drawn workflow showing a Hamiltonian split into representative Pauli terms, compatible terms grouped into measurement settings, repeated plus-one and minus-one samples taken in each setting, individual Pauli expectation values formed from those samples, and the weighted results recombined into one energy estimate. A red annotation emphasizes the large number of circuit executions.
Figure 2: A Hamiltonian is decomposed into Pauli terms. Repeated measurements estimate their expectation values, which are recombined into one energy estimate.

Why observables matter in practice

ObservablesA measurable property represented mathematically by a Hermitian operator. One measurement returns one of its allowed outcomes; repeated measurements can be used to estimate its expectation value. are the bridge between a quantum stateThe mathematical description used to predict the probabilities of different measurement outcomes for a quantum system. and a useful numerical result. They also create practical costs. Estimating one energy may require many Pauli termsOne of the simple building blocks, labelled I, X, Y and Z, that a complicated observable is broken into so hardware can measure it a piece at a time., several measurement settingsOne configuration of the readout, including any basis-rotation gates applied first. Terms needing different settings cannot be measured in the same run., and thousands or millions of circuit executions.4 MeasurementA physical process that produces a classical outcome and updates the quantum state. In an ideal projective measurement, the state is left in an eigenstate associated with the observed outcome. errors, sampling uncertainty, state preparation, circuit noise, and classical post-processing all affect the final estimate.

Ask not only what number a machine obtained, but how many circuit executions it took to make that number reliable.

When you are evaluating a quantum-computing demonstration, a short list of questions does most of the work:

  • What observable was estimated?
  • How was it decomposed into measurable terms?
  • How many measurement settings were required?
  • How many circuit executions were used?
  • What statistical uncertainty was reported?

The central idea stays simple. The observable defines the question. One measurement gives one outcome. Repeated measurements estimate the expectation valueThe probability-weighted average of an observable's possible outcomes for a particular quantum state. It is estimated from many repeated measurements.. So the next time a company reports a quantum result, ask not only what number it obtained, but what had to be measured, and how many circuit executions were required to make that number reliable.

Sources

  1. Mermin, N. D. “From Cbits to Qbits: Teaching Computer Scientists Quantum Mechanics.” American Journal of Physics 71, 23–30 (2003). DOI: 10.1119/1.1522741. Preprint: arXiv:quant-ph/0207118.
  2. McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. “The Theory of Variational Hybrid Quantum-Classical Algorithms.” New Journal of Physics 18, 023023 (2016). DOI: 10.1088/1367-2630/18/2/023023. Preprint: arXiv:1509.04279.
  3. Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M. & Gambetta, J. M. “Hardware-Efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets.” Nature 549, 242–246 (2017). DOI: 10.1038/nature23879. Preprint: arXiv:1704.05018.
  4. Gonthier, J. F., Radin, M. D., Buda, C., Doskocil, E. J., Abuan, C. M. & Romero, J. “Measurements as a Roadblock to Near-Term Practical Quantum Advantage in Chemistry: Resource Analysis.” Physical Review Research 4, 033154 (2022). DOI: 10.1103/PhysRevResearch.4.033154. Preprint: arXiv:2012.04001.