Von Neumann entropy ($S(\rho )$)

The Von Neumann entropy of a density operator isIt vanishes for a pure state and measures mixedness and quantum entanglement in reduced states.

A mixture of two orthogonal pure qubit states has Von Neumann entropy equal to the binary entropy of its weights. Its entropy difference from the first pure state is exactly $h_2(u)$ and its trace distance from that state is $u$. The elementary bound $-(1-u)\ln (1-u)\le u$ gives $h_2(u)\le u{\log }_2(e/u)$; entropy also never exceeds one bit. For small $u$, the leading term is $u{\log }_2(1/u)$.

If a density operator $\sigma$ on an $m\ge 2$ dimensional Hilbert space has overlap $t=⟨\psi |\sigma |\psi ⟩$ with a fixed pure state, then $S(\sigma )\le h(t)+(1-t){\log }_2(m-1)$. Dephase in a basis containing $|\psi ⟩$ and apply the entropy bound with one prescribed probability. The abstract state bound is attained by $\sigma =t|\psi ⟩⟨\psi |+(1-t)(I-|\psi ⟩⟨\psi |)/(m-1)$.

On a $d$-dimensional Hilbert space, the Von Neumann entropy is maximized uniquely by $I/d$. Indeed, nonnegativity of quantum relative entropy giveswith equality exactly when $\rho =I/d$.

If $\sigma ={\sum }_i{P}_i\rho P_i$ is a nonselective projective measurement, then $S(\sigma )\ge S(\rho )$. Equality holds exactly when $\rho$ is already block diagonal in the measurement decomposition, equivalently $[\rho ,P_i]=0$ for every $i$.

For a bipartite pure state, the entanglement entropy is the Von Neumann entropy of either reduced density matrix. The two reduced states have the same nonzero eigenvalues, so their entropies agree.

Every finite-dimensional bipartite pure state has a decomposition $|\psi ⟩={\sum }_{j=1}^r{s}_j|a_j⟩|b_j⟩$ with positive Schmidt coefficients $s_j$ and orthonormal families on each subsystem.

The operator Schmidt rank of a bipartite linear operator is its shortest expansion as a sum of product linear operators. Expanding in local orthonormal operator bases gives a coefficient matrix; its matrix rank is the operator Schmidt rank, by a singular value decomposition. It is unchanged by invertible local operator-basis changes. A rank-one projector onto a bipartite pure state of Schmidt rank $d$ has operator Schmidt rank $d^2$: its expansion contains all $d^2$ independent products of local matrix units.

If two parties share a pure resource of Schmidt rank $R$, a fully refined branch of their local operations has Kraus operator $K_{\mu \nu }={\sum }_{j=1}^R\sqrt{p_j}{A}_{\mu j}\otimes B_{\nu j}$. Hence its operator Schmidt rank is at most $R$. Local ancillas, locally adaptive readouts and refined discarded environments are included in the branch operators. Shared classical randomness only mixes such branches. Classical comparison of records does not increase their operator Schmidt rank.

For the pure state ${\lambda }_0|00⟩+{\lambda }_1|11⟩$ with real normalized Schmidt coefficients, the two-body Pauli correlators form $T=\mathrm{diag}(s,-s,1)$, $s=2{\lambda }_0{\lambda }_1$. Thus arbitrary local axes give $⟨(\mathbf{a}\cdot \sigma )\otimes (\mathbf{b}\cdot \sigma )⟩=s(a_x{b}_x-a_y{b}_y)+a_z{b}_z$. The negative $yy$ component follows from the phases in the Pauli operators.

The Schmidt coefficients of a normalized bipartite pure state are the nonnegative numbers $s_j$ in its Schmidt decomposition; they satisfy ${\sum }_j{s}_j^2=1$ and their squares are the nonzero eigenvalues of either reduced density matrix.

The Schmidt rank is the number of nonzero terms in a Schmidt decomposition, equivalently the rank of either reduced density matrix of a bipartite pure state.

The coefficient matrix of a bipartite pure state has rank equal to its Schmidt rank, by singular value decomposition. A local product operation $A\otimes B$ sends it to $AM{B}^{\mathsf{T}}$, whose rank cannot increase. A complete classical transcript of an LOCC protocol still selects one product Kraus operator, so every nonzero branch obeys the bound. Discarding the transcript gives a mixed output with Schmidt number at most the initial rank.

The Schmidt rank of a bipartite pure state cannot increase under local operations, even in a nonzero selected branch. A coefficient matrix $C$ changes under a local product operator to $AC{B}^T$, whose rank is at most that of $C$. For deterministic exact conversions the same conclusion follows from Nielsen's pure-state conversion theorem: at the input rank $r$, the majorization inequality forces all output coefficients after $r$ to vanish.

A normalized bipartite pure state of Schmidt rank $r$ has quantum fidelity at most $\sqrt{r/d}$ with a fixed maximally entangled state on $d\otimes d$. Its coefficient matrix has matrix rank $r$ and Hilbert-Schmidt norm one. Its overlap is the trace of that matrix divided by $\sqrt{d}$; trace duality and the Cauchy-Schwarz inequality bound this by $\sqrt{r/d}$. Uniform Schmidt coefficients on $r$ matching basis pairs attain equality.

The Schmidt number of a mixed bipartite state is the smallest $k$ for which it has a pure-state ensemble whose members all have Schmidt rank at most $k$. Schmidt number one is equivalent to separability.

For a bipartite density operator ${\rho }_{AB}$,Equality holds exactly when ${\rho }_{AB}={\rho }_A\otimes {\rho }_B$.

For a bipartite density operator ${\rho }_{AB}$,

For a tripartite density operator ${\rho }_{ABC}$,Equivalently, conditioning on an additional quantum system cannot increase quantum conditional entropy.

Apply data-processing inequality for quantum relative entropy to ${\rho }_{ABC}$ and ${\rho }_A\otimes {\rho }_{BC}$, tracing out $C$. The resulting comparison is $D({\rho }_{ABC}\Vert {\rho }_A\otimes {\rho }_{BC})\ge D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$. Expand the logarithms of the product marginals to obtain $S(AB)+S(BC)-S(B)-S(ABC)\ge 0$, exactly Strong subadditivity of Von Neumann entropy.

Every tripartite density operator satisfies $S(AC)+S(BC)\ge S(A)+S(B)$. Purify to $ABCD$ and apply Strong subadditivity of Von Neumann entropy to $ACD$, conditioned on $A$. The Schmidt decomposition gives $S(AD)=S(BC)$ and $S(ACD)=S(B)$, proving the inequality. The original tripartite state need not be pure.

For density operators ${\rho }_1,{\rho }_2$ and $0\le p\le 1$, Von Neumann entropy is concave:

The entropy of a binary mixture also satisfieswhere $H$ is binary entropy. This follows by applying the operator monotonicity of logarithm to $p{\rho }_1\le p{\rho }_1+(1-p){\rho }_2$ and its counterpart for ${\rho }_2$.

The quantum conditional entropy is $H(A|B{)}_{\rho }=S({\rho }_{AB})-S({\rho }_B)$. Unlike its classical counterpart, it may be negative.

For bipartite density operators,This is a consequence of Strong subadditivity of Von Neumann entropy.

If $\epsilon =\Vert {\rho }_{AB}-{\sigma }_{AB}{\Vert }_1/2$ and $d_A=\dim {\mathcal{H}}_A$, thenThe proof couples the two states through their positive and negative differences, then combines concavity of quantum conditional entropy with the entropy bound for a binary mixture.

For every state on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$,The upper bound follows from Subadditivity of Von Neumann entropy and the lower bound from the Araki–Lieb inequality.

The quantum relative entropy is $D(\rho \Vert \sigma )=\mathrm{Tr}[\rho (\log \rho -\log \sigma )]$ when the support of $\rho$ lies in that of $\sigma$.

For density operators, Klein's inequality gives $D(\rho \Vert \sigma )\ge 0$, with equality exactly when $\rho =\sigma$. If the support of a positive operator $\rho$ is not contained in that of $\sigma$, the quantum relative entropy is $+\infty$.

In bits, the quantum Pinsker inequality isThe trace-distance-preserving binary measurement converts the trace distance into classical total variation distance without loss. Apply the data-processing inequality for quantum relative entropy to that measurement, then classical Pinsker's inequality. The factor $\ln 2$ is omitted when relative entropy uses natural logarithms.

For an ensemble of density operators with average $\overline{\omega }={\sum }_j{p}_j{\omega }_j$,Expanding the trace definitions cancels the $\log {\omega }_j$ terms, and the remaining sum combines to a trace against $\overline{\omega }$. Positive-weight ensemble states have their support contained in that of the average, so the first sum on the right is finite in finite dimension. If the reference-state support condition fails, both sides are $+\infty$. The identity is associated with Donald's work on relative entropy.

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.

For every quantum channel $T$,For a partial trace, this says that discarding a subsystem cannot make two density operators more distinguishable in quantum relative entropy.

The quantum relative entropy is additive on tensor products:This follows by expanding $\log ({\sigma }_A\otimes {\sigma }_B)=\log {\sigma }_A\otimes I+I\otimes \log {\sigma }_B$ and its counterpart for $\rho$.

For a bipartite density operator ${\rho }_{AB}$ and a product reference state,The difference between the two sides is the nonnegative quantum mutual information of ${\rho }_{AB}$.

Repeated application of superadditivity of quantum relative entropy gives

If $\Vert {\rho }_n^{\prime }-{\rho }^{\otimes n}{\Vert }_1\to 0$, thenIn finite dimension this follows from multipartite superadditivity of quantum relative entropy, additivity of quantum relative entropy, contraction of trace distance under partial trace, and uniform continuity on the compact state space.

The quantum mutual information of a bipartite state isIts nonnegativity is equivalent to Subadditivity of Von Neumann entropy.

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.

A local quantum channel on either subsystem cannot increase quantum mutual information. Indeed, apply the data-processing inequality for quantum relative entropy to $D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$; the channel maps the product of marginals to the product of the new marginals.

For a channel on $B$, use a Stinespring dilation $B\to B^{\prime }E$. Isometry invariance gives $I(A:B)=I(A:B^{\prime }E)$, so the loss on discarding $E$ is $I(A:B)-I(A:B^{\prime })=I(A:E\mid B^{\prime })$. This quantum conditional mutual information is nonnegative by Strong subadditivity of Von Neumann entropy. The identity holds for arbitrary mixed inputs and identifies the correlations lost to the discarded environment.

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.

The quantum conditional entropy has the relative-entropy representationwhere the minimum ranges over density operators on $B$.