Formula Sheet

This page collects frequently used formulas from composite systems, entanglement, identical particles, Fock space, and second-quantized notation. It is a quick reference, not a substitute for the explanatory pages linked below. For named example states, use Common Composite States. For choosing which entanglement test applies to a state class, use the Entanglement Diagnostic Table. For reproducible numerical checks, use Computational Notebooks.

Unless otherwise stated, logarithms in entropy formulas are natural logarithms.

Tensor Products

For distinguishable subsystems $A$ and $B$,

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

Dimensions multiply:

$$\dim(\mathcal H_A\otimes\mathcal H_B) = \dim\mathcal H_A\,\dim\mathcal H_B.$$

If $\{\lvert i\rangle_A\}$ and $\{\lvert j\rangle_B\}$ are bases, then

$$\{\lvert i\rangle_A\otimes\lvert j\rangle_B\}_{ij}$$

is a product basis. A general pure state is

$$\lvert\Psi\rangle = \sum_{ij} C_{ij}\, \lvert i\rangle_A\lvert j\rangle_B.$$

Local operators are embedded as

$$O_A \mapsto O_A\otimes I_B, \qquad O_B \mapsto I_A\otimes O_B.$$

For an uncoupled Hamiltonian,

$$H_0 = H_A\otimes I_B +I_A\otimes H_B.$$

Interactions add terms that are not local to one factor:

$$H = H_0+V_{AB}.$$

Product, Separable, and Entangled States

A pure product state has the form

$$\lvert\Psi\rangle = \lvert\psi\rangle_A\otimes\lvert\phi\rangle_B.$$

In a product basis, this means

$$C_{ij} = a_i b_j$$

for some coefficient vectors $a_i$ and $b_j$.

A bipartite mixed state is separable if it can be written as

$$\rho_{AB} = \sum_r p_r\, \rho_A^{(r)}\otimes\rho_B^{(r)}, \qquad p_r\ge0, \qquad \sum_r p_r=1.$$

A pure bipartite state is entangled if it is not a product state. A mixed state is entangled if it is not separable.

Density Operators and Partial Trace

For a pure state,

$$\rho = \lvert\Psi\rangle\langle\Psi\rvert.$$

The reduced density operators are

$$\rho_A = \operatorname{Tr}_B\rho_{AB}, \qquad \rho_B = \operatorname{Tr}_A\rho_{AB}.$$

In a product basis,

$$(\rho_A)_{ii'} = \sum_j (\rho_{AB})_{ij,i'j}.$$

Equivalently,

$$\operatorname{Tr}_B \bigl( \lvert i\rangle\langle i'\rvert \otimes \lvert j\rangle\langle j'\rvert \bigr) = \langle j'\vert j\rangle\, \lvert i\rangle\langle i'\rvert.$$

Local expectation values are computed from the reduced state:

$$\operatorname{Tr}_{AB} \bigl[ \rho_{AB} (O_A\otimes I_B) \bigr] = \operatorname{Tr}_A(\rho_A O_A).$$

For a product density operator,

$$\operatorname{Tr}_B(\rho_A\otimes\rho_B) = \rho_A\,\operatorname{Tr}\rho_B = \rho_A.$$

Local Measurement Statistics

For a local observable $M_A$,

$$\operatorname{Tr}_{AB} \bigl[ \rho_{AB}(M_A\otimes I_B) \bigr] = \operatorname{Tr}_A(\rho_A M_A).$$

For local POVM effects $\{E_a\}$ on $A$,

$$p(a) = \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(E_a\otimes I_B) \bigr] = \operatorname{Tr}_A(\rho_A E_a).$$

For local measurements on both subsystems,

$$p(a,b) = \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(E_a\otimes F_b) \bigr],$$

and the marginal is

$$\sum_b p(a,b) = \operatorname{Tr}_A(\rho_A E_a).$$

Marginals and Correlations

The marginal states are

$$\rho_A = \operatorname{Tr}_B\rho_{AB}, \qquad \rho_B = \operatorname{Tr}_A\rho_{AB}.$$

For local observables $M_A$ and $N_B$,

$$C_{MN} = \langle M_A\otimes N_B\rangle = \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(M_A\otimes N_B) \bigr].$$

The connected correlation is

$$C^{\mathrm{conn}}_{MN} = \langle M_A\otimes N_B\rangle - \langle M_A\rangle\langle N_B\rangle.$$

For a product state,

$$\rho_{AB} = \rho_A\otimes\rho_B \quad \Longrightarrow \quad C^{\mathrm{conn}}_{MN}=0$$

for all local observables $M_A,N_B$.

The quantum mutual information is

$$I(A:B) = S(\rho_A)+S(\rho_B)-S(\rho_{AB}).$$

Equivalently,

$$I(A:B) = D \bigl( \rho_{AB} \Vert \rho_A\otimes\rho_B \bigr).$$

Conditional States

For a projective measurement $\{Q_b\}$ on subsystem $B$,

$$p(b) = \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(I_A\otimes Q_b) \bigr].$$

The unnormalized conditional state of $A$ is

$$\widetilde\rho_{A\vert b} = \operatorname{Tr}_B \bigl[ (I_A\otimes Q_b)\rho_{AB}(I_A\otimes Q_b) \bigr].$$

For $p(b)>0$,

$$\rho_{A\vert b} = \frac{\widetilde\rho_{A\vert b}}{p(b)}.$$

The nonselective average returns the reduced state:

$$\sum_b p(b)\rho_{A\vert b} = \rho_A.$$

Purification

If

$$\rho_A = \sum_{k=1}^{r} p_k \lvert k\rangle_A\langle k\rvert_A,$$

then a canonical purification is

$$\lvert\Psi\rangle_{AR} = \sum_{k=1}^{r} \sqrt{p_k}\, \lvert k\rangle_A\lvert k\rangle_R.$$

It satisfies

$$\operatorname{Tr}_R \bigl( \lvert\Psi\rangle\langle\Psi\rvert_{AR} \bigr) = \rho_A.$$

The minimal purifying dimension is

$$\dim\mathcal H_R = \operatorname{rank}\rho_A.$$

Unitary transformations on the purifying system do not change $\rho_A$:

$$(I_A\otimes U_R)\lvert\Psi\rangle_{AR} \quad \text{purifies the same }\rho_A.$$

Bell States

The standard two-qubit Bell states are

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

and

$$\lvert\Psi^\pm\rangle = \frac{1}{\sqrt2} \bigl( \lvert01\rangle \pm \lvert10\rangle \bigr).$$

Each Bell state has maximally mixed one-qubit reductions:

$$\rho_A = \rho_B = \frac12 I.$$

For example,

$$\operatorname{Tr}_B \bigl( \lvert\Phi^+\rangle\langle\Phi^+\rvert \bigr) = \frac12 I.$$

Singlet and Triplet States

With $\lvert0\rangle=\lvert\uparrow\rangle$ and $\lvert1\rangle=\lvert\downarrow\rangle$,

$$\begin{aligned} \lvert1,1\rangle &= \lvert\uparrow\uparrow\rangle,\\ \lvert1,0\rangle &= \frac{1}{\sqrt2} \bigl( \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle \bigr),\\ \lvert1,-1\rangle &= \lvert\downarrow\downarrow\rangle,\\ \lvert0,0\rangle &= \frac{1}{\sqrt2} \bigl( \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle \bigr). \end{aligned}$$

The triplet states are symmetric under exchange, while the singlet is antisymmetric:

$$P_{12}\lvert1,m\rangle = \lvert1,m\rangle, \qquad P_{12}\lvert0,0\rangle = -\lvert0,0\rangle.$$

The singlet correlations are isotropic:

$$\left\langle (\boldsymbol\sigma\cdot\mathbf a) \otimes (\boldsymbol\sigma\cdot\mathbf b) \right\rangle_{\mathrm{singlet}} = -\mathbf a\cdot\mathbf b.$$

GHZ States

The standard $n$-qubit GHZ state is

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

Tracing out any nonempty proper complement of a subset $R$ gives

$$\rho_R = \frac12 \lvert0_R\rangle\langle0_R\rvert + \frac12 \lvert1_R\rangle\langle1_R\rvert.$$

For any nontrivial bipartition $A\vert\bar A$,

$$\operatorname{SR}=2, \qquad S_A=1 \quad \text{bit}.$$

W States

The standard $n$-qubit W state is

$$\lvert W_n\rangle = \frac{1}{\sqrt n} \sum_{k=1}^{n} \lvert0\cdots010\cdots0\rangle,$$

where the $1$ is in slot $k$. A one-qubit reduction is

$$\rho_k = \frac{n-1}{n} \lvert0\rangle\langle0\rvert + \frac1n \lvert1\rangle\langle1\rvert.$$

Across a split with $m$ qubits on one side,

$$\lvert W_n\rangle = \sqrt{\frac{m}{n}}\, \lvert W_m\rangle_A \lvert0\rangle_{\bar A}^{\otimes n-m} + \sqrt{\frac{n-m}{n}}\, \lvert0\rangle_A^{\otimes m} \lvert W_{n-m}\rangle_{\bar A}.$$

The Schmidt probabilities are

$$\frac{m}{n}, \qquad \frac{n-m}{n}.$$

Tracing out one qubit gives

$$\rho_{\mathrm{rest}} = \frac{n-1}{n} \lvert W_{n-1}\rangle\langle W_{n-1}\rvert + \frac1n \lvert0\rangle^{\otimes n-1} \langle0\rvert^{\otimes n-1}.$$

Schmidt Decomposition

Every pure bipartite state in finite-dimensional Hilbert spaces has a Schmidt decomposition

$$\lvert\Psi\rangle = \sum_{k=1}^{r} \lambda_k \lvert u_k\rangle_A \lvert v_k\rangle_B,$$

with

$$\lambda_k>0, \qquad \sum_k\lambda_k^2=1.$$

The sets $\{\lvert u_k\rangle_A\}$ and $\{\lvert v_k\rangle_B\}$ are orthonormal, and $r$ is the Schmidt rank.

Equivalently,

$$r = \operatorname{rank}\rho_A = \operatorname{rank}\rho_B = \operatorname{rank}C,$$

where $C$ is the coefficient matrix in a product basis.

The reduced states are

$$\rho_A = \sum_k \lambda_k^2 \lvert u_k\rangle\langle u_k\rvert, \qquad \rho_B = \sum_k \lambda_k^2 \lvert v_k\rangle\langle v_k\rvert.$$

A pure bipartite state is a product state exactly when

$$r=1.$$

The nonzero eigenvalues of $\rho_A$ and $\rho_B$ are the same:

$$\operatorname{spec}_{>0}(\rho_A) = \operatorname{spec}_{>0}(\rho_B) = \{\lambda_k^2\}.$$

Entropy and Subsystem Entropy

The von Neumann entropy is

$$S(\rho) = -\operatorname{Tr}(\rho\log\rho).$$

If $\rho$ has eigenvalues $p_\alpha$,

$$S(\rho) = -\sum_\alpha p_\alpha\log p_\alpha,$$

with the convention $0\log0=0$.

For subsystem reductions,

$$S_A = S(\rho_A), \qquad S_B = S(\rho_B).$$

For a pure bipartite state,

$$S_A = S(\rho_A), \qquad S_B = S(\rho_B),$$

and

$$S_A=S_B = -\sum_k \lambda_k^2\log\lambda_k^2.$$

For a maximally entangled state of Schmidt rank $d$,

$$S_A=\log d.$$

For mixed joint states, $S_A$ is local mixedness rather than a standalone entanglement measure.

The Renyi entropy of order $\alpha>0$, $\alpha\ne1$, is

$$S_\alpha(\rho) = \frac{1}{1-\alpha} \log \operatorname{Tr}(\rho^\alpha).$$

The second Renyi entropy is

$$S_2(\rho) = -\log \operatorname{Tr}(\rho^2).$$

The purity is

$$\operatorname{Tr}(\rho^2).$$

For a pure state, $\operatorname{Tr}(\rho^2)=1$. For a mixed reduced state, $\operatorname{Tr}(\rho_A^2)<1$.

Continuous-Variable Systems

For two distinguishable particles on a line,

$$\mathcal H_{12} = L^2(\mathbb R)\otimes L^2(\mathbb R) \cong L^2(\mathbb R^2).$$

A product wavefunction has the form

$$\Psi(x_1,x_2) = \psi(x_1)\phi(x_2).$$

The reduced density-kernel of particle 1 is

$$\rho_1(x,x') = \int_{-\infty}^{\infty}dy\, \Psi(x,y)\Psi^*(x',y).$$

For a single oscillator mode,

$$q = \frac{a+a^\dagger}{\sqrt2}, \qquad p = \frac{a-a^\dagger}{i\sqrt2}, \qquad [q,p]=i.$$

For a unitary mode-basis change,

$$\lvert\chi_\alpha\rangle = \sum_i U_{i\alpha}\lvert\varphi_i\rangle, \qquad b_\alpha^\dagger = \sum_i U_{i\alpha}a_i^\dagger.$$

A balanced single-excitation two-mode state is

$$\lvert\Psi_+\rangle = \frac{1}{\sqrt2} \left( \lvert1_A,0_B\rangle + \lvert0_A,1_B\rangle \right),$$

with mode entropy $S_A=\log2$.

The two-mode squeezed vacuum is

$$\lvert\mathrm{TMSV}(r)\rangle = \frac{1}{\cosh r} \sum_{n=0}^{\infty} (\tanh r)^n \lvert n_A,n_B\rangle.$$

Exchange Symmetry

For two identical particles, the exchange operator satisfies

$$P_{12}^2=I.$$

Bosonic states satisfy

$$P_{12}\lvert\Psi\rangle = \lvert\Psi\rangle,$$

while fermionic states satisfy

$$P_{12}\lvert\Psi\rangle = -\lvert\Psi\rangle.$$

For two orthonormal one-particle states $\lvert\alpha\rangle$ and $\lvert\beta\rangle$,

$$\lvert\alpha,\beta\rangle_S = \frac{1}{\sqrt2} \bigl( \lvert\alpha\rangle_1\lvert\beta\rangle_2 + \lvert\beta\rangle_1\lvert\alpha\rangle_2 \bigr),$$

and

$$\lvert\alpha,\beta\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert\alpha\rangle_1\lvert\beta\rangle_2 - \lvert\beta\rangle_1\lvert\alpha\rangle_2 \bigr).$$

The $N$-particle symmetrizer and antisymmetrizer are

$$\Pi_S^{(N)} = \frac{1}{N!} \sum_{\pi\in S_N} U(\pi),$$

and

$$\Pi_A^{(N)} = \frac{1}{N!} \sum_{\pi\in S_N} \operatorname{sgn}(\pi)U(\pi).$$

Pauli exclusion follows from antisymmetry:

$$\lvert\alpha,\alpha\rangle_A=0.$$

If $s=\langle\alpha\vert\beta\rangle$ is not zero, the normalized two-particle combinations are

$$\lvert\alpha,\beta\rangle_\pm = \frac{ \lvert\alpha\rangle_1\lvert\beta\rangle_2 \pm \lvert\beta\rangle_1\lvert\alpha\rangle_2 }{ \sqrt{2(1\pm\lvert s\rvert^2)} },$$

provided the denominator is nonzero.

For orthonormal spin-orbitals $\varphi_1,\ldots,\varphi_N$, the Slater determinant is

$$\Psi(q_1,\ldots,q_N) = \frac{1}{\sqrt{N!}} \det \bigl[ \varphi_j(q_i) \bigr]_{i,j=1}^{N}.$$

For a factorized two-particle state

$$\Psi(q_1,q_2) = \psi(\mathbf x_1,\mathbf x_2)\chi(s_1,s_2),$$

the total exchange parity is

$$\eta_{\mathrm{total}} = \eta_{\mathrm{space}}\eta_{\mathrm{spin}}.$$

Fock Space

For a one-particle Hilbert space $\mathcal h$,

$$\mathcal F_B(\mathcal h) = \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h,$$

and

$$\mathcal F_F(\mathcal h) = \bigoplus_{N=0}^{\infty} \wedge^N\mathcal h.$$

The vacuum sector is

$$\operatorname{Sym}^0\mathcal h \cong \wedge^0\mathcal h \cong \mathbb C,$$

with normalized vector $\lvert0\rangle$.

Bosonic occupations satisfy

$$n_i=0,1,2,\ldots .$$

Fermionic occupations satisfy

$$n_i\in\{0,1\}.$$

The total occupation is

$$N = \sum_i n_i.$$

Number states satisfy

$$N_i \lvert n_1,n_2,\ldots\rangle = n_i \lvert n_1,n_2,\ldots\rangle.$$

Bosonic Creation and Annihilation Operators

For a bosonic mode $i$,

$$a_i^\dagger \lvert\ldots,n_i,\ldots\rangle_B = \sqrt{n_i+1}\, \lvert\ldots,n_i+1,\ldots\rangle_B,$$

and

$$a_i \lvert\ldots,n_i,\ldots\rangle_B = \sqrt{n_i}\, \lvert\ldots,n_i-1,\ldots\rangle_B.$$

The canonical commutation relations are

$$[a_i,a_j^\dagger] = \delta_{ij}I, \qquad [a_i,a_j] =0, \qquad [a_i^\dagger,a_j^\dagger] =0.$$

The normalized bosonic occupation state is

$$\lvert n_1,n_2,\ldots\rangle_B = \prod_i \frac{(a_i^\dagger)^{n_i}}{\sqrt{n_i!}} \lvert0\rangle.$$

Fermionic Creation and Annihilation Operators

For fermions, fix a mode ordering and define

$$S_i = \sum_{k<i}n_k.$$

Creation acts as

$$c_i^\dagger \lvert n_1,\ldots,0_i,\ldots\rangle_F = (-1)^{S_i} \lvert n_1,\ldots,1_i,\ldots\rangle_F,$$

and annihilation acts as

$$c_i \lvert n_1,\ldots,1_i,\ldots\rangle_F = (-1)^{S_i} \lvert n_1,\ldots,0_i,\ldots\rangle_F.$$

If the attempted creation would double-occupy the mode, or the attempted annihilation acts on an empty mode, the result is zero.

The canonical anticommutation relations are

$$\{c_i,c_j^\dagger\} = \delta_{ij}I, \qquad \{c_i,c_j\} =0, \qquad \{c_i^\dagger,c_j^\dagger\} =0.$$

Pauli exclusion in operator form is

$$(c_i^\dagger)^2=0.$$

Number Operators

For bosons,

$$N_i = a_i^\dagger a_i.$$

For fermions,

$$N_i = c_i^\dagger c_i.$$

In both cases,

$$N_i \lvert n_1,n_2,\ldots\rangle = n_i \lvert n_1,n_2,\ldots\rangle.$$

The total number operator is

$$N = \sum_i N_i.$$

For bosons,

$$[N_i,a_j^\dagger] = \delta_{ij}a_j^\dagger, \qquad [N_i,a_j] = -\delta_{ij}a_j.$$

For fermions,

$$[N_i,c_j^\dagger] = \delta_{ij}c_j^\dagger, \qquad [N_i,c_j] = -\delta_{ij}c_j.$$

For a fermionic mode,

$$N_i^2=N_i.$$

Second-Quantized Operator Dictionary

Let $d_i,d_i^\dagger$ denote bosonic or fermionic mode operators, with the appropriate algebra.

A one-body operator $A$ with matrix elements

$$A_{ij} = \langle\varphi_i\vert A\vert\varphi_j\rangle$$

is represented by

$$\widehat A = \sum_{ij} A_{ij} d_i^\dagger d_j.$$

A two-body interaction with matrix elements $V_{ij;kl}$ has the typical number-conserving form

$$\widehat V = \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k,$$

with the ordering convention chosen consistently, especially for fermions.

A common many-particle Hamiltonian with one-body and two-body terms is

$$H = \sum_{ij} h_{ij}d_i^\dagger d_j + \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k.$$

A field operator expanded in a one-particle basis is

$$\psi(\mathbf x) = \sum_i \varphi_i(\mathbf x)d_i, \qquad \psi^\dagger(\mathbf x) = \sum_i \varphi_i^*(\mathbf x)d_i^\dagger.$$

For bosons,

$$[\psi(\mathbf x),\psi^\dagger(\mathbf y)] = \delta(\mathbf x-\mathbf y).$$

For fermions,

$$\{\psi(\mathbf x),\psi^\dagger(\mathbf y)\} = \delta(\mathbf x-\mathbf y).$$

For normal ordering, bosonic mode operators obey

$$:a_i a_j^\dagger: = a_j^\dagger a_i, \qquad a_i a_j^\dagger = :a_i a_j^\dagger: +\delta_{ij}.$$

For fermionic mode operators,

$$:c_i c_j^\dagger: = -c_j^\dagger c_i, \qquad c_i c_j^\dagger = :c_i c_j^\dagger: +\delta_{ij}.$$

With the empty-vacuum contraction convention used in Wick’s theorem preview,

$$C(a_i,a_j^\dagger) = \delta_{ij}, \qquad C(a_i^\dagger,a_j)=0,$$

and similarly

$$C(c_i,c_j^\dagger) = \delta_{ij}, \qquad C(c_i^\dagger,c_j)=0,$$

with fermionic signs supplied by the operator swaps.

Continuous-Variable Wavefunctions, Gaussian, and EPR Correlations

For two distinguishable particles on a line,

$$\mathcal H_{12} = L^2(\mathbb R)\otimes L^2(\mathbb R) \cong L^2(\mathbb R^2).$$

A position-space two-particle state is normalized by

$$\int dx_1\,dx_2\, \lvert\Psi(x_1,x_2)\rvert^2 = 1.$$

Product wavefunctions have

$$\Psi(x_1,x_2) = \psi(x_1)\phi(x_2).$$

For a pure two-particle state, the reduced density kernel of particle 1 is

$$\rho_1(x,x') = \int dy\, \Psi(x,y)\Psi^*(x',y).$$

The diagonal is the marginal position density:

$$\rho_1(x,x) = \int dy\, \lvert\Psi(x,y)\rvert^2.$$

For $N$ bosonic modes, define the quadrature vector

$$R = (q_1,p_1,\ldots,q_N,p_N)^T,$$

with

$$[R_j,R_k] = i\Omega_{jk}, \qquad \Omega = \bigoplus_{\ell=1}^{N} \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}.$$

The Gaussian covariance matrix is

$$V_{jk} = \frac12 \left\langle \Delta R_j\Delta R_k+\Delta R_k\Delta R_j \right\rangle, \qquad \Delta R_j=R_j-\langle R_j\rangle.$$

Physical covariance matrices satisfy

$$V+\frac{i}{2}\Omega \ge 0.$$

An $N$-mode Gaussian Wigner function is

$$W(R) = \frac{1}{(2\pi)^N\sqrt{\det V}} \exp\!\left[ -\frac12 (R-d)^T V^{-1}(R-d) \right].$$

Pure Gaussian states have symplectic eigenvalues $\nu_j=1/2$ and

$$\det V = 2^{-2N}.$$

With the squeezing convention used in this volume,

$$S_1(r) = \exp\!\left[ \frac r2 \left( a^2-a^{\dagger 2} \right) \right],$$

and

$$S_1^\dagger(r)qS_1(r) = e^{-r}q, \qquad S_1^\dagger(r)pS_1(r) = e^r p.$$

The single-mode squeezed-vacuum covariance matrix is

$$V_{\rm sq}(r) = \frac12 \begin{pmatrix} e^{-2r} & 0\\ 0 & e^{2r} \end{pmatrix}.$$

The two-mode squeeze operator is

$$S_2(r) = \exp\!\left[ r \left( a_A^\dagger a_B^\dagger-a_Aa_B \right) \right],$$

and

$$\lvert\mathrm{TMSV}(r)\rangle = \frac{1}{\cosh r} \sum_{n=0}^{\infty} (\tanh r)^n \lvert n_A,n_B\rangle.$$

For two particles on a line,

$$[X_i,P_j] = i\hbar\,\delta_{ij}.$$

The EPR collective observables are

$$X_1-X_2, \qquad P_1+P_2,$$

and they commute:

$$[X_1-X_2,P_1+P_2] = 0.$$

A formal simultaneous generalized eigenfunction is

$$\Psi_{x_0,P}(x_1,x_2) = \mathcal N \exp\!\left[ \frac{iP}{2\hbar}(x_1+x_2) \right] \delta(x_1-x_2-x_0).$$

It obeys

$$(X_1-X_2)\Psi_{x_0,P} = x_0\Psi_{x_0,P}, \qquad (P_1+P_2)\Psi_{x_0,P} = P\Psi_{x_0,P},$$

but it is not normalizable in $L^2(\mathbb R^2)$.

A finite Gaussian approximation is

$$\Psi_{\sigma,\Sigma} = \frac{1}{\sqrt{\pi\sigma\Sigma}} \exp\!\left[ -\frac{(x_1-x_2-x_0)^2}{4\sigma^2} -\frac{(x_1+x_2)^2}{4\Sigma^2} +\frac{iP}{2\hbar}(x_1+x_2) \right].$$

With this convention,

$$\operatorname{Var}(X_1-X_2) = \sigma^2, \qquad \operatorname{Var}(P_1+P_2) = \frac{\hbar^2}{\Sigma^2}.$$

For dimensionless two-mode quadratures,

$$q_j = \frac{a_j+a_j^\dagger}{\sqrt2}, \qquad p_j = \frac{a_j-a_j^\dagger}{i\sqrt2},$$

the two-mode squeezed vacuum satisfies, in a common phase convention,

$$\operatorname{Var}(q_A-q_B) = e^{-2r}, \qquad \operatorname{Var}(p_A+p_B) = e^{-2r}.$$

The ideal EPR limit corresponds to singular limits such as $\sigma\to0$, $\Sigma\to\infty$, or $r\to\infty$.

Quantum Information Resource Shorthands

The standard Bell pair is

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt2} \left( \lvert00\rangle+\lvert11\rangle \right).$$

Its one-qubit reduced states are

$$\rho_A = \rho_B = \frac12 I,$$

and its entanglement entropy is

$$S(\rho_A) = 1 \quad \text{bit}.$$

This is one ebit of bipartite pure-state entanglement. In the quantum-information bridge, the standard resource shorthand is

$$1\ \text{ebit} + 2\ \text{classical bits} \longrightarrow 1\ \text{qubit transmission}$$

for teleportation, and

$$1\ \text{ebit} + 1\ \text{qubit transmission} \longrightarrow 2\ \text{classical bits}$$

for superdense coding. These are protocol resource statements, not local signaling mechanisms.

Foundations Shorthands

For the Bell state

$$\lvert\Phi^+\rangle = \frac{ \lvert00\rangle+\lvert11\rangle }{\sqrt2},$$

the one-qubit reduced states are

$$\rho_A = \rho_B = \frac12 I.$$

Local outcome probabilities are computed from the reduced state:

$$p(a) = \operatorname{Tr}_A(\rho_A E_a).$$

An ordinary trace-preserving operation on the remote subsystem cannot change the unconditioned local reduced state:

$$\operatorname{Tr}_B \left[ (I_A\otimes\mathcal E_B)(\rho_{AB}) \right] = \rho_A.$$

Conditional states can change after a remote outcome is known:

$$\rho_{A\vert b} = \frac{\sigma_{A\vert b}}{p(b)}.$$

A standard CHSH expression has the local-hidden-variable bound

$$\left| E(a,b) + E(a,b') + E(a',b) - E(a',b') \right| \le 2,$$

while suitable quantum states and measurements can reach $2\sqrt2$. See the foundations bridge for the distinction between entanglement, steering, Bell nonlocality, no-signaling, and measurement update.

Quantum Optics Shorthands

A low-gain pair source can be written schematically as

$$\lvert\Psi\rangle \approx \lvert0\rangle + \epsilon \sum_{\mu,\nu} f_{\mu\nu}\, a_\mu^\dagger b_\nu^\dagger \lvert0\rangle + O(\epsilon^2).$$

After selecting the one-pair component, signal-idler entanglement is present when the normalized amplitude $F_{\mu\nu}$ is not factorizable:

$$\lvert\psi_2\rangle = \sum_{\mu,\nu} F_{\mu\nu} \lvert1_\mu\rangle_s \lvert1_\nu\rangle_i.$$

A polarization Bell-like state is

$$\lvert\Phi_\phi\rangle_{\mathrm{pol}} = \frac{ \lvert H\rangle_s\lvert H\rangle_i + e^{i\phi} \lvert V\rangle_s\lvert V\rangle_i }{\sqrt2}.$$

A single-photon path-entangled state is

$$\lvert\psi\rangle = \frac{ \lvert1\rangle_A\lvert0\rangle_B + e^{i\phi} \lvert0\rangle_A\lvert1\rangle_B }{\sqrt2}.$$

Homodyne detection measures phase-dependent quadratures such as

$$q_\theta = \frac{ a e^{-i\theta} + a^\dagger e^{i\theta} }{\sqrt2}.$$

For the two-mode squeezed vacuum convention used above,

$$\operatorname{Var}(q_A-q_B) = e^{-2r}, \qquad \operatorname{Var}(p_A+p_B) = e^{-2r}.$$

See the quantum-optics bridge for mode labels, down-conversion cautions, polarization and path entanglement, and homodyne interpretation.

Many-Body Entanglement Shorthands

For a spatial, site, orbital, or mode region $A$,

$$\mathcal H \cong \mathcal H_A\otimes\mathcal H_{\bar A}, \qquad \rho_A = \operatorname{Tr}_{\bar A}\rho.$$

For a pure global state, the entanglement entropy across the $A\vert\bar A$ cut is

$$S_A = -\operatorname{Tr}(\rho_A\log\rho_A).$$

In the many-body bridge, common scaling shorthands are

$$S_A \sim \alpha\,\lvert\partial A\rvert \quad \text{area law},$$

and

$$S_A \propto \lvert A\rvert \quad \text{volume law}.$$

The entanglement Hamiltonian is defined, up to additive constants, by

$$\rho_A = \frac{e^{-H_E}}{Z_E}, \qquad Z_E = \operatorname{Tr}(e^{-H_E}).$$

For a matrix product state with bond dimension $\chi$ across a cut,

$$S_A \le \log\chi.$$

For two-dimensional topologically ordered gapped phases, a common preview form is

$$S(A) = \alpha\,\lvert\partial A\rvert - \gamma + \cdots, \qquad \gamma = \log\mathcal D$$

in standard cases, where $\mathcal D$ is the total quantum dimension. Extracting $\gamma$ requires region combinations that cancel nonuniversal boundary terms.

QFT Entanglement Shorthands

For a regulated spatial region $A$ in a field theory,

$$\rho_A = \operatorname{Tr}_{\bar A} \lvert0\rangle\langle0\rvert, \qquad S_A = -\operatorname{Tr}(\rho_A\log\rho_A).$$

In $d>1$ spatial dimensions, a common leading cutoff behavior is

$$S_A \sim \alpha\, \frac{\operatorname{Area}(\partial A)}{\epsilon^{d-1}} + \cdots.$$

For a one-dimensional conformal field theory interval of length $\ell$,

$$S_A = \frac{c}{3} \log\frac{\ell}{\epsilon} + \mathrm{const}.$$

The modular Hamiltonian shorthand is

$$K_A = -\log\rho_A, \qquad \rho_A = e^{-K_A},$$

up to additive constants if a separate normalization is used. Relative entropy can be written as

$$D(\rho\Vert\sigma) = \Delta\langle K_\sigma\rangle - \Delta S \ge 0, \qquad K_\sigma=-\log\sigma.$$

In holographic theories, the simplest static classical relation is

$$S_A = \frac{\operatorname{Area}(\gamma_A)}{4G_N}.$$

These are preview shorthands. See Entanglement in QFT Preview for regulator, region, algebra, modular-Hamiltonian, and holography cautions.

Quantum Chemistry Shorthands

In an orthonormal spin-orbital basis, an $N$-electron determinant is

$$\lvert\Phi\rangle = a_{i_1}^\dagger a_{i_2}^\dagger \cdots a_{i_N}^\dagger \lvert0\rangle.$$

A correlated finite-basis state can be expanded as

$$\lvert\Psi\rangle = \sum_I C_I\lvert\Phi_I\rangle.$$

Relative to a reference determinant $\lvert\Phi_0\rangle$, common excitation notation is

$$\lvert\Phi_i^a\rangle = a_a^\dagger a_i \lvert\Phi_0\rangle, \qquad \lvert\Phi_{ij}^{ab}\rangle = a_a^\dagger a_b^\dagger a_j a_i \lvert\Phi_0\rangle.$$

For a spatial orbital $p$ with local basis

$$\lvert0\rangle, \qquad \lvert\uparrow\rangle, \qquad \lvert\downarrow\rangle, \qquad \lvert\uparrow\downarrow\rangle,$$

the one-orbital entropy is

$$s_p = -\operatorname{Tr}(\rho_p\log\rho_p), \qquad \rho_p = \operatorname{Tr}_{\bar p}\rho.$$

For orbitals $p,q$, the orbital mutual information is

$$I(p:q) = s_p+s_q-s_{pq}, \qquad s_{pq} = -\operatorname{Tr}(\rho_{pq}\log\rho_{pq}).$$

These are basis-dependent mode-entanglement diagnostics. See the quantum-chemistry bridge for cautions about antisymmetry, electron correlation, orbital basis choices, and spin entanglement.

Cross-Links

• Composite Systems and Entanglement
• Common Composite States
• Entanglement Diagnostic Table
• Computational Notebooks
• Tensor Products of Hilbert Spaces
• Operators on Composite Systems
• Tensor Product Exercises
• Separable Mixed States
• Bell States
• Singlet and Triplet States
• Partial Trace
• Partial Trace Exercises
• Local Measurement Statistics
• Conditional States
• Purification
• Marginals and Correlations
• Subsystem Entropy
• Schmidt Decomposition
• Schmidt Rank
• Entanglement Entropy
• Renyi Entropies
• Mutual Information
• GHZ States
• W States
• Symmetrization Postulate
• Symmetric and Antisymmetric Wavefunctions
• Identical Particle Exercises
• Fock Space Exercises
• Slater Determinants
• Spin and Spatial Wavefunctions
• Pauli Exclusion Principle
• Occupation-Number Basis
• Number States
• Bosonic Fock Space
• Fermionic Fock Space
• Fock Space Examples
• Creation and Annihilation Operators
• Number Operators
• Mode Expansions
• Field Operators
• One-Body Operators
• Two-Body Operators
• Many-Particle Hamiltonians
• Normal Ordering
• Wick’s Theorem Preview
• Bosonic Commutation Relations
• Fermionic Anticommutation Relations
• Second Quantization: Bridge to QFT
• Continuous-Variable Systems
• Position-Space Two-Particle States
• Mode Decompositions
• Two-Mode Entanglement
• EPR State Preview
• Gaussian States Preview
• Squeezed States as Entangled Modes
• Entanglement in Quantum Information
• Entanglement in Many-Body Physics
• Entanglement in Quantum Chemistry
• Entanglement in Quantum Optics
• Entanglement in Foundations
• Entanglement in QFT Preview

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