Stabilizer States Preview

A stabilizer description specifies a quantum state by naming observables for which the state has definite eigenvalue $+1$. For many qubit states, especially Bell, GHZ, cluster, and graph states, this is more compact than writing all amplitudes in the computational basis.

For a pure state $\lvert\psi\rangle$, an operator $S$ stabilizes the state if

$$S\lvert\psi\rangle = \lvert\psi\rangle.$$

In the stabilizer formalism, the operators $S$ are usually products of Pauli matrices and identities. This page gives only the state-description preview needed in composite quantum mechanics. Stabilizer Formalism is the canonical home for Pauli groups, binary symplectic data, Clifford gates, stabilizer circuits, stabilizer codes, logical operators, and syndrome algebra.

Stabilizers as Commuting Observables

The basic building blocks are Pauli operators acting on named qubits:

$$X_j, \qquad Y_j, \qquad Z_j.$$

A Pauli product is a tensor product of single-qubit Pauli matrices and identities, possibly with an overall sign. For example,

$$Z_1X_2Z_3$$

means

$$Z\otimes X\otimes Z$$

on qubits $1,2,3$, with identities on any other qubits left implicit.

To specify a state by simultaneous eigenvalue equations, the chosen observables must commute. If

$$[S_a,S_b]=0,$$

then the two observables can have simultaneous eigenstates. A stabilizer state is a pure state specified as the unique common $+1$ eigenstate of a sufficiently large commuting family of Pauli products.

For $n$ qubits, $n$ independent commuting Pauli-product constraints usually specify one pure stabilizer state:

$$S_j\lvert\psi\rangle = \lvert\psi\rangle, \qquad j=1,\ldots,n.$$

If there are fewer independent generators, the common $+1$ eigenspace is larger. That larger space is the entry point to stabilizer codes.

Simple One-Qubit Examples

The state $\lvert0\rangle$ is stabilized by $Z$:

$$Z\lvert0\rangle = \lvert0\rangle.$$

The state $\lvert1\rangle$ is stabilized by $-Z$:

$$(-Z)\lvert1\rangle = \lvert1\rangle.$$

The state $\lvert+\rangle$ is stabilized by $X$:

$$X\lvert+\rangle = \lvert+\rangle, \qquad \lvert+\rangle = \frac{1}{\sqrt2} \bigl( \lvert0\rangle+\lvert1\rangle \bigr).$$

The state $\lvert-\rangle$ is stabilized by $-X$.

For two qubits, the product state $\lvert00\rangle$ is the unique simultaneous $+1$ eigenstate of

$$Z_1, \qquad Z_2.$$

Thus stabilizer language is not only for entangled states. It is a way of recording exact Pauli measurement outcomes.

Bell States as Stabilizer States

The Bell state

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt2} \bigl( \lvert00\rangle+\lvert11\rangle \bigr)$$

is stabilized by two commuting Pauli products:

$$X_1X_2\lvert\Phi^+\rangle = \lvert\Phi^+\rangle, \qquad Z_1Z_2\lvert\Phi^+\rangle = \lvert\Phi^+\rangle.$$

The first equation says the two qubits have the same $X$-basis value. The second says they have the same $Z$-basis value. These two statements are compatible because $X_1X_2$ and $Z_1Z_2$ commute: there is one $X$-$Z$ anticommutation on each qubit, so the two minus signs cancel.

The four Bell states can be labeled by their eigenvalues under the commuting pair $X_1X_2$ and $Z_1Z_2$:

$$\begin{array}{ccc} \text{state} & X_1X_2 & Z_1Z_2 \\ \lvert\Phi^+\rangle & +1 & +1\\ \lvert\Phi^-\rangle & -1 & +1\\ \lvert\Psi^+\rangle & +1 & -1\\ \lvert\Psi^-\rangle & -1 & -1 \end{array}$$

This is the stabilizer version of saying that Bell states are distinguished by joint correlations rather than by one-qubit reduced states.

GHZ States as Stabilizer States

The $n$-qubit GHZ plus state is

$$\lvert\mathrm{GHZ}_n^+\rangle = \frac{1}{\sqrt2} \bigl( \lvert0\rangle^{\otimes n} + \lvert1\rangle^{\otimes n} \bigr).$$

A convenient stabilizer generating set is

$$X_1X_2\cdots X_n, \qquad Z_1Z_k \quad (k=2,\ldots,n).$$

The $Z_1Z_k$ stabilizers encode equality of all computational-basis bits:

$$Z_1Z_k \lvert\mathrm{GHZ}_n^+\rangle = \lvert\mathrm{GHZ}_n^+\rangle.$$

The global $X$ stabilizer encodes the coherence between the all-zero and all-one branches:

$$X_1X_2\cdots X_n \lvert\mathrm{GHZ}_n^+\rangle = \lvert\mathrm{GHZ}_n^+\rangle.$$

Together these $n$ independent generators specify the pure GHZ plus state. Changing the sign of the global $X$ generator selects the minus-phase GHZ state.

Graph States as Stabilizer States

Graph States are stabilizer states with generators attached to graph vertices. For a graph $G=(V,E)$, the generator at vertex $v$ is

$$K_v = X_v \prod_{u\in N(v)} Z_u.$$

The graph state satisfies

$$K_v\lvert G\rangle = \lvert G\rangle \qquad \text{for every }v\in V.$$

For the three-qubit line $1-2-3$, the generators are

$$K_1=X_1Z_2, \qquad K_2=Z_1X_2Z_3, \qquad K_3=Z_2X_3.$$

This illustrates why stabilizer descriptions are economical. A state with eight computational-basis amplitudes is specified by three local Pauli constraints.

Stabilizer Codes Preview

If $r$ independent commuting stabilizer generators act on $n$ qubits, and no contradiction such as $-I$ is generated, then the common $+1$ eigenspace has dimension

$$2^{n-r}.$$

When $r=n$, this space is one-dimensional: a stabilizer state. When $r<n$, the common eigenspace can encode quantum information. This is the stabilizer-code idea.

The same Pauli measurement logic becomes error detection. If an error anticommutes with a stabilizer generator, it flips that generator’s measured eigenvalue from $+1$ to $-1$. The pattern of flipped signs is called a syndrome in quantum error correction.

This page stops at the state and eigenspace viewpoint. Full stabilizer codes, logical operators, Clifford circuits, decoding, surface codes, and fault-tolerance thresholds require a separate quantum-information treatment.

Stabilizers and Witnesses

Because stabilizers are measurable Pauli products, they often provide practical entanglement diagnostics. For a target stabilizer state, one can measure selected stabilizers or products of stabilizers and compare the results with bounds satisfied by separable states.

This is one route from the structural definition of entanglement to Entanglement Witnesses. The witness page treats the general convex-geometry idea; stabilizer witnesses are a structured family of examples.

Common Mistakes

• Thinking every state has a short stabilizer description. Stabilizer states are a special family.
• Forgetting that stabilizer generators must commute if they are to define simultaneous eigenvalue constraints.
• Confusing a stabilizer state with a stabilizer code. A state is the one-dimensional case; a code is a larger stabilized subspace.
• Treating $-Z$ as a different measurement from $Z$ rather than the same observable with the eigenvalue convention reversed.
• Assuming stabilizer measurements reveal a hidden classical bit string. They reveal eigenvalues of commuting quantum observables.
• Using stabilizer language without specifying the qubit ordering and tensor-factor supports.

Cross-Links

• Pauli Matrices
• Pauli Matrix Table
• Operators on Composite Systems
• Bell States
• GHZ States
• Graph States
• Entanglement Sharing
• Multipartite Systems
• Local Unitary Equivalence
• Local Measurement Statistics
• Entanglement Witnesses
• Projective Measurement
• Stabilizer Formalism

References

• D. Gottesman, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology, 1997.
• D. Gottesman, “The Heisenberg Representation of Quantum Computers,” in Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, International Press, 1999.
• R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer,” Physical Review Letters 86, 5188-5191, 2001.
• M. Hein, J. Eisert, and H. J. Briegel, “Multi-Party Entanglement in Graph States,” Physical Review A 69, 062311, 2004.
• S. Aaronson and D. Gottesman, “Improved Simulation of Stabilizer Circuits,” Physical Review A 70, 052328, 2004.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Verify the one-qubit stabilizers $Z\lvert0\rangle=\lvert0\rangle$ and $X\lvert+\rangle=\lvert+\rangle$.

Solution

Using

$$Z\lvert0\rangle = \lvert0\rangle, \qquad X\lvert0\rangle = \lvert1\rangle, \qquad X\lvert1\rangle = \lvert0\rangle,$$

we get

$$X\lvert+\rangle = \frac{1}{\sqrt2} \bigl( X\lvert0\rangle+X\lvert1\rangle \bigr) = \frac{1}{\sqrt2} \bigl( \lvert1\rangle+\lvert0\rangle \bigr) = \lvert+\rangle.$$

2. Show that $X_1X_2$ and $Z_1Z_2$ commute.

Solution

On one qubit, $X$ and $Z$ anticommute:

$$XZ=-ZX.$$

For two qubits,

$$(X_1X_2)(Z_1Z_2) = (X_1Z_1)(X_2Z_2).$$

Moving each $Z$ past an $X$ gives two minus signs:

$$(X_1Z_1)(X_2Z_2) = (-Z_1X_1)(-Z_2X_2) = (Z_1Z_2)(X_1X_2).$$

Thus the two Pauli products commute.

3. Verify that $\lvert\Phi^-\rangle$ has eigenvalues $-1$ under $X_1X_2$ and $+1$ under $Z_1Z_2$.

Solution

For

$$\lvert\Phi^-\rangle = \frac{1}{\sqrt2} \bigl( \lvert00\rangle-\lvert11\rangle \bigr),$$

the operator $X_1X_2$ swaps $\lvert00\rangle$ and $\lvert11\rangle$:

$$X_1X_2\lvert\Phi^-\rangle = \frac{1}{\sqrt2} \bigl( \lvert11\rangle-\lvert00\rangle \bigr) = -\lvert\Phi^-\rangle.$$

The operator $Z_1Z_2$ gives eigenvalue $+1$ to both $\lvert00\rangle$ and $\lvert11\rangle$, so

$$Z_1Z_2\lvert\Phi^-\rangle = \lvert\Phi^-\rangle.$$

4. Check that $X_1X_2X_3$ stabilizes $\lvert\mathrm{GHZ}_3^+\rangle$.

Solution

The operator flips every computational-basis bit:

$$X_1X_2X_3\lvert000\rangle = \lvert111\rangle, \qquad X_1X_2X_3\lvert111\rangle = \lvert000\rangle.$$

Therefore

$$X_1X_2X_3 \frac{\lvert000\rangle+\lvert111\rangle}{\sqrt2} = \frac{\lvert111\rangle+\lvert000\rangle}{\sqrt2} = \lvert\mathrm{GHZ}_3^+\rangle.$$

5. Suppose $r$ independent commuting stabilizer generators act on $n$ qubits. Why is the common $+1$ eigenspace dimension $2^{n-r}$?

Solution

The full $n$-qubit Hilbert space has dimension $2^n$. A nontrivial Pauli-product stabilizer has eigenvalues $\pm1$ with equal-dimensional eigenspaces, so imposing one independent $+1$ constraint halves the dimension. Imposing $r$ independent commuting constraints halves the dimension $r$ times:

$$2^n \longmapsto \frac{2^n}{2^r} = 2^{n-r}.$$

Commutation is needed so the constraints can be imposed simultaneously.