Subsystems and Entanglement Formula Sheet

Use this sheet to check a tensor-product calculation, a reduced state, or an entanglement measure. Common composite states lists named examples; the diagnostic table explains when each test is conclusive.

Helpful background. Tensor products and partial trace explain the notation. Finite-dimensional, normalized states are assumed unless a continuous-variable section says otherwise. Entropy logarithms are base two, so entropies are in bits. Operator products involving unbounded observables additionally need compatible domains and finite moments.

Tensor products and coefficient matrices

For distinguishable factors, $\mathcal H_{AB}=\mathcal H_A\otimes\mathcal H_B$. In orthonormal bases,

$$\lvert\psi\rangle=\sum_{i=0}^{d_A-1}\sum_{j=0}^{d_B-1}C_{ij}\lvert i\rangle_A\lvert j\rangle_B, \qquad \sum_{ij}|C_{ij}|^2=1.$$

Using the flat order $k=i d_B+j$, the amplitude vector is the row-major flattening of $C$. A pure state is product exactly when $C$ has rank one. Local operations and basis changes have different formulas:

$$C\xrightarrow{U_A\otimes U_B}U_A C U_B^T,\qquad C_{\rm new}=V_A^\dagger C V_B^*.$$

In the second formula, columns of $V_A,V_B$ are the new basis vectors expressed in the old bases. See product bases and local-unitary equivalence.

For bounded operators, $(A\otimes B)(C\otimes D)=AC\otimes BD$. Local observables are $A\otimes I_B$ and $I_A\otimes B$; they commute. An uncoupled Hamiltonian is $H_A\otimes I_B+I_A\otimes H_B$, with joint energies $E_a+E_b$ when the factors have energy eigenvectors. Interactions add terms coupling the factors.

Product and separable density operators

A product state is $\rho_A\otimes\rho_B$. A finite-dimensional separable state admits

$$\rho_{AB}=\sum_k p_k\rho_A^{(k)}\otimes\rho_B^{(k)},\qquad p_k\ge0,\quad \sum_kp_k=1.$$

It can have correlations even though it has no entanglement. In infinite dimension, separability uses the trace-norm closure of finite convex combinations. See separable mixed states.

Partial trace and local observations

The basis formulas are

$$(\rho_A)_{ii'}=\sum_j\rho_{ij,i'j},\qquad (\rho_B)_{jj'}=\sum_i\rho_{ij,ij'}.$$

For the pure-state coefficient convention above,

$$\rho_A=CC^\dagger,\qquad \rho_B=C^TC^*=(C^\dagger C)^T.$$

The transpose on the second formula matters for complex amplitudes. For example, $C=\tfrac1{\sqrt2}\begin{pmatrix}1&i\\0&0\end{pmatrix}$ gives $\rho_B=\tfrac12\begin{pmatrix}1&-i\\i&1\end{pmatrix}$, the state of $\lvert+y\rangle$.

For local effects $E_a,F_b$, with each set summing to its identity,

$$p(a,b)=\operatorname{Tr}[\rho_{AB}(E_a\otimes F_b)],\qquad p(a)=\operatorname{Tr}(\rho_A E_a).$$

Similarly $\langle A\otimes I\rangle=\operatorname{Tr}(\rho_A A)$ for bounded $A$. Connected correlations are $\langle A\otimes B\rangle-\langle A\rangle\langle B\rangle$. They vanish for every pair of local observables exactly when the density operator is product. See local measurement statistics and marginals and correlations.

Conditional states and no-signaling

For a remote POVM effect $F_b$ on $B$, the subnormalized state left on $A$ is

$$\sigma_{A|b}=\operatorname{Tr}_B[\rho_{AB}(I_A\otimes F_b)],\qquad p_b=\operatorname{Tr}\sigma_{A|b}.$$

When $p_b>0$, $\rho_{A|b}=\sigma_{A|b}/p_b$. Though the product inside the partial trace need not look Hermitian, the resulting operator is positive; it equals the expression obtained by placing $\sqrt{F_b}$ on both sides of $\rho_{AB}$ before tracing. Any instrument with the same effect gives this same remote state. The measured subsystem and the retained joint state can still depend on the instrument.

$$\sum_b\sigma_{A|b}=\rho_A,\qquad \operatorname{Tr}_B[(\operatorname{id}_A\otimes\mathcal E_B)(\rho_{AB})]=\rho_A$$

for a trace-preserving local channel $\mathcal E_B$. Outcome selection requires a classical record. See conditional states.

Schmidt decomposition and purification

If $C=U\operatorname{diag}(s_k)V^\dagger$ is a singular-value decomposition,

$$\lvert\psi\rangle=\sum_k s_k\lvert u_k\rangle_A\lvert v_k^*\rangle_B, \qquad p_k=s_k^2,\qquad \sum_kp_k=1.$$

The star is complex conjugation in the chosen $B$ basis; columns of $V$ themselves are not generally the second-factor Schmidt kets. The common nonzero reduced spectrum is $\{p_k\}$, and the Schmidt rank equals $\operatorname{rank}C$.

For $\rho_A=\sum_kp_k\lvert k\rangle\langle k\rvert$, a purification is $\sum_k\sqrt{p_k}\lvert k\rangle_A\lvert k\rangle_R$. Its smallest possible reference dimension is $\operatorname{rank}\rho_A$. Local unitaries on $R$ leave $\rho_A$ fixed; comparisons of unequal-size purifiers use isometries on their occupied supports. See Schmidt decomposition and purification.

Entropies and total correlation

For density eigenvalues $p_k$,

$$S(\rho)=-\sum_k p_k\log_2p_k,\qquad S_\alpha(\rho)=\frac{\log_2\sum_kp_k^\alpha}{1-\alpha} \quad(\alpha>0,\ \alpha\ne1),$$

with $0\log_20=0$. Purity is $\sum_kp_k^2$ and $S_2=-\log_2\operatorname{Tr}\rho^2$. For a globally pure bipartite state, $S_A=S_B$ measures entanglement; it is zero exactly for product vectors and at most $\log_2\min(d_A,d_B)$. For mixed global states, local entropy alone does not measure entanglement.

$$I(A:B)=S_A+S_B-S_{AB} =D(\rho_{AB}\Vert\rho_A\otimes\rho_B),$$

where $D(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\log_2\rho-\log_2\sigma)]$, with value $+\infty$ if the support condition fails. Mutual information is total correlation and vanishes exactly for a product state. In infinite dimension, use the relative-entropy definition when the entropy subtraction would involve infinities. See entropy, Rényi entropies, and mutual information.

Partial transpose, negativity, and concurrence

Transpose only the $B$ indices:

$$(\rho^{T_B})_{ij,i'j'}=\rho_{ij',i'j},\qquad \mathcal N(\rho)=\frac{\|\rho^{T_B}\|_1-1}{2},\qquad E_N=\log_2\|\rho^{T_B}\|_1.$$

A negative partial-transpose eigenvalue proves entanglement. PPT is sufficient for separability for $2\times2$ and $2\times3$ systems, but not in general for larger bipartitions. For a pure state with Schmidt coefficients $s_k$, $\mathcal N=[(\sum_ks_k)^2-1]/2$.

For a normalized two-qubit pure state with coefficient matrix $M$, concurrence is $C(\psi)=2|\det M|$. For a mixed two-qubit state, let $\widetilde\rho=(\sigma_y\otimes\sigma_y)\rho^*(\sigma_y\otimes\sigma_y)$ and let $\lambda_1\ge\cdots\ge\lambda_4$ be the nonnegative square roots of the eigenvalues of $\rho\widetilde\rho$. Then

$$C(\rho)=\max\{0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\}.$$

See negativity and PPT, concurrence, and witnesses. A witness $W$ is nonnegative on every separable state, so $\operatorname{Tr}(W\rho)<0$ certifies entanglement; the opposite sign is generally inconclusive.

Multipartite reference states

For $n\ge2$,

$$\lvert\mathrm{GHZ}_n\rangle=\frac{\lvert0\rangle^{\otimes n}+\lvert1\rangle^{\otimes n}}{\sqrt2},\qquad \lvert W_n\rangle=\frac1{\sqrt n}\sum_{j=1}^n\lvert0\cdots1_j\cdots0\rangle.$$

A nonempty proper GHZ reduction is the equal mixture of its all-zero and all-one branches. Every nontrivial pure-state cut has entropy one bit. Across an $m:(n-m)$ cut of $W_n$, the Schmidt probabilities are $m/n$ and $(n-m)/n$, for $1\le m<n$.

For a qubit graph state, the entropy of a cut is the rank over $\mathbb F_2$ of the cross-cut adjacency block. It is not generally the number of crossing edges. See GHZ, W, graph states, and the named-state list for Bell and spin states.

Continuous-variable wavefunctions and covariance

A normalized two-particle wavefunction obeys $\int dx\,dy\,|\Psi(x,y)|^2=1$, and

$$\rho_A(x,x')=\int dy\,\Psi(x,y)\Psi^*(x',y).$$

For $N$ modes, $q=(a+a^\dagger)/\sqrt2$, $p=(a-a^\dagger)/(i\sqrt2)$, and vacuum variance is $1/2$. Put $R=(q_1,p_1,\ldots,q_N,p_N)^T$, $d_j=\langle R_j\rangle$, and

$$V_{jk}=\tfrac12\langle\Delta R_j\Delta R_k+\Delta R_k\Delta R_j\rangle,\qquad \Omega=\bigoplus_{j=1}^N\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Finite physical covariances obey $V+i\Omega/2\ge0$. For Gaussian states only, $d,V$ determine the full state, with

$$\operatorname{Tr}\rho^2=\frac1{\sqrt{\det(2V)}},\qquad S(\rho)=\sum_jg(\nu_j-1/2),$$

where $g(n)=(n+1)\log_2(n+1)-n\log_2n$ and $\nu_j$ are symplectic eigenvalues. Partial transposition flips the transposed modes’ momenta. Gaussian PPT is sufficient for separability across a one-mode-versus-$N$-mode cut, but not across general multimode cuts. See Gaussian states.

Squeezing and EPR correlations

With the phase convention

$$S_1(r)=e^{r(a^2-a^{\dagger2})/2},\qquad S_2(r)=e^{r(a_A^\dagger a_B^\dagger-a_Aa_B)},$$

$S_1(r)|0\rangle$ has variances $(e^{-2r}/2,e^{2r}/2)$. For finite $r\ge0$,

$$S_2(r)|0,0\rangle=\frac1{\cosh r}\sum_{n=0}^\infty(\tanh r)^n|n,n\rangle,\qquad \bar n=\sinh^2r.$$

The EPR variances are $\operatorname{Var}(q_A-q_B)=\operatorname{Var}(p_A+p_B)=e^{-2r}$. Every separable state with finite second moments has their sum at least two. Exact zero spreads require a singular limit, not a normalized EPR vector. See squeezed modes and EPR states.

Identical particles and applications

For normalized one-particle vectors with overlap $s=\langle\alpha|\beta\rangle$, normalized symmetric/antisymmetric combinations use the denominator $\sqrt{2(1\pm|s|^2)}$, whenever nonzero. Antisymmetry concerns complete particle labels; it does not by itself quantify entanglement between laboratories. See exchange symmetry and identical-particle cautions.

For operator formulas beyond the composite-state calculations above, use creation and annihilation operators, one-body operators, and two-body operators. Applications with their necessary assumptions are developed in quantum information, many-body physics, chemistry, optics, foundations, and QFT.

References

• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Reviews of Modern Physics 81, 865–942 (2009), doi:10.1103/RevModPhys.81.865.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), author’s text.
• C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Reviews of Modern Physics 84, 621–669 (2012), doi:10.1103/RevModPhys.84.621.