Common Quantum States

To use a named quantum state in a calculation, first fix its basis, subsystem labels, and oscillator conventions. This page collects useful normalized representatives and a diagnostic for each. Density Operators for QI develops the state formalism; Composite Systems and Entanglement explains correlations between subsystems.

Qubit states and stabilizers

All qubit kets below use the computational basis. A global phase is irrelevant to the state; relative phases are retained. Bell-state subscripts refer to the declared tensor factors, and $n\geq2$ for GHZ and W states.

State	Definition and diagnostic
Z basis	$\lvert0\rangle$, $\lvert1\rangle$ Z eigenvalues $+1,-1$.
X basis	$\lvert\pm\rangle=(\lvert0\rangle\pm\lvert1\rangle)/\sqrt2$ X eigenvalues $+1,-1$.
Y basis	$\lvert\pm i\rangle=(\lvert0\rangle\pm i\lvert1\rangle)/\sqrt2$ Y eigenvalues $+1,-1$.
Bell Φ states	$\lvert\Phi^\pm\rangle=(\lvert00\rangle\pm\lvert11\rangle)/\sqrt2$ Each one-qubit reduction is $I/2$.
Bell Ψ states	$\lvert\Psi^\pm\rangle=(\lvert01\rangle\pm\lvert10\rangle)/\sqrt2$ The minus state is the two-qubit singlet.
GHZ	$\lvert\mathrm{GHZ}_n\rangle=(\lvert0\rangle^{\otimes n}+\lvert1\rangle^{\otimes n})/\sqrt2$ Coherence between the two branches distinguishes this state from their statistical mixture.
W	$\lvert W_n\rangle=n^{-1/2}\sum_{j=1}^n\lvert0\cdots1_j\cdots0\rangle$ Exactly one excitation, coherently distributed.
Uniform mixture	$\rho=I/2^n$ Maximally mixed n-qubit state: entropy n bits and no preferred basis.

A pure stabilizer state is the unique common $+1$ eigenvector of $n$ independent commuting Hermitian Pauli operators whose generated group excludes $-I$. A graph state on a simple graph with vertices $V$ and edges $E$ is

$$\lvert G\rangle=\prod_{(u,v)\in E}\operatorname{CZ}_{uv}\, \lvert+\rangle^{\otimes |V|}.$$

Its generators are $K_v=X_v\prod_{u\in N(v)}Z_u$. Cluster states are graph states on specified lattice graphs; the graph and boundaries are part of the state definition. See Stabilizer Formalism and Measurement-Based Quantum Computation.

Oscillator states and finite energy

For a single bosonic mode, $[a,a^\dagger]=1$, $q=(a+a^\dagger)/\sqrt2$, and $p=(a-a^\dagger)/(i\sqrt2)$, so $[q,p]=i$ and vacuum variances are $1/2$. Translate covariance conventions before importing formulas from a different reference.

Coherent state. The normalized eigenstate $a\lvert\alpha\rangle=\alpha\lvert\alpha\rangle$ is

$$\lvert\alpha\rangle=e^{-|\alpha|^2/2} \sum_{m=0}^{\infty}\frac{\alpha^m}{\sqrt{m!}}\lvert m\rangle.$$

It has mean occupation $|\alpha|^2$. Distinct coherent states are generally nonorthogonal.

Squeezed vacuum. For real $r$, choose $S(r)=\exp[r(a^2-a^{\dagger2})/2]$. Then $S(r)\lvert0\rangle$ has $\operatorname{Var}(q)=e^{-2r}/2$ and $\operatorname{Var}(p)=e^{2r}/2$, with mean occupation $\sinh^2r$. Squeezing reduces one variance while increasing the conjugate variance; it does not reduce both below the uncertainty bound. See Squeezing.

Even and odd cats. For $\alpha\ne0$,

$$\lvert C_\pm(\alpha)\rangle= \frac{\lvert\alpha\rangle\pm\lvert-\alpha\rangle} {\sqrt{2(1\pm e^{-2|\alpha|^2})}}.$$

The two signs have even and odd photon-number parity. Replacing the denominator by $\sqrt2$ is an approximation requiring small overlap. State preparation alone is distinct from a protected Cat Code.

Ideal square-lattice GKP state. With the quadratures above, the logical position combs, labeled by $\mu=0,1$, are formally

$$\langle q\vert\mu_L\rangle\propto \sum_{s\in\mathbb Z}\delta\bigl(q-(2s+\mu)\sqrt\pi\bigr).$$

These ideal combs are not normalizable finite-energy states. Physical approximations require finite peak width and an envelope, which introduce residual displacement errors. Gottesman, Kitaev, and Preskill supply the encoding construction; GKP Codes explains its error model.

A reduced-state check

Tracing out the third qubit of $\lvert\mathrm{GHZ}_3\rangle$ gives the separable mixture $(\lvert00\rangle\langle00\rvert+\lvert11\rangle\langle11\rvert)/2$. Tracing out the third qubit of $\lvert W_3\rangle$ gives

$$\rho_{12}=\frac13\lvert00\rangle\langle00\rvert +\frac23\lvert\Psi^+\rangle\langle\Psi^+\rvert.$$

Its partial transpose has the negative eigenvalue $(1-\sqrt5)/6$, so this pair remains entangled. The two examples show why a named multipartite resource cannot be characterized solely by one-qubit entropy.

References

• Gottesman, Daniel, Alexei Kitaev, and John Preskill, “Encoding a qubit in an oscillator,” Physical Review A 64, 012310 (2001), doi:10.1103/PhysRevA.64.012310 — ideal and approximate oscillator code states.
• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010, doi:10.1017/CBO9780511976667 — qubit states and stabilizer constructions.
• Weedbrook, Christian, et al., “Gaussian quantum information,” Reviews of Modern Physics 84, 621–669 (2012), doi:10.1103/RevModPhys.84.621 — Gaussian states and quadrature conventions, translated here to vacuum variance one half.