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Improper and proper mixtures with the same reduced preparation: the singlet marginal versus a fair random mixture

Physical scope

An exact comparison in the declared framework between the reduced preparation of the two-factor spin singlet and a fair classical randomization of the two spin-up and spin-down preparations of one factor: the two density operators coincide, the composite descriptions differ, and the joint directional correlations distinguish them. The example asserts no measured correlation, no experiment and no empirical fact, and it makes no claim about any interpretation of the mixture.

Example

Adopt the declared internal-factor composite of two spin-half factors and the singlet preparation $ρ = T_{Ψ, Ψ}$ of The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals, with $V = C^{2}$ , orthonormal basis $(e_{+}, e_{-})$ , spin matrices $S_{i} = ℏ J_{i}$ and $S_{n} = \sum_{i} n_{i} S_{i}$ for unit $n \in R^{3}$ (Finite spin matrices, Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities), and let Reduced preparations of a declared composite: the subsystem states of the framework and Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion supply the reading of the reduced preparations. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), as required by the cited spin-outcome and singlet suppliers. Declare:

(a) the improper mixture of factor $A$ : the reduced preparation $ρ_{A} = {Tr}_{B} ρ$ of the singlet;

(b) the proper mixture of factor $A$ : the declared preparation in which a fair classical randomization selects the pure preparation $T_{e_{+}, e_{+}}$ with probability $\frac{1}{2}$ and $T_{e_{-}, e_{-}}$ with probability $\frac{1}{2}$ , so that $ρ_{p r o p e r} : = \frac{1}{2} T_{e_{+}, e_{+}} + \frac{1}{2} T_{e_{-}, e_{-}};$

(c) the EPR-type product mixture of the composite: the declared preparation $M : = \frac{1}{2} T_{e_{+} \otimes e_{-}, e_{+} \otimes e_{-}} + \frac{1}{2} T_{e_{-} \otimes e_{+}, e_{-} \otimes e_{+}},$ in which each factor carries a definite spin along the $z$ -axis and the two spins are opposite.

Then:

(i) $ρ_{A} = ρ_{p r o p e r} = \frac{1}{2} I$ and the marginals of $M$ are ${Tr}_{B} M = \frac{1}{2} I = {Tr}_{A} M$ : the two preparations of factor $A$ and the marginal of $M$ are the same density operator, and $ρ$ and $M$ are not the same composite preparation;

(ii) for unit directions $a, b$ , writing $a_{3} = a \cdot e_{z}$ and $b_{3} = b \cdot e_{z}$ with $e_{z}$ the direction of the $z$ -axis of the basis $(e_{+}, e_{-})$ , the directional correlation of the singlet is $E_{ρ} (a, b) = - \cos θ = - a \cdot b$ while that of the product mixture is $E_{M} (a, b) = - \frac{1}{2} a_{3} b_{3} - \frac{1}{2} a_{3} b_{3} = - a_{3} b_{3};$ in particular $E_{ρ} (z, z) = E_{M} (z, z) = - 1$ but $E_{ρ} (x, x) = - 1$ and $E_{M} (x, x) = 0$ for the orthogonal direction $x$ ;

(iii) $M$ is a convex combination of the two pure product states $T_{e_{+} \otimes e_{-}, e_{+} \otimes e_{-}}$ and $T_{e_{-} \otimes e_{+}, e_{-} \otimes e_{+}}$ , hence a separable preparation in the sense of clause (b) of Product vectors, separable states and entangled states of a two-factor composite, while $ρ$ is a non-product pure preparation; and every local readout law of factor $A$ is the same in $ρ$ , in $ρ_{p r o p e r}$ extended by any preparation of $B$ , and in $M$ , because all three have the reduced preparation $\frac{1}{2} I$ on $A$ ;

(iv) consequently the density operator $\frac{1}{2} I$ alone does not determine whether the preparation of factor $A$ arises as the reduction of an entangled composite preparation (the improper case) or from a classical randomization of factor preparations (the proper case): the two cases are distinguished by the joint statistics with the other factor, not by any local statistic of $A$ .

The comparison is a conditional computation of the declared framework; it asserts no measured correlation, no experiment and no empirical fact.

Verification

Given: The internal-factor composite $V \hat{\otimes} V$ ; the singlet preparation $ρ$ and its reductions; the proper mixture $ρ_{p r o p e r}$ ; the product mixture $M$ ; unit directions $a, b$ ; the Axiom of Countable Choice and the Axiom of Choice.

[L1] Singlet data. $ρ_{A} = ρ_{B} = \frac{1}{2} I$ and $E_{ρ} (a, b) = - a \cdot b$ for all unit $a, b$ (The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals clauses (a), (c)).

[L2] Product reductions and local laws. For unit $x, y$ one has ${Tr}_{B} T_{κ (x \otimes y), κ (x \otimes y)} = T_{x, x}$ and ${Tr}_{A} T_{κ (x \otimes y), κ (x \otimes y)} = T_{y, y}$ (Separable partial trace and its basic properties clause (c)). On $V$ , the projections $P_{n, \pm} = (I \pm n \cdot σ) / 2$ and the law $μ_{ψ}^{S_{n}} ({\pm ℏ / 2}) = (1 \pm ⟨ (n \cdot σ) ψ, ψ ⟩) / 2$ are the finite-factor deduction in [F2] of The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals, using Pauli matrices and P2, rather than a transfer of a law on $L^{2} (R^{3}; C^{2})$ . The matrices in Finite spin matrices clause (e) give $⟨ (n \cdot σ) e_{\pm}, e_{\pm} ⟩ = \pm n_{3}$ . Joint probabilities in a unit product preparation factorize by Factorization of local statistics for product preparations, and its failure for non-product pure preparations clause (a). The rank-one action is $T_{q, q} z = ⟨ z, q ⟩ q$ (Rank-one operator calculus).

[L3] Local readout law. If two preparations of the composite have the same reduced preparation on $A$ , then every local readout law of $A$ agrees in the two preparations (Local readout laws and reduced preparations: the subsystem state determines, and is determined by, the local statistics clause (a)).

[L4] Mixtures and separability. A convex combination $\sum_{j} w_{j} T_{Ψ_{j}, Ψ_{j}}$ of pure product preparations with weights $w_{j} \geq 0$ , $\sum_{j} w_{j} = 1$ is a preparation of the composite and a separable state in the sense of clause (b) of Product vectors, separable states and entangled states of a two-factor composite; outcome laws are affine in the preparation (Product vectors, separable states and entangled states of a two-factor composite clause (c), Density operators and scalar spectral probability clause (b)).

[L5] The density expansion into an at most countable positive orthonormal rank-one sum and trace-norm completeness are Density operators and scalar spectral probability clause (a) and Trace class is a two sided Banach operator ideal clause 3. Partial trace is trace-norm continuous by Separable partial trace and its basic properties clause (a).

[L6] Choice hypothesis. The Axiom of Choice is assumed by the cited spin-outcome and singlet suppliers for their spectral PVM constructions (The Axiom of Choice).

Proof technique: direct; compute the two marginals and the two correlation functions from the product structure, and compare the local readout laws through the common reduced preparation.

Given: The data of the statement and [L1]-[L4].

1.1givenL1L2L6algebra

[Marginals] By [L2], ${Tr}_{B} T_{e_{+} \otimes e_{-}, e_{+} \otimes e_{-}} = T_{e_{+}, e_{+}}$ and ${Tr}_{B} T_{e_{-} \otimes e_{+}, e_{-} \otimes e_{+}} = T_{e_{-}, e_{-}}$ , so the linearity of the partial trace gives ${Tr}_{B} M = \frac{1}{2} T_{e_{+}, e_{+}} + \frac{1}{2} T_{e_{-}, e_{-}} = \frac{1}{2} I$ ; likewise ${Tr}_{A} M = \frac{1}{2} I$ . Together with $ρ_{A} = \frac{1}{2} I$ from [L1] and $ρ_{p r o p e r} = \frac{1}{2} I$ by definition these are the marginal identities in (i). Put $q_{1} = κ (e_{+} \otimes e_{-})$ , $q_{2} = κ (e_{-} \otimes e_{+})$ ; they are orthonormal and $Ψ = (q_{1} - q_{2}) / \sqrt{2}$ . Thus the displayed mixture and the rank-one action in [L2] give $M Ψ = Ψ / 2$ , whereas $ρ Ψ = T_{Ψ, Ψ} Ψ = Ψ$ . Since $Ψ$ is unit, $ρ \neq M$ , completing (i).

1.2givenL1L2L4L6algebra

[Correlations of the product mixture] By [L2] and [L4] the joint outcome law of the local measurements of $S_{a} \otimes I$ and $I \otimes S_{b}$ in $M$ is the weighted average of the two product laws, so $E_{M} (a, b) = \frac{1}{2} ⟨ (a \cdot σ) e_{+}, e_{+} ⟩ ⟨ (b \cdot σ) e_{-}, e_{-} ⟩ + \frac{1}{2} ⟨ (a \cdot σ) e_{-}, e_{-} ⟩ ⟨ (b \cdot σ) e_{+}, e_{+} ⟩ = \frac{1}{2} (a_{3}) (- b_{3}) + \frac{1}{2} (- a_{3}) (b_{3}) = - a_{3} b_{3}$ ; with [L1] this is (ii).

2.1givenL1L2L3L4L5L6step 1.1step 1.2algebra

[Local indistinguishability and separability] Let $τ_{B}$ be any preparation of $B$ . Its density expansion is $τ_{B} = \sum_{j} q_{j} T_{z_{j}, z_{j}}$ in trace norm, with unit $z_{j}$ , $q_{j} > 0$ and $\sum_{j} q_{j} = 1$ , by [L5]. The independent extension of the proper mixture is the trace-norm sum $Ω = \sum_{j} (q_{j} / 2) (T_{e_{+} \otimes z_{j}, e_{+} \otimes z_{j}} + T_{e_{-} \otimes z_{j}, e_{-} \otimes z_{j}})$ : every summand is positive, and its trace norm equals its nonnegative trace, so the series converges with trace one by [L4], [L5]. The rank-one reduction formula [L2] and trace-norm continuity [L5] give ${Tr}_{B} Ω = (T_{e_{+}, e_{+}} + T_{e_{-}, e_{-}}) / 2 = I_{A} / 2$ and ${Tr}_{A} Ω = τ_{B}$ , which need not equal $I_{B} / 2$ . Since $ρ$ and $M$ also have reduction $I_{A} / 2$ by [L1] and step 1.1, [L3] proves all local- $A$ readout laws agree for $ρ$ , $M$ and this arbitrary- $B$ extension. The displayed convex decomposition makes $M$ separable by [L4], while $ρ$ is non-product and pure by [L1] and the purity criterion. Their joint laws differ at $(x, x)$ by step 1.2. This proves (iii) and (iv) with the full arbitrary-extension claim.

3.1step 1.1step 1.2step 2.1∎

[Assembly] Step 1.1 is (i), step 1.2 is (ii), step 2.1 is (iii) and (iv); only the declared framework readings and the cited computations were used, so no measured correlation or empirical fact is asserted.

Remarks

The reduction is not an ensemble decomposition. Nothing in the example presents $ρ_{A} = \frac{1}{2} I$ as produced by a classical randomization; the reduction is the partial trace of the composite preparation, and the randomization of the proper mixture is a separate declared preparation of factor $A$ that happens to assign the same density operator.
Same local state, different composite. The comparison is a precise instance of the general fact that the reduced preparation determines all local statistics of its own factor while the joint statistics depend on the composite preparation.

Improper and proper mixtures with the same reduced preparation: the singlet marginal versus a fair random mixture

Example

Verification

Remarks

Depends on

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