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arXiv:quant-ph/0211037 · Physical Review Letters

Environment-Assisted Invariance, Entanglement, and Probabilities in Quantum Physics

Wojciech Hubert Zurek

Published Mar 26, 2003• 203 citations
arXivHTMLPDF

Abstract

I introduce environment-assisted invariance or envariance-a symmetry exhibited by correlated quantum systems and related to causality-and describe how it can be used to understand the nature of ignorance and, hence, the origin and interpretation of Born's rule for quantum probabilities.
DOI
10.1103/physrevlett.90.120404

Related papers

Probabilities from entanglement, Born’s rule p k = ∣ ψ k ∣ 2 from envariance

Wojciech Hubert Zurek

I show how probabilities arise in quantum physics by exploring the implications of environment-assisted invariance or envariance, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like'' states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Born's rule ${p}_{k}\ensuremath{\propto}{\ensuremath{\mid}{\ensuremath{\psi}}_{k}\ensuremath{\mid}}^{2}$ for the outcomes associated with Schmidt states. The probabilities derived in this manner are an objective reflection of the underlying state of the system---they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge'' of the observer. This envariance-based approach is compared with and found to be superior to prequantum definitions of probability including the standard definition based on the ``principle of indifference'' due to Laplace and the relative frequency approach advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Born's rule: Envariance is enough to establish the dynamical independence of preferred branches of the evolving state vector of the composite system and, thus, to arrive at the environment-induced superselection (einselection) of pointer states, which was usually derived by an appeal to decoherence. The envariant origin of Born's rule for probabilities sheds light on the relation between ignorance (and, hence, information) and the nature of quantum states.

arXiv:quant-ph/0405161•Physical Review A•May 25, 2005•261 citations

Experimental test of environment-assisted invariance

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Envariance, or environment-assisted invariance, is a recently identified symmetry for maximally entangled states in quantum theory with important ramifications for quantum measurement, specifically for understanding Born's rule. We benchmark the degree to which nature respects this symmetry by using entangled photon pairs. Our results show quantum states can be $(99.66\ifmmode\pm\else\textpm\fi{}0.04)%$ envariant as measured using the quantum fidelity, and $(99.963\ifmmode\pm\else\textpm\fi{}0.005)%$ as measured using a modified Bhattacharya coefficient, as compared with a perfectly envariant system which would be 100% in either measure. The deviations can be understood by the less-than-maximal entanglement in our photon pairs.

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Envariance, or environment-assisted invariance, is a recently identified symmetry for maximally entangled states in quantum theory with important ramifications for quantum measurement, specifically for understanding Born's rule. We benchmark the degree to which nature respects this symmetry by using entangled photon pairs. Our results show quantum states can be 99.66(4)% envariant as measured using the quantum fidelity, and 99.963(5)% as measured using a modified Bhattacharya Coefficient, as compared with a perfectly envariant system which would be 100% in either measure. The deviations can be understood by the less-than-maximal entanglement in our photon pairs.

arXiv:1408.7087•Aug 29, 2014•0 citations

Born’s Rule from the Symmetries of Entanglement

Wojciech Hubert Zurek

Chapter 3 describes how quantum entanglement leads to probabilities based on a symmetry, but—in contrast to subjective equal likelihood based solely on ignorance—it is an objective symmetry of known quantum states. Entanglement-assisted invariance (or envariance for short) relies on quantum correlations: One can know the quantum state of the whole and use this to quantify the resulting ignorance of the states of parts. Thus, quantum probability is, in effect, an objective consequence of the Heisenberg-like indeterminacy between global and local observables. This derivation of Born’s rule is based on the consistent subset of quantum postulates. It justifies statistical interpretation of reduced density matrices, an indispensable tool of decoherence theory. Hence, it gives one the mandate to explore—in Part II of this book—the fundamental implications of decoherence and its consequences using reduced density matrices and other customary tools.

Cambridge University Press eBooks•Mar 6, 2025•0 citations

No-signalling-based version of Zurek's derivation of quantum probabilities: A note on "Environment-assisted invariance, entanglement, and probabilities in quantum physics"

Howard Barnum

Zurek has derived the quantum probabilities for Schmidt basis states of bipartite quantum systems in pure joint states, from the assumption that they should be not be affected by one party's action if the action can be undone by the other party (``envariance of probability'') and an auxiliary assumption. We argue that a natural generalization of the auxiliary assumption is actually strong enough to yield the Born rule itself, but that Zurek's argument and protocol can be adapted to do without this assumption, at the cost of using envariance of probability in both directions. We consider alternative motivations for envariance, one based on the no-signalling constraint that actions on one subsystem of a quantum system not allow signalling to another subsystem entirely distinct from the first, and another which is perhaps strongest in the context of a relative-state interpretation of quantum mechanics. In part because of this, we argue that the relative appeal of our version and the original version of Zurek's argument depends in part upon whether one interprets the quantum formalism in terms of relative states or definite measurement outcomes.

arXiv:quant-ph/0312150•Dec 17, 2003•18 citations

Quantum probabilities from quantum entanglement: experimentally unpacking the Born rule

Jérémie Harris, Frédéric Bouchard, Enrico Santamato +3 more

The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality.

arXiv:1604.01471•New Journal of Physics•May 11, 2016•13 citations
I show how probabilities arise in quantum physics by exploring the implications of environment-assisted invariance or envariance, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like'' states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Born's rule ${p}_{k}\ensuremath{\propto}{\ensuremath{\mid}{\ensuremath{\psi}}_{k}\ensuremath{\mid}}^{2}$ for the outcomes associated with Schmidt states. The probabilities derived in this manner are an objective reflection of the underlying state of the system---they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge'' of the observer. This envariance-based approach is compared with and found to be superior to prequantum definitions of probability including the standard definition based on the ``principle of indifference'' due to Laplace and the relative frequency approach advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Born's rule: Envariance is enough to establish the dynamical independence of preferred branches of the evolving state vector of the composite system and, thus, to arrive at the environment-induced superselection (einselection) of pointer states, which was usually derived by an appeal to decoherence. The envariant origin of Born's rule for probabilities sheds light on the relation between ignorance (and, hence, information) and the nature of quantum states.
Envariance, or environment-assisted invariance, is a recently identified symmetry for maximally entangled states in quantum theory with important ramifications for quantum measurement, specifically for understanding Born's rule. We benchmark the degree to which nature respects this symmetry by using entangled photon pairs. Our results show quantum states can be $(99.66\ifmmode\pm\else\textpm\fi{}0.04)%$ envariant as measured using the quantum fidelity, and $(99.963\ifmmode\pm\else\textpm\fi{}0.005)%$ as measured using a modified Bhattacharya coefficient, as compared with a perfectly envariant system which would be 100% in either measure. The deviations can be understood by the less-than-maximal entanglement in our photon pairs.
Envariance, or environment-assisted invariance, is a recently identified symmetry for maximally entangled states in quantum theory with important ramifications for quantum measurement, specifically for understanding Born's rule. We benchmark the degree to which nature respects this symmetry by using entangled photon pairs. Our results show quantum states can be 99.66(4)% envariant as measured using the quantum fidelity, and 99.963(5)% as measured using a modified Bhattacharya Coefficient, as compared with a perfectly envariant system which would be 100% in either measure. The deviations can be understood by the less-than-maximal entanglement in our photon pairs.
Chapter 3 describes how quantum entanglement leads to probabilities based on a symmetry, but—in contrast to subjective equal likelihood based solely on ignorance—it is an objective symmetry of known quantum states. Entanglement-assisted invariance (or envariance for short) relies on quantum correlations: One can know the quantum state of the whole and use this to quantify the resulting ignorance of the states of parts. Thus, quantum probability is, in effect, an objective consequence of the Heisenberg-like indeterminacy between global and local observables. This derivation of Born’s rule is based on the consistent subset of quantum postulates. It justifies statistical interpretation of reduced density matrices, an indispensable tool of decoherence theory. Hence, it gives one the mandate to explore—in Part II of this book—the fundamental implications of decoherence and its consequences using reduced density matrices and other customary tools.
Zurek has derived the quantum probabilities for Schmidt basis states of bipartite quantum systems in pure joint states, from the assumption that they should be not be affected by one party's action if the action can be undone by the other party (``envariance of probability'') and an auxiliary assumption. We argue that a natural generalization of the auxiliary assumption is actually strong enough to yield the Born rule itself, but that Zurek's argument and protocol can be adapted to do without this assumption, at the cost of using envariance of probability in both directions. We consider alternative motivations for envariance, one based on the no-signalling constraint that actions on one subsystem of a quantum system not allow signalling to another subsystem entirely distinct from the first, and another which is perhaps strongest in the context of a relative-state interpretation of quantum mechanics. In part because of this, we argue that the relative appeal of our version and the original version of Zurek's argument depends in part upon whether one interprets the quantum formalism in terms of relative states or definite measurement outcomes.
The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality.