Local Measurement Statistics

Local measurement statistics are the probabilities seen by an observer who can act only on one subsystem of a composite quantum system. The reduced density operator is physically meaningful because it reproduces exactly those statistics.

If the joint state on $\mathcal H_A\otimes\mathcal H_B$ is $\rho_{AB}$ and

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

then every measurement performed only on $A$ can be computed from $\rho_A$ alone. The discarded subsystem $B$ can still carry correlations with $A$, but it is not needed for local outcome probabilities on $A$.

The shorter Core-level bridge is Subsystems and Local Observables.

Local Observables

An operator is local to subsystem $A$ if it has the form

$$M_A\otimes I_B.$$

The expectation value in the joint state is

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

The defining property of the partial trace gives

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

This identity is the operational meaning of $\rho_A$: it is the unique density operator on $\mathcal H_A$ that gives the same expectation values for all observables accessible on subsystem $A$.

Projective Measurements on a Subsystem

For a projective measurement on $A$ with projectors $\{P_a\}$,

$$P_aP_{a'} = \delta_{aa'}P_a, \qquad \sum_a P_a=I_A,$$

the corresponding measurement on the composite system uses projectors

$$P_a\otimes I_B.$$

The Born-rule probability for outcome $a$ is

$$\begin{aligned} p(a) &= \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(P_a\otimes I_B) \bigr]\\ &= \operatorname{Tr}_A(\rho_A P_a). \end{aligned}$$

Thus the same local probabilities are obtained whether one starts from the full joint state and inserts an identity on $B$, or first reduces to $\rho_A$ and then applies the ordinary density-operator Born rule.

General Measurement Effects

The same statement holds for generalized measurement probabilities. Let $\{E_a\}$ be positive operators on $\mathcal H_A$ satisfying

$$E_a\ge0, \qquad \sum_a E_a=I_A.$$

The operators $E_a$ are the effects of a POVM. If this measurement is performed locally on $A$, the corresponding effects on the joint system are

$$E_a\otimes I_B.$$

The probability of outcome $a$ is

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

This is why reduced density operators are not tied to projective measurements only. They summarize all statistics obtainable from any measurement whose effects act on the local subsystem.

What Local Statistics Determine

If enough local measurements are performed on identically prepared copies, their statistics can reconstruct $\rho_A$. For a qubit, for example,

$$\rho_A = \frac12 \bigl( I+\mathbf r\cdot\boldsymbol\sigma \bigr),$$

where the Bloch vector components are

$$r_i = \operatorname{Tr}(\rho_A\sigma_i), \qquad i\in\{x,y,z\}.$$

Measurements of $\sigma_x$, $\sigma_y$, and $\sigma_z$ on subsystem $A$ determine $\mathbf r$ and therefore determine $\rho_A$.

They do not determine $\rho_{AB}$. Many different joint states can have the same reduced state. Local tomography reconstructs the local state, not the global state.

Same Local Statistics, Different Global States

The Bell state

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

has

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

The separable mixed state

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

has the same reduced states:

$$\operatorname{Tr}_B\rho_{\mathrm{cc}} = \operatorname{Tr}_A\rho_{\mathrm{cc}} = \frac12 I.$$

Therefore every local measurement on either one qubit alone has the same statistics for these two global states. The difference appears only in joint measurements. For example, $x$-basis measurements on both qubits are perfectly correlated in $\lvert\Phi^+\rangle$, but are uncorrelated for $\rho_{\mathrm{cc}}$.

This is not a flaw in the reduced-state description. It is exactly what a reduced state is supposed to do: describe local statistics, not all correlations with discarded degrees of freedom.

Joint Measurements and Marginals

If a local measurement on $A$ has effects $\{E_a\}$ and a local measurement on $B$ has effects $\{F_b\}$, then the joint probability is

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

The marginal probability for $A$ is obtained by summing over $B$:

$$\begin{aligned} p_A(a) &= \sum_b p(a,b)\\ &= \operatorname{Tr}_{AB} \left[ \rho_{AB} \left( E_a\otimes\sum_b F_b \right) \right]\\ &= \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(E_a\otimes I_B) \bigr]\\ &= \operatorname{Tr}_A(\rho_A E_a). \end{aligned}$$

The local marginal does not depend on which complete measurement $\{F_b\}$ is chosen on $B$. The joint distribution can depend strongly on both measurements, but the unconditioned local distribution on $A$ cannot.

No-Signaling Preview

The marginal identity is the algebraic core of no-signaling in ordinary quantum mechanics. If one party changes which measurement is performed on $B$, the unconditioned outcome probabilities on $A$ remain

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

This does not say that measurements on $B$ are irrelevant. If a particular outcome $b$ is obtained and communicated, the state assigned to $A$ conditional on that outcome can change. That conditional update is developed in Conditional States and is a different question from the unconditioned local distribution. No-Cloning and No-Signaling gives the full channel-level theorem and its communication consequences.

The useful separation is:

• reduced states determine local statistics before conditioning on remote outcomes;
• the joint state determines correlations;
• conditional states describe what one assigns after a remote outcome is known.

Local Operations Without Readout

A local operation on $B$ that is performed without revealing or selecting an outcome cannot change $\rho_A$. In the simplest measurement case, suppose $B$ is projectively measured with projectors $\{Q_b\}$ and the outcome is ignored. The post-measurement joint state is

$$\rho'_{AB} = \sum_b (I_A\otimes Q_b)\rho_{AB}(I_A\otimes Q_b).$$

The reduced state of $A$ remains unchanged:

$$\operatorname{Tr}_B\rho'_{AB} = \operatorname{Tr}_B\rho_{AB} = \rho_A.$$

The joint correlations may change because the nonselective measurement can destroy coherence involving $B$. But without conditioning on an outcome, the statistics of measurements on $A$ alone are unchanged.

Common Mistakes

• Thinking $\rho_A$ contains the full information in $\rho_{AB}$.
• Thinking a local observer can distinguish a Bell state from a classically correlated state using only one-subsystem measurements.
• Forgetting the identity factor when writing a local measurement on a composite space.
• Confusing a marginal probability with a conditional probability after a remote outcome is known.
• Treating no-signaling as saying that entanglement has no observable consequences. Entanglement affects joint statistics, not controllable local marginals.
• Assuming that a nonselective measurement on $B$ and a selective measurement with a known outcome have the same implication for $A$.

Cross-Links

• Reduced Density Operators
• Subsystems and Local Observables
• Partial Trace
• Partial Trace Exercises
• Conditional States
• Marginals and Correlations
• Classical Correlation versus Entanglement
• Local Unitary Equivalence
• Bell States
• Entanglement in Foundations
• Operators on Composite Systems
• Born Rule
• Projective Measurement
• State Update Rule
• Density Operators

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.

Exercises

1. Prove the local probability identity for projective measurements.

Solution

Let $\rho_A=\operatorname{Tr}_B\rho_{AB}$. The partial trace is characterized by

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

Set $X_{AB}=\rho_{AB}$ and $M_A=P_a$. Then

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

This is the Born-rule probability for the local outcome $a$.

2. Show that every spin-direction measurement on one qubit of $\lvert\Phi^+\rangle$ gives equal probabilities.

Solution

For either qubit of $\lvert\Phi^+\rangle$,

$$\rho_A = \frac12 I.$$

A projective measurement along a unit direction $\mathbf n$ has projectors

$$P_\pm = \frac12 \bigl( I\pm\mathbf n\cdot\boldsymbol\sigma \bigr).$$

Therefore

$$p(\pm) = \operatorname{Tr}(\rho_A P_\pm) = \frac12\operatorname{Tr}(P_\pm) = \frac12,$$

because each rank-one qubit projector has trace $1$.

3. Compare $x$-basis joint measurements for $\lvert\Phi^+\rangle$ and $\rho_{\mathrm{cc}}$.

Solution

The Bell state can be written in the $x$ basis as

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt2} \bigl( \lvert++\rangle+\lvert--\rangle \bigr).$$

Thus the outcomes agree: $++$ and $--$ each occur with probability $1/2$.

For

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

each product state gives independent $x$-basis outcomes with probability $1/4$ for each pair. The mixture therefore gives

$$p(++) = p(+-) = p(-+) = p(--) = \frac14.$$

The two states have the same one-qubit local statistics but different joint statistics.

4. Explain why changing the measurement on $B$ cannot change the unconditioned probability for outcome $a$ on $A$.

Solution

Let $\{F_b\}$ be any complete POVM on $B$, so $\sum_b F_b=I_B$. Then

$$\begin{aligned} \sum_b \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(E_a\otimes F_b) \bigr] &= \operatorname{Tr}_{AB} \left[ \rho_{AB} \left( E_a\otimes\sum_b F_b \right) \right]\\ &= \operatorname{Tr}_{AB} \bigl[ \rho_{AB}(E_a\otimes I_B) \bigr]\\ &= \operatorname{Tr}_A(\rho_A E_a). \end{aligned}$$

The final expression contains no reference to the particular measurement chosen on $B$.