Envariance

Envariance, or environment-assisted invariance, is a symmetry of an entangled state: an operation on one subsystem can be undone by an operation on another. Its proposed foundational use is to relate local outcome probabilities to symmetries of the joint state. The symmetry is a mathematical fact; identifying its consequences for probability requires additional interpretive assumptions.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Reversible changes in an entangled state

Let a system $S$ and environment $E$ have Schmidt form

$$\lvert\Psi\rangle=\sum_k c_k\lvert s_k\rangle\lvert e_k\rangle.$$

A unitary $U_S$ is envariant for this state if some $U_E$ satisfies

$$(U_S\otimes U_E)\lvert\Psi\rangle=\lvert\Psi\rangle.$$

A phase change of a Schmidt term on $S$ can be cancelled by the opposite phase on $E$. For two equal-magnitude coefficients, a swap of system labels can likewise be cancelled by a swap of environment labels with suitable phase adjustments.

For the state $(\lvert00\rangle+\lvert11\rangle)/\sqrt2$, applying $X$ to $S$ produces $(\lvert10\rangle+\lvert01\rangle)/\sqrt2$. Applying $X$ to $E$ restores the initial state. Neither operation alone restores it.

From swaps to equal probabilities

The probability argument assumes that local outcome probabilities do not change under an operation on the remote environment alone. It also requires that swapping the system alternatives swaps their probabilities. Because a remote counterswap restores the full state, equal Schmidt coefficients are then assigned equal probabilities.

One must not silently use the Born interpretation of $\rho_S=\operatorname{Tr}_E|\Psi\rangle\langle\Psi|$ as the proof of the very probability rule under discussion. Reduced states provide a compact check within standard quantum mechanics, whereas the proposed foundational derivation should state its local-state and invariance assumptions independently.

Unequal coefficients by fine-graining

Take

$$\lvert\Psi\rangle= \sqrt{\frac23}\lvert0\rangle\lvert e_0\rangle+ \sqrt{\frac13}\lvert1\rangle\lvert e_1\rangle.$$

Introduce an ancillary record $C$ and a suitable enlarged environment, so that the correlated state becomes

$$\lvert\widetilde\Psi\rangle=\frac{1}{\sqrt3} \bigl( \lvert0,c_1,e_1'\rangle+ \lvert0,c_2,e_2'\rangle+ \lvert1,c_3,e_3'\rangle \bigr),$$

where the three $c_j$ and the three $e_j'$ are orthonormal. Across the partition $SC\,|\,E$, this is a Schmidt expansion with three equal coefficients. Swapping the joint $SC$ alternatives can now be undone on $E$. These are not swaps of $S$ alone: the original unequal coefficients have not acquired a new symmetry merely by being written as a sum.

If the refinement leaves the original system probabilities unchanged and probabilities add under grouping, the two alternatives associated with system outcome 0 yield $p(0)=2/3$.

This uses more than a swap: physically admissible refinement, additivity, and independence from irrelevant details of the extension. Continuity extends the rational construction to arbitrary squared coefficients.

Scope and limitations

Envariance explains why phases of Schmidt coefficients cannot affect local outcome weights under the stated assumptions. It also provides a symmetry-based account of equal coefficients. It does not select a single outcome, determine what probability means, or remove the need to define subsystems and measurement alternatives.

The environment here need not be a thermal bath. It is any purifying system with the relevant correlations. Conversely, a generic unitary on $S$ is not envariant: unequal Schmidt magnitudes cannot generally be swapped and repaired solely by acting on $E$.

Compare decision theory, which uses a related refinement idea but constrains preferences, and decoherence, which concerns suppressed local interference.

Exercises

Why can local phase shifts always be undone in Schmidt form, while an arbitrary swap of unequal Schmidt coefficients cannot?

Solution

An opposite phase on each orthogonal environment vector cancels the system phase term by term. Environment unitaries preserve the system’s Schmidt spectrum and cannot restore a system swap that changes which unequal weight belongs to a fixed system vector. Equal weights permit the swap symmetry; unequal weights generally do not.

References

• M. Schlosshauer and A. Fine, “On Zurek’s derivation of the Born rule,” Foundations of Physics 35, 197–213 (2005), doi:10.1007/s10701-004-1941-6.
• W. H. Zurek, “Probabilities from entanglement, Born’s rule from envariance,” Physical Review A 71, 052105 (2005), author manuscript.