Consistent Histories Interpretation

The awkwardness at the heart of the Copenhagen picture is the special, unanalyzed status of "measurement" — a moment when the smooth Schrödinger evolution is interrupted by a collapse, triggered by something the theory never defines. Consistent histories removes the special moment by changing what the basic objects of the theory are. Instead of a state at one instant, the fundamental object is a history: an assertion that the system had property $a_1$ at $t_1$ , then $a_2$ at $t_2$ , and so on — each property a projection operator, the history their time-ordered product. The theory then asks a sharp question: to which sets of histories can you assign ordinary probabilities that add and multiply like a classical stochastic process? The answer is governed by a single object, the decoherence functional $D(\alpha,\beta)=\mathrm{Tr}[C_\alpha\,\rho\,C_\beta^\dagger]$ , a generalized inner product between two histories. Its diagonal is the would-be probability of a history; its off-diagonals are interference terms — the same obstruction as the double slit, where $|A_1+A_2|^2\neq|A_1|^2+|A_2|^2$ . A family of histories is consistent exactly when those interference terms vanish, $\mathrm{Re}\,D(\alpha,\beta)=0$ for $\alpha\neq\beta$ ; then and only then do the probabilities $p(\alpha)=\mathrm{Tr}[C_\alpha\rho C_\alpha^\dagger]$ add classically. The projectors evolve in the Heisenberg picture (recalled from Properties and Time Evolution of the Density Matrix and Measurement and Wavefunction Collapse by name); there is no collapse anywhere. This article owns the consistency condition and the machinery that makes it useful. It derives the decoherence functional, pins the consistency condition on an explicit two-time qubit family (showing a consistent choice and an inconsistent one, with the interference term computed), and cashes the chapter's central mechanism: environmental decoherence supplies consistency automatically for coarse-grained, quasiclassical variables. The honest cost — the set-selection problem, that many incompatible consistent families coexist and the formalism picks none — is stated plainly, because it is where the real debate lives.

Key takeaways

The consistent-histories framework takes time-ordered sequences of projectors as its basic objects and does not require a fundamental measurement postulate or wavefunction collapse. Between the specified times, the quantum state evolves unitarily, while probabilities are assigned only to histories belonging to a consistent family.
A history is a class operator $C_\alpha = P^{(n)}_{a_n}(t_n)\cdots P^{(1)}_{a_1}(t_1)$ ; the decoherence functional is $D(\alpha,\beta)=\mathrm{Tr}[C_\alpha\rho C_\beta^\dagger]$ .
Consistency condition (owned): $\mathrm{Re}\,D(\alpha,\beta)=0$ for $\alpha\neq\beta$ makes $p(\alpha)=\mathrm{Tr}[C_\alpha\rho C_\alpha^\dagger]$ obey classical additivity. When it fails, probabilities do not add — the double-slit obstruction.
Worked (owned): a $(z,z)$ qubit family is consistent (all off-diagonal $\mathrm{Re}\,D=0$ ); an $(x,z)$ family on $|0\rangle$ is inconsistent, with cross term $\mathrm{Re}\,D=-0.25$ and an additivity gap of $-0.5$ .
Decoherence promotes consistency (owned): when an environment records information about coarse-grained histories, the off-diagonal terms of $D$ can become small, making the corresponding probabilities approximately additive. Stronger environmental records suppress interference more effectively, but decoherence does not guarantee exact consistency for every coarse-graining.
Honest cost: many incompatible consistent families coexist (the single-framework rule), and the formalism selects none — the open set-selection problem.
Key idea: promote histories to fundamental objects and assign classical probabilities only to families where the decoherence functional is diagonal — recovering classical reasoning, and measurement, without collapse or observers.

The core insight

Consistent histories replaces the state-at-an-instant with time-ordered chains of projectors (histories) and assigns Born-rule probabilities $p(\alpha)=\mathrm{Tr}[C_\alpha\rho C_\alpha^\dagger]$ only to families whose decoherence functional $D(\alpha,\beta)=\mathrm{Tr}[C_\alpha\rho C_\beta^\dagger]$ is diagonal, so probabilities add classically. Environmental decoherence makes quasiclassical families automatically consistent — no collapse, no privileged measurement — at the cost of many incompatible frameworks the theory does not choose between.

Where this lives today

Consistent histories is most alive where there is no external observer to invoke collapse: quantum cosmology. In a closed universe there is no outside apparatus, so the Copenhagen split is unavailable, and the decoherent-histories program of Gell-Mann and Hartle was built precisely to assign probabilities to histories of the universe as a whole — the emergence of a quasiclassical domain of hydrodynamic variables (energy density, momentum flow) whose coarse-grained histories decohere and behave classically. The framework also feeds the modern study of the quantum-to-classical transition: it makes precise the sense in which classical trajectories are the consistent, decoherent histories of a system coupled to its environment, complementing the pointer-basis account. On the foundational side, the set-selection problem remains an active research topic — attempts to supplement the formalism with a rule preferring "quasiclassical" or maximally-refined families — and the framework is a standard tool for cleanly analyzing which counterfactual questions are legitimate in scenarios like the quantum eraser and delayed-choice experiments.

The physical picture

Think of a history as a movie of a quantum system: a list of yes/no properties at successive frames, each property a projector answering a question. Classical physics lets you ask any question at any frame and add up the probabilities of the different movies. Quantum mechanics does not — the double-slit experiment shows you cannot always ask "which slit?" and "where on the screen?" and have the probabilities add, because the amplitudes for the two slits interfere. Consistent histories makes this precise: only certain sets of movies are self-consistent, in the sense that their frame-by-frame probabilities behave like an ordinary stochastic process. For those sets, and only those, you may reason classically. The decoherence functional is the referee. It is a generalized inner product $D(\alpha,\beta)$ between two histories: its diagonal $D(\alpha,\alpha)$ gives the candidate probability of history $\alpha$ , while its off-diagonal elements quantify interference between different histories. The consistency condition concerns the real parts of these off-diagonal terms. When the off-diagonals vanish, the histories are like the slits of a which-path-marked interferometer: distinguishable, non-interfering, classically additive. When they do not, you have an ordinary interference pattern and the very notion of "the probability that history $\alpha$ happened" is ill-defined. The physical mechanism that often drives real systems toward approximate consistency is environmental decoherence: entanglement of a coarse-grained variable with a large environment suppresses the off-diagonal elements of $D$ , making suitable quasiclassical families approximately consistent.

Figure

The decoherence functional: diagonal for a consistent family, off-diagonal for an inconsistent one

The consistency condition

A history is a chain of projectors, one per time; the class operator collects them into a single object by evolving each to a common time. The decoherence functional is then the natural bilinear pairing of two histories with the initial state $\rho$ . In the Heisenberg picture, the projectors evolve as $P(t)=U^\dagger(t)PU(t)$ . Projectors also appear in the Lüders measurement rule, but the consistent-histories framework uses them to define alternative histories and does not require a physical wavefunction-collapse postulate.

Figure

A history is a chain of projectors; two histories interfere unless the cross term vanishes

Derivation AThe decoherence functional and the consistency condition

We build the class operator for a history, define the decoherence functional, and derive the condition under which the history probabilities obey the classical additivity rule. The strategy is to coarse-grain two exclusive histories into one, compare the probability of the coarse-grained history with the sum of the fine-grained probabilities, and identify the interference term that must vanish in its real part. This is the same mathematical obstruction that appears in the double-slit experiment.

Step 1. Build the class operator. A history $\alpha=(a_1,\dots,a_n)$ with property $a_k$ at time $t_k$ is encoded by the time-ordered chain

C_\alpha = P^{(n)}_{a_n}(t_n)\,\cdots\,P^{(1)}_{a_1}(t_1),\qquad P(t)=U^\dagger(t)\,P\,U(t) \qquad (1)

the projectors in the Heisenberg picture, latest on the left. Why this order: reading right to left is reading forward in time — first project onto $a_1$ at $t_1$ , propagate, project onto $a_2$ , and so on — so $C_\alpha$ is the amplitude operator for the whole ordered sequence.

Step 2. Define the decoherence functional. For two histories $\alpha,\beta$ and initial state $\rho$ ,

D(\alpha,\beta) = \mathrm{Tr}\!\big[C_\alpha\,\rho\,C_\beta^\dagger\big] \qquad (2)

a Hermitian form, $D(\alpha,\beta)=D(\beta,\alpha)^*$ , with $\sum_{\alpha\beta}D(\alpha,\beta)=\mathrm{Tr}\,\rho=1$ over a complete family. Why it is the right object: its diagonal $D(\alpha,\alpha)=\mathrm{Tr}[C_\alpha\rho C_\alpha^\dagger]$ is the Born-rule probability of the sequence $\alpha$ , and its off-diagonal is the amplitude overlap of two different sequences — the interference between histories.

Step 3. Demand additivity and read off the condition. Coarse-grain two fine histories $\alpha,\beta$ into one, $\alpha\vee\beta$ , with class operator $C_{\alpha\vee\beta}=C_\alpha+C_\beta$ . Then

p(\alpha\vee\beta) = \mathrm{Tr}[(C_\alpha+C_\beta)\rho(C_\alpha+C_\beta)^\dagger] = p(\alpha)+p(\beta) + 2\,\mathrm{Re}\,D(\alpha,\beta) \qquad (3)

so probabilities add classically, $p(\alpha\vee\beta)=p(\alpha)+p(\beta)$ , when $\mathrm{Re}\,D(\alpha,\beta)=0$ . This is the consistency condition used here. A stronger condition, often called medium decoherence, requires $D(\alpha,\beta)=0$ for $\alpha\neq\beta$ . Thus consistency concerns the real part of the interference term, while medium decoherence suppresses the full complex interference term.

Step 4. Assign probabilities. For a consistent family, the candidate numbers become genuine probabilities: $p(\alpha)=\mathrm{Tr}[C_\alpha\rho C_\alpha^\dagger]\ge0$ . They sum to one for a complete family and obey the classical sum rules under coarse-graining. Why this is the payoff: the framework therefore does not require collapse as a fundamental dynamical process; measurement can be described within a suitable family of system–apparatus–environment histories. The single-framework rule then requires that statements from incompatible families not be combined. *Verified numerically: for a qubit, the $(z,z)$ family on $|+\rangle$ has all off-diagonal $\mathrm{Re}\,D=0$ (max $<10^{-12}$ ) with probabilities $(0.5,0.5)$ summing to $1$ ; the functional is Hermitian and sums to $\mathrm{Tr}\,\rho=1$ over the complete family.*

When consistency fails, and how decoherence restores it

Go deeper · Derivation B · 2 minAn inconsistent family, and decoherence making it consistent

Derivation BAn inconsistent family, and decoherence making it consistent

We exhibit an explicit family whose histories interfere — so probabilities do not add — and then show that entangling the coarse variable with an environment suppresses the interference term. In the ideal limit of orthogonal environmental records the family becomes exactly consistent; for partial records, decoherence instead makes the family approximately consistent. The reduced-density-matrix machinery is recalled from The Reduced Density Matrix and Partial Trace and Quantum Decoherence and the Density Matrix by name.

Step 1. Build an interfering family. Start a qubit in $|0\rangle$ (a coherent superposition in the $x$ -basis) and ask "which $x$ at $t_1$ ?" then "which $z$ at $t_2$ ?" (trivial evolution $U=I$ ). The two $x$ -branches feeding the $z{=}1$ outcome have class operators $C_{x_+,z_1}$ and $C_{x_-,z_1}$ , and their cross term is

\mathrm{Re}\,D(x_+z_1,\,x_-z_1) = -0.25 \qquad (4)

Why it is nonzero: $|0\rangle$ is a genuine superposition of $|x_+\rangle$ and $|x_-\rangle$ , so the two "which- $x$ " branches are coherent, and re-asking a $z$ -question exposes their interference — exactly the double slit.

Step 2. See additivity break. The coarse-grained history "reach $z{=}1$ " has probability $p(z_1|0)=|\langle1|0\rangle|^2=0$ , yet the two fine branches carry $p(x_+,z_1)+p(x_-,z_1)=0.5$ . The additivity gap is

p(z_1) - \big[p(x_+,z_1)+p(x_-,z_1)\big] = 2\,\mathrm{Re}\,D = -0.5 \qquad (5)

Why this forbids classical reasoning: the branch probabilities sum to $0.5$ but the actual coarse probability is $0$ — the interference cancels them. There is no consistent assignment of "the probability the particle went through $x_+$ and landed at $z_1$ ." The family is inconsistent.

Step 3. Turn on an environment. Copy the "which- $x$ " information into a two-state environment, so the branches acquire record states with overlap $\langle e_+|e_-\rangle=\epsilon$ . The cross term (4) is multiplied by $\epsilon$ : a partial record $\epsilon=0.5$ gives $|D_{\rm cross}|=0.125$ (halved), and an orthogonal record $\epsilon=0$ gives $\mathrm{Re}\,D=0$ — the family is now consistent. Why this is the quasiclassical world: a coarse variable like center-of-mass position leaks its value into $\sim10^{23}$ environmental degrees of freedom, so $\epsilon\to0$ instantly, and the histories of macroscopic variables are automatically consistent — no fine-tuning, no observer. *Verified numerically: $\mathrm{Re}\,D_{\rm cross}=-0.25$ ; additivity gap $=-0.5=2\,\mathrm{Re}\,D$ ; $p(z_1|0)=0$ vs branch sum $0.5$ ; environment overlap $\epsilon=0.5\Rightarrow|D|=0.125$ (halved), $\epsilon=0\Rightarrow\mathrm{Re}\,D=0$ ; the functional stays Hermitian and normalized throughout.*

Limiting cases

case	decoherence functional	reading
single time (one projector)	trivially diagonal	any single-time family is consistent
commuting projectors at all times	$D$ diagonal	classical variables — always consistent
coherent superposition, incompatible questions	$\mathrm{Re}\,D\neq0$	double-slit obstruction; no probabilities
orthogonal environment records	$\mathrm{Re}\,D\to0$	decoherence supplies consistency
coarse-grained hydrodynamic variables	$D$ approximately diagonal when decohered	the quasiclassical domain
many incompatible consistent families	each internally diagonal	the set-selection problem — theory picks none

Common trapConsistent histories does not tell you which history actually happened — the set-selection problem

The most seductive misreading is that consistent histories solves the measurement problem by delivering definite outcomes. It does not, and its proponents are careful to say so. What it delivers is a rule for when you may reason with classical probabilities about quantum histories — the consistency condition — and a proof that measurement is an ordinary consistent family, not a primitive. But two things it does not do: it does not collapse the state (evolution is unitary throughout), and it does not select a unique consistent family. For the same system and state, many mutually incompatible consistent families exist, and the formalism offers no rule choosing one — this is the set-selection problem, and it is a genuine open issue (sharpened by the Dowker–Kent critique of 1995, which showed the profusion of frameworks is severe). The single-framework rule forbids combining propositions from incompatible families, which avoids apparent contradictions such as those arising in EPR or quantum-eraser scenarios. However, the framework does not provide a unique rule for selecting one consistent family as the physically realized description. Whether this should be regarded as a satisfactory resolution of the measurement problem or as a reformulation of it remains an interpretational debate.

Worked examples

Step 1. The $(z,z)$ family. Histories are $(z_0,z_0),(z_0,z_1),(z_1,z_0),(z_1,z_1)$ . Every off-diagonal of the decoherence functional has $\mathrm{Re}\,D=0$ (machine-verified, max $<10^{-12}$ ), so the family is consistent. Its probabilities are $p(z_0,z_0)=\tfrac12$ , $p(z_1,z_1)=\tfrac12$ , and the "jump" histories $(z_0,z_1),(z_1,z_0)$ have probability $0$ — a qubit in $|+\rangle$ has a definite $z$ -value at each frame and does not change it, so the movie is classical.

Step 2. The $(x,z)$ family. Now start in $|0\rangle$ and ask $x$ at $t_1$ , $z$ at $t_2$ . The two branches ending at $z{=}1$ have cross term $\mathrm{Re}\,D=\boxed{-0.25}$ (machine-verified), so the family is inconsistent.

Step 3. Read the failure. The coarse probability of reaching $z{=}1$ is $p(z_1|0)=|\langle1|0\rangle|^2=0$ , but the two fine branches carry $p(x_+,z_1)+p(x_-,z_1)=0.5$ . They do not add — the additivity gap is $2\,\mathrm{Re}\,D=\boxed{-0.5}$ , and the interference cancels the branch probabilities exactly. You cannot consistently say "the particle went through $x_+$ and landed at $z_1$ "; the question is not classically answerable. That is the double slit in the language of histories.

Example 2 — decoherence suppresses interference and drives an inconsistent family toward consistency (exam level). Take the inconsistent $(x,z)$ family from Example 1 and couple the "which- $x$ " property to an environment, so the two $x$ -branches acquire record states with overlap $\epsilon=\langle e_+|e_-\rangle$ . Track the cross term.

Step 1. Partial record. With $\epsilon=0.5$ , the cross term is multiplied by the overlap: $|D_{\rm cross}|=0.5\times0.25=\boxed{0.125}$ (machine-verified) — halved. The family is more consistent but not yet fully.

Step 2. Orthogonal record. With $\epsilon=0$ (the environment perfectly records which $x$ -branch occurred), the cross term vanishes: $\mathrm{Re}\,D=0$ (machine-verified). The family is now consistent, and the history probabilities add classically: $p(x_+,z_1)+p(x_-,z_1)=0.5$ becomes the genuine probability of reaching $z{=}1$ , because the environment has "measured" the path and destroyed the interference.

Step 3. Scale it up. A macroscopic coarse variable leaks its value into $\sim10^{23}$ environmental degrees of freedom, so $\epsilon\to0$ essentially instantly and irreversibly. Histories built from such variables — pointer positions, the trajectory of a planet — are therefore automatically consistent, with no need to choose a special family by hand. This is why the everyday world of definite classical facts is a decoherent (consistent) family, and why the framework recovers classical reasoning as an emergent limit rather than a postulate.

The logic of quantum histories

Origins · 1 minJames Hartle and how this was discovered

James Hartle

1939–2023 · Santa Barbara

Consistent histories has three founders and one great systematizer. Robert Griffiths introduced consistent histories in 1984, giving the consistency condition and the single-framework rule that dissolves paradoxes without collapse; Roland Omnès developed the logical structure through the late 1980s into a full "interpretation as extended classical logic." Independently, Murray Gell-Mann and James Hartle recast the idea in 1990 as decoherent histories, aimed squarely at quantum cosmology — a quantum mechanics for closed systems with no external observer, in which a quasiclassical domain of coarse-grained hydrodynamic variables emerges because environmental decoherence makes its histories consistent. Hartle, the Santa Barbara relativist better known to many for the Hartle–Hawking "no-boundary" wavefunction of the universe, spent decades arguing that this is the right framework for cosmology precisely because it needs no measurers. The program's honesty about its own limitation — the profusion of incompatible frameworks, and the absence of a rule to select one — is part of what makes it a serious foundational proposal rather than a slogan.

History

1984

— Griffiths introduces consistent histories and the consistency condition

1988

— Omnès develops the logical framework and the single-framework rule

1990

— Gell-Mann and Hartle recast it as decoherent histories for quantum cosmology

1992

— Omnès reviews the interpretation as an extended classical logic (RMP 64, 339)

1995

— Dowker and Kent sharpen the set-selection problem (PRL 75, 3038)

Going further

References. Griffiths, J. Stat. Phys. 36, 219 (1984) · Griffiths, Consistent Quantum Theory (Cambridge, 2002) · Omnès, Rev. Mod. Phys. 64, 339 (1992); The Interpretation of Quantum Mechanics (Princeton, 1994) · Gell-Mann & Hartle, in Complexity, Entropy, and the Physics of Information, ed. Zurek (1990); Hartle, Les Houches lectures (1992) · Dowker & Kent, Phys. Rev. Lett. 75, 3038 (1995) (set-selection critique) · Hartle & Hawking, Phys. Rev. D 28, 2960 (1983).

Provenance. The base article is standard and correct. The Heisenberg-picture evolution of projectors $P(t)=U^\dagger PU$ is owned by Properties and Time Evolution of the Density Matrix; the projection/Lüders language by Measurement and Wavefunction Collapse; the environment-tracing mechanism by The Reduced Density Matrix and Partial Trace and Quantum Decoherence and the Density Matrix — all recalled by name. Owned and machine-verified here: (1) the decoherence functional $D(\alpha,\beta)=\mathrm{Tr}[C_\alpha\rho C_\beta^\dagger]$ , its Hermiticity, and normalization $\sum D=\mathrm{Tr}\,\rho=1$ ; (2) the consistency condition $\mathrm{Re}\,D(\alpha,\beta)=0\Leftrightarrow$ additivity $p(\alpha\vee\beta)=p(\alpha)+p(\beta)$ (from the $2\,\mathrm{Re}\,D$ cross term); (3) an explicit consistent $(z,z)$ family on $|+\rangle$ (all off-diagonal $\mathrm{Re}\,D=0$ , probabilities $(0.5,0.5)$ ) and an inconsistent $(x,z)$ family on $|0\rangle$ (cross term $\mathrm{Re}\,D=-0.25$ , additivity gap $-0.5$ , coarse $p(z_1)=0$ vs branch sum $0.5$ ); (4) decoherence supplies consistency — an environment record of overlap $\epsilon$ scales the cross term, $\epsilon=0.5\Rightarrow|D|=0.125$ , $\epsilon=0\Rightarrow\mathrm{Re}\,D=0$ . The measurement problem itself is owned by The Quantum Measurement Problem and the collapse it replaces by Von Neumann Measurement and the Projection Postulate; the set-selection problem is stated as open — all by name.

Builds on: The Density Operator and Mixed States · Properties and Time Evolution of the Density Matrix · The Reduced Density Matrix and Partial Trace · Pure versus Mixed States and the Purity Criterion

Leads to: The Quantum Measurement Problem · Emergence of Classicality from Quantum Mechanics · The Copenhagen Interpretation

How is "history" here different from just the trajectory of a particle?

A classical trajectory assigns a definite position at every instant, automatically and without restriction. A quantum history assigns a property (a projector — which can be coarse, like "in the left half") at a chosen set of times, and crucially it is only meaningful within a consistent family: a set of histories whose decoherence functional is diagonal, so their probabilities add. You cannot freely combine questions across incompatible families (the single-framework rule). So a history is like a trajectory, but the theory tells you which sets of trajectory-questions are legitimate to ask together — and for a coherent superposition, incompatible questions (which-path and where-on-screen) are not jointly askable.

Does the theory have wavefunction collapse?

No. This is its defining feature: the state evolves unitarily throughout, with no collapse anywhere. Probabilities are assigned to histories via the decoherence functional, not produced by a physical reduction of the state. "Measurement" is just a consistent family in which system, apparatus, and environment become correlated. What replaces the projection postulate is the choice of a consistent framework, and within that framework the ordinary Born-rule probabilities of the sequence hold.

If many consistent families exist, isn't the theory hopelessly ambiguous?

This is the set-selection problem, and it is the framework's honest weak point. For the same system and state, many mutually incompatible consistent families coexist, and the formalism does not pick one. Proponents respond in two ways: pragmatically, that in practice environmental decoherence singles out the quasiclassical family of coarse hydrodynamic variables as overwhelmingly robust; and philosophically, that the single-framework rule (never mix incompatible families) is enough to keep the theory consistent and paradox-free even without a unique choice. Critics (notably Dowker and Kent) argue the profusion is severe enough to undercut the claim that it explains classicality. The debate is live; the article states it as open rather than solved.

How does consistent histories dissolve the EPR paradox?

By the single-framework rule. EPR-style paradoxes arise from combining incompatible descriptions — e.g., assigning both a definite spin- $z$ and spin- $x$ to a particle based on measurements on its entangled partner. In consistent histories those correspond to different, incompatible consistent families, and the rule forbids chaining inferences across them. Within any one family the reasoning is unambiguous and local; the "paradox" only appears if you illegitimately merge two frameworks. This is not collapse and it is not nonlocal influence — it is a discipline about which questions may be combined, which the formalism makes precise.

What is the difference between "consistency" (Re D = 0) and "medium decoherence" (full D = 0)?

The minimal requirement for probabilities to add is that the real part of the off-diagonal decoherence functional vanishes, $\mathrm{Re}\,D(\alpha,\beta)=0$ — that is exactly what kills the $2\,\mathrm{Re}\,D$ cross term in the additivity relation. "Medium decoherence" demands the stronger condition that the full complex $D(\alpha,\beta)=0$ . The stronger condition is more physically robust: it survives further coarse-graining and refinement without reintroducing interference, and it is what environmental decoherence actually delivers (orthogonal records give $D=0$ , not just $\mathrm{Re}\,D=0$ ). Most physical quasiclassical families satisfy the stronger condition automatically.

Is this the same as decoherence, then?

They are deeply linked but not identical. Decoherence, owned by Quantum Decoherence and the Density Matrix by name, is a physical process — a system entangling with an environment so its reduced state loses coherence. Consistent histories is a framework for assigning probabilities to time-sequences without collapse. The connection is that decoherence is what makes realistic, coarse-grained families consistent (it drives the off-diagonal $D$ to zero), which is why the two are usually discussed together as "decoherent histories." But you can state the consistency condition abstractly for any family, decohering or not; decoherence is the physical engine that makes the everyday quasiclassical family satisfy it.