Mutual Information

Quantum mutual information measures the total correlation between two subsystems of a quantum state. For a bipartite density operator $\rho_{AB}$,

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

where

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

and

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

is the von Neumann entropy.

The density-operator first encounter with this entropy is Entropy Overview.

Mutual information is a correlation measure, not an entanglement measure for arbitrary mixed states. It counts all correlations: classical correlation, quantum correlation, and pure-state entanglement contributions. It vanishes exactly for product density operators.

Classical Analogy

For classical random variables $X$ and $Y$ with joint distribution $p(x,y)$, the classical mutual information is

$$I(X:Y) = H(X)+H(Y)-H(X,Y),$$

where $H$ is Shannon entropy.

Equivalently,

$$I(X:Y) = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p_X(x)p_Y(y)}.$$

This form shows that mutual information is a relative entropy comparing the joint distribution with the product of its marginals. It is zero exactly when

$$p(x,y)=p_X(x)p_Y(y).$$

Quantum mutual information is the density-operator analogue. It compares $\rho_{AB}$ with $\rho_A\otimes\rho_B$.

Quantum Definition

For a finite-dimensional bipartite state $\rho_{AB}$, define

$$I(A:B)_\rho = S_A+S_B-S_{AB},$$

with

$$S_A=S(\rho_A), \qquad S_B=S(\rho_B), \qquad S_{AB}=S(\rho_{AB}).$$

The logarithm base fixes the units. Base $2$ gives bits, and natural logarithms give nats.

Quantum mutual information satisfies

$$I(A:B)_\rho\ge0,$$

and

$$I(A:B)_\rho=0 \quad \Longleftrightarrow \quad \rho_{AB}=\rho_A\otimes\rho_B.$$

The nonnegativity is equivalent to subadditivity of von Neumann entropy:

$$S_{AB}\le S_A+S_B.$$

Thus mutual information measures exactly how far the joint entropy falls below the sum of the local entropies.

Product States

For a product density operator,

$$\rho_{AB} = \rho_A\otimes\rho_B,$$

the entropy is additive:

$$S(\rho_A\otimes\rho_B) = S(\rho_A)+S(\rho_B).$$

Therefore

$$I(A:B) = S_A+S_B-S_{AB} = 0.$$

This matches the operational meaning: a product state contains no correlations between measurements on $A$ and measurements on $B$.

For a pure product state, the same conclusion is immediate because

$$S_A=S_B=S_{AB}=0.$$

Classically Correlated Separable State

Consider

$$\rho_{\mathrm{cc}} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

The reduced states are

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

so, in bits,

$$S_A=S_B=1.$$

The joint state has two nonzero eigenvalues, $1/2$ and $1/2$, so

$$S_{AB}=1.$$

Therefore

$$I(A:B) = 1+1-1 = 1$$

bit.

This state is separable. Its nonzero mutual information is ordinary classical correlation: the two computational-basis outcomes share one bit of common information.

Entangled Pure States

If $\rho_{AB}$ is pure, then

$$S_{AB}=0.$$

For a bipartite pure state, the reduced-state entropies are equal:

$$S_A=S_B.$$

Thus

$$I(A:B) = 2S_A = 2S_B.$$

For pure bipartite states, mutual information is twice the entanglement entropy. The factor of two is important: a Bell state has one bit of entanglement entropy but two bits of quantum mutual information.

For

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

we have

$$\rho_A=\rho_B=\frac12 I, \qquad S_A=S_B=1, \qquad S_{AB}=0.$$

Therefore

$$I(A:B)=2$$

bits.

Same Marginals, Different Mutual Information

The Bell state, the classically correlated state $\rho_{\mathrm{cc}}$, and the product mixed state

$$\rho_{\mathrm{prod}} = \frac12 I_A\otimes\frac12 I_B$$

all have

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

But their mutual informations differ:

$$\begin{array}{c|c|c|c|c} \text{state} & S_A & S_B & S_{AB} & I(A:B)\\ \hline \text{Bell pure state} & 1 & 1 & 0 & 2\\ \rho_{\mathrm{cc}} & 1 & 1 & 1 & 1\\ \rho_{\mathrm{prod}} & 1 & 1 & 2 & 0 \end{array}$$

All entries are in bits. The local marginals are identical; the joint state determines the correlation content.

Relation to Relative Entropy

The classical comparison measure is Relative Entropy. The finite-dimensional quantum relative entropy is

$$D(\rho\Vert\sigma) = \operatorname{Tr} \bigl[ \rho(\log\rho-\log\sigma) \bigr],$$

when the support of $\rho$ is contained in the support of $\sigma$.

Quantum mutual information can be written as

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

This identity makes the interpretation precise: mutual information is the relative entropy distance from the joint state to the uncorrelated product state with the same marginals.

It also explains why

$$I(A:B)_\rho\ge0,$$

because quantum relative entropy is nonnegative, and why equality occurs only for

$$\rho_{AB} = \rho_A\otimes\rho_B.$$

This is a preview of a broader information-theoretic toolkit. Full proofs of relative-entropy monotonicity and strong subadditivity belong to quantum information and mathematical quantum theory.

Conditional Entropy Identities

Define the quantum conditional entropy

$$S(A\vert B) = S_{AB}-S_B.$$

Then

$$I(A:B) = S_A-S(A\vert B) = S_B-S(B\vert A).$$

Unlike classical conditional entropy, quantum conditional entropy can be negative. For a Bell state,

$$S(A\vert B) = 0-1 = -1$$

bit. This negativity is a quantum-information signal tied to entanglement, but mutual information itself still measures total correlation rather than only entanglement.

Many-Body and QFT Preview

In many-body physics, $I(A:B)$ is used to quantify total correlation between spatial regions, lattice blocks, sites, modes, or subsystems. It includes relationships not visible in one chosen two-point function. Connected Correlation Functions owns the complementary operator-cumulant hierarchy and explains why one selected connected correlator is neither a complete total-correlation measure nor an entanglement witness.

In thermal states, mutual information can be nonzero because of thermal and classical correlations even when entanglement is absent or small. In ground states, it can help compare correlation structure across regions and system sizes.

In QFT, individual entanglement entropies of spatial regions are often regulator-dependent. Mutual information between separated regions is often better behaved because some local boundary divergences cancel, though adjacent regions and continuum limits still require care. The finite-dimensional formula above is the clean starting point, not the full field-theoretic analysis.

For the broader many-body setting where mutual information is compared with entanglement entropy, area laws, thermalization, and localization, see Entanglement in Many-Body Physics. For the continuum role of mutual information beside modular Hamiltonians and relative entropy, see Entanglement in QFT Preview.

Common Mistakes

• Treating mutual information as an entanglement measure for arbitrary mixed states.
• Forgetting the factor of two for pure bipartite states: $I(A:B)=2S_A$.
• Computing $S_A+S_B$ but forgetting to subtract $S_{AB}$.
• Looking only at marginals and assuming mutual information is determined.
• Comparing values without specifying the logarithm base.
• Treating zero connected correlation for one observable pair as equivalent to zero mutual information.
• Assuming QFT mutual information is always finite without checking the geometry and regulator assumptions.

Cross-Links

• Marginals and Correlations
• Subsystem Entropy
• Entropy Overview
• Entanglement Entropy
• Renyi Entropies
• Concurrence for Two Qubits
• Negativity and PPT Criterion
• Entanglement Witnesses
• LOCC Preview
• Classical Correlation versus Entanglement
• Product States
• Separable Mixed States
• Bell States
• Reduced Density Operators
• Entanglement in Many-Body Physics
• Thermal Entropy vs Entanglement Entropy
• Mutual Information in Many-Body Systems — spatial geometries, thermal cancellation, correlator bounds, constrained ensembles, and extraction methods.
• Entanglement in QFT Preview
• Formula Sheet

References

• C. E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal 27, 379-423 and 623-656, 1948.
• H. Araki and E. H. Lieb, “Entropy Inequalities,” Communications in Mathematical Physics 18, 160-170, 1970.
• A. Wehrl, “General Properties of Entropy,” Reviews of Modern Physics 50, 221-260, 1978.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley, 2006.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

Exercises

1. Show that a product state has zero mutual information.

Solution

If $\rho_{AB}=\rho_A\otimes\rho_B$, then

$$S_{AB} = S(\rho_A\otimes\rho_B) = S_A+S_B.$$

Therefore

$$I(A:B) = S_A+S_B-S_{AB} = 0.$$

2. Compute $I(A:B)$ for $\rho_{\mathrm{cc}}$ in bits.

Solution

For

$$\rho_{\mathrm{cc}} = \frac12\lvert00\rangle\langle00\rvert + \frac12\lvert11\rangle\langle11\rvert,$$

we have $\rho_A=\rho_B=I/2$, so $S_A=S_B=1$ bit. The joint state has two nonzero eigenvalues $1/2$ and $1/2$, so $S_{AB}=1$ bit. Hence

$$I(A:B) = 1+1-1 = 1$$

bit.

3. Compute $I(A:B)$ for a Bell state in bits.

Solution

A Bell state is pure, so $S_{AB}=0$. Each one-qubit reduction is $I/2$, so $S_A=S_B=1$ bit. Therefore

$$I(A:B) = 1+1-0 = 2$$

bits.

4. For a pure bipartite state, show that $I(A:B)=2S_A$.

Solution

If $\rho_{AB}$ is pure, then $S_{AB}=0$. The reduced states of a pure bipartite state have the same nonzero eigenvalues, so $S_A=S_B$. Thus

$$I(A:B) = S_A+S_B-S_{AB} = S_A+S_A-0 = 2S_A.$$

5. Why can a separable state have nonzero mutual information?

Solution

Separable means the state can be written as a classical mixture of product states. The shared classical label in the mixture can correlate the subsystems. Mutual information counts total correlation, including classical correlation, so it can be nonzero even when the state is not entangled.