Quantum Darwinism Preview

Quantum Darwinism is the program that studies how information about pointer states is redundantly recorded in many fragments of the environment. Decoherence suppresses local interference. Quantum Darwinism asks an additional question: why can many observers independently learn the same classical-looking facts by sampling different pieces of the environment?

This account concerns information accessible in specified environmental fragments. Suppressed system interference alone does not imply redundant, readable records, and total mutual information alone does not establish that the records are classical.

Helpful background. Pointer states identify robust alternatives; mutual information measures correlations using entropy.

From Decoherence to Records

For normalized pure initial states and an initially uncorrelated environment, a simple record-forming interaction gives

$$\sum_a c_a\lvert a\rangle\lvert E_0\rangle \longmapsto \sum_a c_a\lvert a\rangle\lvert E_a\rangle.$$

Tracing out the environment suppresses system coherences when

$$\langle E_b|E_a\rangle\approx0 \qquad a\ne b.$$

Quantum Darwinism refines the environment into fragments:

$$E=E_1E_2\cdots E_N.$$

One useful additional assumption is a product of fragment states conditional on the pointer label:

$$\lvert E_a\rangle = \bigotimes_{k=1}^N \lvert E_a^{(k)}\rangle.$$

If many fragments separately carry information about the same pointer label $a$, then many observers can consult different fragments and infer the same pointer alternative without directly measuring the system.

This is the “environment as witness” idea: the environment is not merely a sink that destroys interference; it is also a communication channel that broadcasts selected information.

What Is Selected

Quantum Darwinism does not say that arbitrary quantum information is copied into the environment. That would conflict with the no-cloning principle. Instead, the proliferated information is information about robust pointer alternatives: states or sectors that the environment can monitor repeatedly without rapidly destroying the record.

The selected information is usually classical information about a pointer observable:

$$\{P_a\}.$$

Across repeated preparations, fragment measurements may support estimates of the probabilities

$$p_a=\operatorname{Tr}(P_a\rho_S),$$

or may reveal a particular recorded alternative after conditioning on an observed record. Superpositions of different pointer alternatives are not redundantly copied as coherent quantum states. Their phases are precisely what become inaccessible in the reduced description.

For the selection mechanism, see Pointer States and Einselection.

Operational Objectivity

An operational notion of objectivity is:

1. many observers can independently access the information;
2. they can do so by measuring different environmental fragments;
3. they can agree on the same pointer property;
4. their observations do not require each observer to disturb the system appreciably.

For example, people rarely observe a macroscopic object by interacting with the object directly. They intercept scattered photons. If many photons redundantly encode the object’s coarse position, many observers can learn that position from different photon subsets.

This does not mean the environment literally thinks, observes, or chooses. It means the environment stores records that can be sampled.

Mutual information and readable pointer information

For a system $S$ and environment fragment $F$,

$$\mathcal I(S:F)=S(\rho_S)+S(\rho_F)-S(\rho_{SF}), \qquad S(\rho)=-\operatorname{Tr}(\rho\log_2\rho).$$

Entropies are in bits. A plot of $\mathcal I(S:F)$ against fragment size often rises toward a plateau near $S(\rho_S)$. Such a plateau is a useful diagnostic, but quantum mutual information counts all correlations, including correlations that are not locally readable as the chosen pointer label.

To isolate that label, imagine a classical variable $A$ with probabilities $p_a$ and conditional fragment states $\rho_{F|a}$. Its Holevo information is

$$\chi(A:F)=S\!\left(\sum_a p_a\rho_{F|a}\right) -\sum_a p_aS(\rho_{F|a}).$$

The mutual information obtainable by measuring $F$ is bounded above by $\chi(A:F)$ and by the label entropy $H(A)$. A large Holevo bound does not in general guarantee that one measurement attains it. Mutually orthogonal conditional supports do: the fragment can then reveal $a$ without confusing alternatives. These distinctions matter when interpreting a mutual-information plateau (Le and Olaya-Castro, 2019).

Worked example: perfect redundant records

Take $N\ge2$ environment qubits and

$$|\Psi\rangle_{SE} =\frac{|0\rangle_S|0\rangle^{\otimes N} +|1\rangle_S|1\rangle^{\otimes N}}{\sqrt2}.$$

For any nonempty proper fragment containing $1\le m<N$ qubits, tracing out the remaining environment removes the cross terms. Then $S(\rho_S)=S(\rho_F)=S(\rho_{SF})=1$, so $\mathcal I(S:F)=1$ bit. Measuring even one fragment qubit in the $Z$ basis reads the same pointer label, and $\chi(A:F)=1$.

For the whole environment, $\rho_{SE}$ is pure and $\mathcal I(S:E)=2$ bits. The additional bit is not a second copy of the classical label; it reflects quantum correlations in the pure joint state. Replacing the joint superposition by the corresponding incoherent mixture leaves every proper-fragment calculation unchanged but reduces the whole-environment mutual information to one bit. Thus redundant local records do not distinguish those two global states.

Counting independently readable records

Choose $0<\delta<1$ and an operational criterion, such as accessible information at least $(1-\delta)H(A)$. If statistically equivalent fragments of $m_\delta$ out of $N$ elementary parts meet that criterion, one often quotes

$$f_\delta=\frac{m_\delta}{N},\qquad R_\delta=\frac1{f_\delta}.$$

This estimates how many disjoint useful fragments fit into the environment. A small specially selected fragment is not enough to establish that all fragments of that size work. In a heterogeneous environment, count or characterize disjoint qualifying fragments directly; state whether a plot averages over fragment choices and whether it uses total, Holevo, or accessible information.

A useful ideal form for a system and several observed fragments is

$$\rho_{SF_1\cdots F_k} =\sum_a p_a|a\rangle\langle a| \otimes\rho_{F_1|a}\otimes\cdots\otimes\rho_{F_k|a},$$

where the supports of $\rho_{F_j|a}$ and $\rho_{F_j|b}$ are orthogonal for $a\ne b$. This spectrum broadcast structure makes independent, nondisturbing readout explicit. It is a stronger statement than a plateau in total mutual information. Real fragment correlations and imperfect distinguishability must be examined rather than assumed away.

Example: Scattered Photons

A dust grain, pointer needle, or macroscopic object scatters ambient photons. Different object positions produce different outgoing photon states:

$$\lvert x\rangle\lvert \gamma_{\mathrm{in}}\rangle \longmapsto \lvert x\rangle\lvert \gamma_x\rangle.$$

If many scattered photons carry partial information about $x$, then small photon fragments can reveal the object’s coarse position. At the same time, the unobserved photon environment suppresses spatial coherences in the reduced density matrix:

$$\rho(x,x';t) \approx D(x,x';t)\rho(x,x';0).$$

Thus the same physical process can both decohere position superpositions and distribute records of position.

This is why quantum Darwinism is naturally tied to Environment-Induced Decoherence rather than being a separate measurement postulate.

What redundant records do not settle

The mechanism explains how a selected property can become readable in many places. It neither copies arbitrary quantum states nor guarantees that every environment produces useful records. Fragment structure, initial environment states, and the interaction all matter. For example, the mixed-environment dephasing model destroys system coherence while leaving the two conditional environment states identical: it creates no readable pointer-label record in that environment.

Redundancy also does not select one global branch, derive outcome probabilities without probabilistic assumptions, or remove the distinction between proper and improper mixtures. See What Decoherence Does Not Solve for those separate questions.

Exercises

1. Imperfect records and a readout threshold. Consider equal pointer probabilities and the normalized joint state

$$|\Psi\rangle=\frac{|0\rangle|e_0\rangle^{\otimes N} +|1\rangle|e_1\rangle^{\otimes N}}{\sqrt2}, \qquad \langle e_0|e_1\rangle=c,\quad 0<c<1.$$

An observer may perform any joint measurement on an $m$-qubit fragment and must guess the pointer label on every run. Using binary state discrimination, find the minimum error probability. For $c=0.9$ and $N=100$, how many disjoint fragments can each achieve error at most $\varepsilon=0.05$? Compare with an environment whose first conditional qubit states have overlap $c^N$ and whose other conditional states are identical for both labels.

Solution

Conditioned on a pointer-basis outcome, the fragment signals are two pure states with overlap $c^m$. The equal-prior Holevo–Helstrom result therefore gives (Watrous, 2018, Theorem 3.4)

$$p_{\rm err}(m)=\frac{1-\sqrt{1-c^{2m}}}{2}.$$

This optimum permits a collective measurement within each fragment; merely counting qubits does not specify an optimal readout. For $0<\varepsilon<1/2$, the condition $p_{\rm err}\le\varepsilon$ is equivalent to $c^{2m}\le4\varepsilon(1-\varepsilon)$, hence

$$m_{\varepsilon} =\left\lceil\frac{\ln[4\varepsilon(1-\varepsilon)]}{2\ln c}\right\rceil.$$

Here $m_{\varepsilon}=8$: the errors for seven and eight qubits are about $0.0609$ and $0.0487$. Thus $\lfloor100/8\rfloor=12$ disjoint qualifying fragments fit. This is a guessing-error threshold, distinct from the entropy-deficit convention for $R_\delta$ above.

In both environments, the system coherence multiplier is the product of all conditional overlaps, $c^{100}\simeq2.66\times10^{-5}$. In the second environment, however, every fragment excluding the first qubit has identical conditional states and error $1/2$. Only one member of a disjoint collection can contain the informative qubit. Strong decoherence can therefore coexist with either many readable records or only one: the distribution of information matters as well as the total overlap.

2. No-cloning check. Why does redundant recording of pointer information not violate the no-cloning theorem?

Solution

The no-cloning theorem forbids copying an arbitrary unknown quantum state. Quantum Darwinism concerns redundant records of selected pointer alternatives, which are effectively classical and approximately distinguishable. The environment does not copy an arbitrary superposition as a coherent quantum state; it records information about a preferred observable while suppressing coherence between different pointer alternatives.

3. Redundancy from fragment size. If statistically equivalent fragments of fraction $f_\delta=0.02$ each satisfy the chosen accessible-information threshold, what is $R_\delta$? Is $1-\delta$ necessarily the probability of guessing the pointer label correctly?

Solution

Using

$$R_\delta=\frac{1}{f_\delta},$$

one finds

$$R_\delta=\frac{1}{0.02}=50.$$

In this convention, about fifty disjoint fragments can satisfy the information threshold. The fraction $1-\delta$ is a fraction of entropy, not a success probability; converting an information deficit to a discrimination error requires another bound or an explicit measurement model.

4. Not all observables. Why can position-like information be redundantly recorded by scattered photons while phase coherence between separated positions is not equally public?

Solution

Scattered photons can carry which-position information because their outgoing states depend on the object’s position. That same correlation suppresses off-diagonal position coherences when the photons are ignored. The relative phase between macroscopically separated components is stored in nonlocal correlations involving the system and many environmental degrees of freedom; it is not locally available in small fragments. Thus the environment makes position-like information public while hiding phase coherence from local access.

5. Objectivity boundary. State one thing quantum Darwinism helps explain and one thing it does not settle.

Solution

It helps explain why many observers can independently infer the same pointer property by sampling different environmental fragments. It does not, by itself, prove that exactly one outcome occurs in an interpretation-neutral sense. That stronger claim requires additional interpretive, collapse-dynamical, or operational assumptions.

References

• R. Blume-Kohout and W. H. Zurek, “Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information,” Physical Review A 73, 062310 (2006). DOI.
• T. P. Le and A. Olaya-Castro, “Strong quantum Darwinism and strong independence are equivalent to spectrum broadcast structure,” Physical Review Letters 122, 010403 (2019). DOI.
• H. Ollivier, D. Poulin, and W. H. Zurek, “Objective properties from subjective quantum states: Environment as a witness,” Physical Review Letters 93, 220401 (2004). DOI.
• C. J. Riedel and W. H. Zurek, “Quantum Darwinism in an everyday environment: Huge redundancy in scattered photons,” Physical Review Letters 105, 020404 (2010). DOI.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), Theorem 3.4. Author manuscript.
• W. H. Zurek, “Quantum Darwinism,” Nature Physics 5, 181–188 (2009). DOI.