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For a density operator $\rho$ on a finite-dimensional Hilbert space, the Von Neumann entropy iswith $0\log 0=0$. Its concavity says that, for $0\le p\le 1$,

If $\epsilon =0$, then ${\rho }_{AB}={\sigma }_{AB}$ and the desired continuity bound is immediate, so assume $\epsilon >0$. Apply the positive-negative decomposition of a Hermitian operator toBecause $\mathrm{Tr}X=0$ and $\Vert X{\Vert }_1=2\epsilon$, one hasThus ${\Delta }_{AB}=X_{+}/\epsilon$ is positive with trace one, hence is a density operator. DefineIt is a convex combination of states. The equation $\epsilon {\Delta }_{AB}^{\prime }=(1+\epsilon ){\omega }_{AB}-{\rho }_{AB}$ giveswhich is likewise positive and has trace one.

For a subsystem $A$ of a multipartite system with density operator ${\rho }_{A{E}_1\cdots E_n}$, its reduced density matrix is the unique operatorsuch that $\mathrm{Tr}(M_A{\rho }_A)=\mathrm{Tr}[(M_A\otimes I_E){\rho }_{AE}]$ for every observable $M_A$ on $A$. In an orthonormal basis $\{|e_j⟩\}$ of the other subsystems, the partial trace iswhich is independent of the chosen basis.

For a two-qubit density operator, the real three-by-three matrix $T$ encodes the spin-product expectation values: unit measurement axes $\mathbf{a},\mathbf{b}$ have $E(\mathbf{a},\mathbf{b})={\mathbf{a}}^{T}T\mathbf{b}$. The tensor does not include local Bloch-vector data and alone need not specify the complete density operator. The CHSH optimum from the correlation tensor depends on the two largest eigenvalues of $T^{T}T$.

A finite-dimensional bipartite density operator ${\rho }_{AB}$ is a separable quantum state exactly when $({\mathrm{id}}_A\otimes \Phi )({\rho }_{AB})\ge 0$ for every positive linear map from operators on $B$ to operators on $A$. In combination with the Størmer-Woronowicz decomposability theorem, it makes the positive partial transpose criterion sufficient for qubit-qutrit states. The criterion is established in the Horodeckis' separability paper.

A bipartite density operator has positive partial transpose when ${\rho }^{T_B}\ge 0$. Every separable quantum state has this property, since transposing one factor of a positive product state preserves positivity and the separable sum preserves it as well. The property does not depend on which local basis defines the transpose.

Every density operator $\sigma$ with positive partial transpose has squared quantum fidelity at most $1/d$ with a fixed maximally entangled state on $d\otimes d$. The partial transpose of a maximally entangled projector and the trace-adjoint identity for partial transpose express that squared overlap as $\mathrm{Tr}(S{\sigma }^{T_B})/d$. Since ${\sigma }^{T_B}$ is positive of trace one and the swap operator has operator norm one, trace duality gives the bound. A matching product state attains equality.

A bipartite density operator is a product state when it factors as ${\rho }_A\otimes {\rho }_B$. A pure product state factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$; its reduced density matrices have rank one and zero entanglement entropy. Mixed product states can have nonzero marginal Von Neumann entropy.

A pure quantum state is represented by a unit vector $|\psi ⟩$ up to global phase, or equivalently by the rank-one density operator $|\psi ⟩⟨\psi |$.

For a density operator $\rho$ and pure state $P=|\psi ⟩⟨\psi |$, the diagonal absolute-sum bound for the trace norm in a basis containing $|\psi ⟩$ gives $D(\rho ,P)\ge 1-⟨\psi |\rho |\psi ⟩=1-F(\rho ,P{)}^2$. The quantum fidelity is unsquared. Block-diagonal states with respect to the target and its complement attain equality.

QMA is the class of promise problems with a polynomial-size quantum witness checked by a uniform polynomial-size quantum circuit. YES instances have an accepting witness with probability at least $2/3$; NO instances have acceptance at most $1/3$ for every witness, including mixed states. The verifier's acceptance is linear in the witness density operator. Inverse-polynomial or exponentially small error can be obtained through QMA error reduction.

For a tripartite density operator, the quantum conditional mutual information is $I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)$. It is nonnegative by Strong subadditivity of Von Neumann entropy. It measures the coherent information lost when a channel environment is discarded after an isometric dilation.

For any tripartite density operator, expanding the three quantum mutual information terms in Von Neumann entropy gives $I(X:BD)=I(X:B)+I(XB:D)-I(B:D)$. This algebraic identity allows separate bounds on two information contributions, while nonnegativity of quantum relative entropy allows the last term to be dropped in an upper bound.

Quantum state tomography estimates enough expectation values on many identically prepared systems to reconstruct their density operator. A single ordinary specimen cannot provide arbitrarily many independent samples without disturbance.

A normalized quantum state is a linear combination of orthonormal basis vectors in a Hilbert space. Its density operator is $|\psi ⟩⟨\psi |$. When at least two basis amplitudes are nonzero, it has off-diagonal terms and differs from the mixed state ${\sum }_j|a_j{|}^2|j⟩⟨j|$. Quantum gates act linearly on this superposition and preserve its phase information.

For a density operator $\pi ={\sum }_i{q}_i|{\varphi }_i⟩⟨{\varphi }_i|$, the quantum typical subspace is spanned by product eigenvectors of ${\pi }^{\otimes n}$ whose eigenvalues $q_{i_1}\cdots q_{i_n}$ satisfyIt applies the classical weakly typical sequence definition to the spectrum. If $P_{\epsilon }^{(n)}$ is its orthogonal projection, thenThe first statement follows from the typical-set cardinality bounds, and the second from the weak law of large numbers. This subspace holds nearly all the source probability while using exponentially fewer dimensions than the whole space when $S(\pi )<{\log }_{2}d$.

A quantum witness is a polynomial-size quantum state supplied to a verifier as evidence for a YES instance. The verifier fixes its own ancilla qubits independently. Pure quantum witnesses suffice when maximizing a linear acceptance functional on density operators. Quantum witnesses are not generally restricted to basis vectors or product states.

If $\rho$ is a density operator with decreasing eigenvalues ${\lambda }_j$ and $T$ has matrix rank at most $k$, let $\Pi$ project onto the image of $T$. The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product givesThe Hermitian effect variational principle bounds $\mathrm{Tr}(\rho \Pi )$ by ${\sum }_{j<k}{\lambda }_j$, proving the claimed estimate without assuming $\rho$ is invertible.

The ensemble average $\overline{\omega }$ uniquely minimizes ${\sum }_j{p}_{j}D({\omega }_j\Vert \rho )$ over density operators $\rho$. Donald's identity separates this objective into a constant plus $D(\overline{\omega }\Vert \rho )$, and nonnegativity of quantum relative entropy proves the claim. The minimum is the Holevo quantity $S(\overline{\omega })-{\sum }_j{p}_{j}S({\omega }_j)$, equivalently the quantum mutual information of the associated classical-quantum state.