Normal Product States and Trace-Norm Closure

A normal bipartite state is a product exactly when every bounded local correlation factorizes. This is an operator-level statement involving all local tests, not a check on a few chosen moments. The canonical Product States article explains the physical definition and elementary factorization tests; here the factors may be infinite-dimensional.

Helpful background. Trace-Class Operators supplies the normal-state pairing. Trace and Partial Trace establishes the trace-class reduction map. Hilbert Space Tensor Products supplies bounded local operators.

Products of density operators remain density operators

Work with separable Hilbert spaces $\mathcal H_A,\mathcal H_B$ and normal states represented by density operators. If $\rho_A=\sum_i p_i|u_i\rangle\langle u_i|$ and $\rho_B=\sum_j q_j|v_j\rangle\langle v_j|$, then

$$\rho_A\otimes\rho_B =\sum_{i,j}p_iq_j |u_i\otimes v_j\rangle\langle u_i\otimes v_j|.$$

All terms are positive and $\sum_{i,j}p_iq_j=1$. The sum converges in trace norm, proving positivity and normalization of the product state.

More generally, for trace-class $R,S$,

$$\|R\otimes S\|_1=\|R\|_1\|S\|_1, \qquad \operatorname{Tr}(R\otimes S) =\operatorname{Tr}R\,\operatorname{Tr}S.$$

One obtains the norm identity from the products of singular values, whose double sum is absolutely convergent. Finite-rank approximation then proves the trace identity. This argument avoids manipulating an unconditionally unjustified double series.

The two partial traces of $\rho_A\otimes\rho_B$ are its declared factors. Consequently a product joint state, if it exists with given marginals, is unique.

Factorization of all bounded local tests is sufficient

For any bounded $X$ on $A$ and $Y$ on $B$, the product state obeys

$$\operatorname{Tr}\bigl[(\rho_A\otimes\rho_B)(X\otimes Y)\bigr] =\operatorname{Tr}(\rho_AX)\operatorname{Tr}(\rho_BY).$$

Conversely, let $\rho$ have marginals $\rho_A,\rho_B$ and suppose this identity holds with $\rho$ on the left for every bounded $X,Y$. Put $T=\rho-\rho_A\otimes\rho_B$. All pairings $\operatorname{Tr}[T(X\otimes Y)]$ vanish.

Choose matrix units in orthonormal bases of the two factors. Their tensor products test every matrix element of $T$ in the product basis. Each matrix element therefore vanishes, so $T=0$. Thus

$$\rho=\rho_A\otimes\rho_B \quad\Longleftrightarrow\quad \text{all bounded local correlations factorize}.$$

It suffices to test all local effects: their complex linear spans contain every bounded operator. Testing all local projection probabilities also suffices through spectral approximation.

By contrast, equality for one pair of observables, or even vanishing covariance for a short list, does not establish product structure. A restricted measurement family may miss the correlations entirely.

Product structure is stable under trace-norm limits

For density operators $\rho_A,\rho_B,\sigma_A,\sigma_B$, add and subtract $\sigma_A\otimes\rho_B$ to obtain

$$\|\rho_A\otimes\rho_B-\sigma_A\otimes\sigma_B\|_1 \leq\|\rho_A-\sigma_A\|_1+\|\rho_B-\sigma_B\|_1.$$

The tensor trace-norm identity and unit traces prove the bound. It gives a quantitative estimate for independently approximated subsystem states.

It also proves that the set of normal product states is trace-norm closed. Suppose product states $\rho_A^{(n)}\otimes\rho_B^{(n)}$ converge in trace norm to $\rho$. Contractivity of partial trace forces the factors to converge to $\rho_A,\rho_B$. The displayed estimate then makes the products converge to $\rho_A\otimes\rho_B$, so uniqueness of limits gives $\rho=\rho_A\otimes\rho_B$.

Closedness is different from convexity. Mixing products can create classical correlations; the larger closed convex set is the separable set. Its definition and infinite-dimensional closure requirement are developed in Separable and Entangled Mixed States.

Correlated number states with thermal marginals

Let $(|n\rangle)_{n\geq0}$ be number bases and $0<q<1$. Consider

$$\rho_{\mathrm{corr}} =(1-q)\sum_{n=0}^\infty q^n |n,n\rangle\langle n,n|.$$

It is a trace-norm convergent mixture of product projectors. Both marginals are the geometric state $\tau_q=(1-q)\sum_nq^n|n\rangle\langle n|$, but the joint state is not $\tau_q\otimes\tau_q$.

A bounded test already detects the difference. With $P_0=|0\rangle\langle0|$,

$$\operatorname{Tr}[\rho_{\mathrm{corr}}(P_0\otimes P_0)] =1-q,$$

whereas the product of the two marginal probabilities is $(1-q)^2$. The state is separable and correlated.

The number covariance is also nonzero, but number is unbounded. Here its moments are finite because of the geometric tail, so that calculation is legitimate. For a general normal state, unbounded means or products may diverge. The bounded-effect criterion avoids that unnecessary integrability assumption.

Pure products and mixed products

The product state’s purity factorizes:

$$\operatorname{Tr}[(\rho_A\otimes\rho_B)^2] =\operatorname{Tr}(\rho_A^2)\operatorname{Tr}(\rho_B^2).$$

Each factor lies in $(0,1]$, with equality to one exactly for a rank-one density operator. Thus a product density operator is pure exactly when both local factors are pure. A mixed product can have arbitrarily mixed marginals while carrying no correlations; mixedness alone is not entanglement.

Exercises

1. If $\|\rho-\sigma_A\otimes\sigma_B\|_1\leq\varepsilon$, bound the distances between $\rho$‘s marginals and $\sigma_A,\sigma_B$.

Solution

Each marginal distance is at most $\varepsilon$ by trace-norm contractivity of partial trace. Consequently $\|\rho_A\otimes\rho_B-\sigma_A\otimes\sigma_B\|_1\leq2\varepsilon$, and the triangle inequality gives $\|\rho-\rho_A\otimes\rho_B\|_1\leq3\varepsilon$. This is a useful sufficient bound, without a claim of optimal constants.

2. Find the number covariance in the correlated geometric example.

Solution

The two numbers are equal in every term. Their covariance is therefore the geometric variance $q/(1-q)^2$. For the product of the same marginals the covariance is zero. These statements use finite second moments, unlike the preceding bounded-projector test.

References

• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, second revised and expanded edition, De Gruyter, 2019 — normal states, tensor products, and trace-class reductions. Publisher.
• Simon, Barry, Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs 120, American Mathematical Society, 2005 — singular values and trace-ideal estimates. Publisher.