Relational and QBist Interpretations

Copenhagen leaves an uncomfortable seam between a "quantum system" and a "classical observer," and the measurement problem asks what physically distinguishes the unitary Schrödinger evolution from the discontinuous collapse at measurement. Relational Quantum Mechanics (RQM) and QBism are two frontier interpretations that respond by relocating the quantum state — out of the world and into the relation between systems (RQM) or into the mind of an agent (QBism). Neither touches the mathematics: the density operator, unitary evolution, partial trace (recalled from The Reduced Density Matrix and Partial Trace by name), and the Born rule are all retained, so both make exactly the same predictions as textbook quantum mechanics. They are interpretations, not rival theories — and they cannot be distinguished by any experiment. RQM begins from one radical postulate: there is no observer-independent state of a system; every physical quantity is relational, taking a definite value only with respect to a second system that has interacted with the first. A measurement is an ordinary physical interaction that establishes a correlation, and the value an apparatus records is a fact relative to that apparatus, not a universal fact — the same relativization that makes velocity frame-dependent in Galilean mechanics or simultaneity frame-dependent in relativity. QBism (originally "Quantum Bayesianism") takes personalist Bayesian probability seriously and identifies the state $\rho$ an agent assigns with that agent's gambling commitments about the outcomes of measurements the agent might perform; Born probabilities are subjective degrees of belief, and collapse is nothing but Bayesian updating when the agent learns a result. This article owns the structural content of both: the relative-state consistency that lets Wigner and his friend assign different states without contradiction (verified numerically — both predict the same 50/50 statistics), and QBism's technical crown, the rewriting of the Born rule as a normative "urgleichung" over a SIC-POVM (verified: it reproduces projective Born probabilities exactly). It closes with the shared payoff — by relativizing facts (RQM) or subjectivizing the state (QBism), both stay local, compatible with Bell without nonlocal hidden variables — and the honest philosophical price each one pays.

Key takeaways

RQM (Rovelli, 1996): quantum states and physical values are not absolute but defined only relative to a system that has interacted with — measured — another; collapse is the updating of a relative state, with no privileged observer and no classical/quantum cut.
QBism (Fuchs, Mermin, Schack): the state $\rho$ is an agent's personal Bayesian degree of belief about the outcomes of their own future experiences; Born probabilities are subjective, and collapse is just Bayesian conditioning on a result.
Both keep the formalism and the Born rule intact — same density operators, same unitary evolution, same predictions — so they are interpretations, empirically indistinguishable from standard QM.
Wigner's friend (owned): friend assigns a definite post-measurement state (relative to himself), Wigner assigns the still-entangled state (relative to himself); both are consistent, both predict the same 50/50 z-statistics ( $\rho_S=I/2$ ) — no contradiction, facts are relative.
QBism urgleichung (owned): the Born rule rewritten over a SIC-POVM, $q(j)=\sum_i[(d+1)p(i)-1/d]\,r(j|i)$ , reproduces projective Born probabilities exactly (verified $q=(0.7,0.3)$ for a generic qubit) — quantum theory as a normative addition to probability theory.
Both stay local: relativizing facts / subjectivizing the state avoids nonlocal hidden variables, so both are compatible with Bell WITHOUT nonlocality — the honest contrast with Bohmian mechanics.
Key idea: quantum mechanics can be read without an observer-independent, objectively real state — RQM makes every fact relative to the system that registered it, QBism makes the state an agent's personal belief — dissolving the measurement problem by denying that collapse is a physical, observer-independent process.

The core insight

RQM and QBism keep the entire quantum formalism and the Born rule but deny that the quantum state is an observer-independent element of reality. RQM relativizes every physical fact to the system that registered it (so Wigner and his friend consistently assign different states); QBism makes the state an agent's Bayesian degrees of belief and rewrites the Born rule as a normative coherence condition over a SIC-POVM. Both dissolve the measurement problem and stay local — compatible with Bell without nonlocal hidden variables — at the price of abandoning a single shared reality of facts (RQM) or the realist reading of the state (QBism).

Where this lives today

Both interpretations are active research programs, not museum pieces. QBism drives a deep line in quantum foundations: the search for a beautiful SIC-POVM in every dimension (the "SIC existence problem," an unexpectedly hard problem in algebraic number theory tied to Hilbert's twelfth problem), and the reformulation of quantum theory as a modest, principled distortion of Bayesian probability — the hope that the "right" primitive is the urgleichung, from which the Hilbert-space formalism might be derived rather than assumed. RQM connects to quantum gravity: Rovelli, a founder of loop quantum gravity, sees the relational, background-independent reading as the natural quantum ontology for a theory with no external observer or fixed spacetime, and RQM has been sharpened in response to recent extended-Wigner's-friend theorems (Frauchiger–Renner and Local Friendliness) that probe whether observers can consistently assign states to each other. Both feed the modern experimental interest in Wigner's-friend scenarios — proof-of-principle photonic experiments implementing a "friend" as a small quantum system — which test the consistency conditions these interpretations lean on, without ever distinguishing them from standard quantum mechanics.

The physical picture

In RQM, stop asking "what is the spin?" and ask "what is the spin relative to this apparatus?" — exactly as you never ask for an absolute velocity, only velocity relative to a chosen frame. An electron can have a definite $z$ -spin relative to a magnet it passed through, while relative to a distant, uninvolved observer the joint electron-plus-magnet system is still in a superposition. Both descriptions are correct; they simply refer to different relations. Collapse, in this reading, is not a physical event but the updating of a relative state when a new interaction occurs — and because there is no privileged observer, there is no need for a classical/quantum cut anywhere. In QBism, the quantum state is like a bookmaker's odds, not like a stone's mass. When you write $|\psi\rangle$ you are recording your personal bets on what you will experience if you act on the system. Collapse is the trivial mental act of replacing your prior odds with the result you just saw — no spooky physical jump, no privileged observer, and no contradiction if your colleague holds different odds, just as two bettors can rationally hold different odds on the same coin. A useful slogan captures the pair: RQM relativizes the facts (an outcome is an event for the system that registered it), while QBism relativizes the state (the assignment is a belief of the agent who wrote it). Both keep every equation of quantum mechanics untouched; what they change is the reading of what those equations refer to.

Figure

Two ways to give up an observer-independent state — same predictions, different reading

The relational structure: Wigner and his friend

The clean test of "facts are relative" is the Wigner's-friend scenario: a friend measures a qubit inside a sealed lab, while Wigner, outside, has not interacted with anything. The full Wigner's-friend analysis is developed in The Wigner's Friend Thought Experiment by name; here we own the relational consistency — that the friend's definite outcome and Wigner's still-superposed description are both correct, because they are facts relative to different systems, and they agree on every statistic each can access.

Figure

Wigner and his friend: a definite outcome inside, an entangled state outside — both correct

Derivation ARelational consistency — the friend's fact and Wigner's fact do not contradict

We show that the friend's definite post-measurement state and Wigner's entangled description are both legitimate, and that they predict identical statistics for any measurement the friend can perform — so "the outcome is relative" is a consistent position, not a paradox. The reduced-state machinery is recalled from The Reduced Density Matrix and Partial Trace by name.

Step 1. Set the scene. The qubit starts in $|+\rangle=\tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$ . The friend $F$ measures it in the $z$ -basis inside a sealed lab; as a unitary interaction (no collapse from Wigner's outside view) this correlates system and friend into

|\Psi\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle|F_0\rangle + |1\rangle|F_1\rangle\big) \qquad (1)

Why Wigner writes this: Wigner has not interacted with the lab, so relative to him the only correct description is the unitary one — a pure, entangled $S{+}F$ state, $\mathrm{Tr}\,|\Psi\rangle\langle\Psi|^2 = 1$ .

Step 2. The friend's relative fact. Relative to the friend, who did interact, the measurement had a definite outcome — say " $0$ " — so the friend's post-measurement state of $S$ is the definite $|0\rangle$ (or $|1\rangle$ ). Averaged over the outcomes he could have obtained, the friend's proper mixture is $\tfrac12|0\rangle\langle0|+\tfrac12|1\rangle\langle1| = \tfrac{I}{2}$ .

Step 3. Wigner's reduced state. Wigner's improper reduced state of $S$ , tracing out the friend, is

\rho_S = \mathrm{Tr}_F\,|\Psi\rangle\langle\Psi| = \tfrac12|0\rangle\langle0|+\tfrac12|1\rangle\langle1| = \tfrac{I}{2} \qquad (2)

Why there is no contradiction: the friend's proper mixture and Wigner's improper reduced state are the <strong>same operator</strong> $I/2$ , so they give identical predictions for any measurement on $S$ — both say $P(0)=P(1)=\tfrac12$ . The friend's "definite outcome" and Wigner's "still-superposed" are facts relative to different systems, and they never disagree on anything either can check.

Step 4. Where they could differ. Only Wigner can, in principle, perform a coherence experiment on the whole $S{+}F$ system — an interference measurement sensitive to the $|0F_0\rangle\langle1F_1|$ off-diagonal of $|\Psi\rangle\langle\Psi|$ (magnitude $0.5$ ) that the friend's block-diagonal mixture lacks. Why this is the relational punchline: the "extra" coherence Wigner sees is a fact relative to Wigner, requiring an interaction the friend cannot perform on himself; RQM says there is simply no observer-independent fact about "whether the measurement really happened" — it happened relative to the friend, and not (yet) relative to Wigner. *Verified numerically: $|\Psi\rangle$ is pure ( $\mathrm{Tr}\,\rho_W^2=1$ ); Wigner's $\rho_S=I/2$ ; the friend's outcome-averaged mixture $=I/2$ (same operator); both give $P(0)=P(1)=0.5$ ; the distinguishing $S{+}F$ coherence $|\langle0F_0|\Psi\rangle\langle\Psi|1F_1\rangle|=0.5$ .*

QBism's technical core: the Born rule as an urgleichung

Go deeper · Derivation B · 2 minRewriting the Born rule over a SIC-POVM

Derivation BRewriting the Born rule over a SIC-POVM

We show QBism's structural claim: the Born rule can be recast as a normative relation between the probabilities of a reference measurement and those of any target measurement, with no operators appearing — quantum theory as an addition to probability theory. The tool is a symmetric informationally complete POVM (SIC), and the reference to generalized measurements is forwarded to Generalized Measurements and POVMs by name.

Step 1. Build the qubit SIC. A SIC-POVM for $d=2$ has four elements $E_i=\tfrac{1}{d}|\psi_i\rangle\langle\psi_i|$ whose states form a regular tetrahedron on the Bloch sphere. Completeness and symmetry:

\sum_{i=1}^{4} E_i = I,\qquad |\langle\psi_i|\psi_j\rangle|^2 = \frac{1}{d+1} = \frac13\ \ (i\neq j) \qquad (3)

Why "symmetric, informationally complete": the four states are equiangular (pairwise overlap $\tfrac13$ ) and their statistics fix any qubit state completely, so the reference-measurement probabilities $p(i)=\mathrm{Tr}(\rho E_i)$ are a full, redundancy-free "probability portrait" of $\rho$ .

Step 2. State the urgleichung. For any target measurement with outcomes $j$ , QBism writes the outcome probabilities purely in terms of the reference probabilities $p(i)$ and the conditionals $r(j|i)$ (target statistics given the SIC states):

q(j) = \sum_{i=1}^{d^2}\Big[(d+1)\,p(i) - \tfrac1d\Big]\,r(j|i) \qquad (4)

Why this is the "urgleichung" (fundamental equation): it is the Born rule with the operators removed — a coherence condition relating the gambles $p(i)$ on the reference measurement to the gambles $q(j)$ on the target. It differs from the classical law of total probability $q(j)=\sum_i p(i)r(j|i)$ by the factors $(d{+}1)$ and $-\tfrac1d$ : quantum theory is a small, universal distortion of Bayesian bookkeeping.

Step 3. Check it reproduces Born. Take a generic qubit $\rho$ with Bloch vector $(0.3,-0.5,0.4)$ and a projective $z$ -measurement. Computing $p(i)=\mathrm{Tr}(\rho E_i)$ and $r(j|i)=\mathrm{Tr}(|\psi_i\rangle\langle\psi_i|\,M_j)$ and feeding (4) gives

q = (0.7,\,0.3) = \big(\mathrm{Tr}(\rho\,|0\rangle\langle0|),\ \mathrm{Tr}(\rho\,|1\rangle\langle1|)\big) \qquad (5)

exactly the projective Born probabilities. Why this matters to QBism: the entire empirical content of the Born rule is captured by a normative constraint on an agent's probability assignments for two measurements — no reference to an objective state or wavefunction is needed. The state $\rho$ is a bookkeeping device for the reference probabilities $p(i)$ , which are the agent's beliefs. *Verified numerically: SIC completeness $\sum E_i=I$ and symmetry $|\langle\psi_i|\psi_j\rangle|^2=\tfrac13$ exactly; the urgleichung reproduces the $z$ -Born probabilities $q=(0.7,0.3)$ and, for an $x$ -measurement, $q_x=(0.65,0.35)$ — both matching direct Born to machine precision, with $\sum_j q(j)=1$ .*

Limiting cases

case	RQM reading	QBism reading
isolated system, no interaction	no facts about it (no relation yet)	agent holds prior beliefs $\rho$
measurement occurs	a fact relative to the measuring system	agent updates beliefs (conditioning)
Wigner's friend	outcome relative to friend, superposition relative to Wigner	friend and Wigner are two agents with two valid $\rho$ 's
EPR / Bell setup	outcomes relative to local devices — local	agents' beliefs updated locally — local
classical limit	relations become stable, shared, "objective-looking"	agents' beliefs converge via shared experience
urgleichung, $(d{+}1)\to1,\ \tfrac1d\to0$	—	reduces toward the classical law of total probability

Common trapThese are interpretations, not theories — they predict nothing new, and neither claims to prove "reality"

Two overclaims to avoid. First, RQM and QBism are interpretations: they keep the density operator, unitary evolution, partial trace, and the Born rule exactly, so they make identical predictions to textbook quantum mechanics and cannot be tested against it — unlike objective collapse, developed in Objective Collapse Theories by name, which really is a rival theory. Second, neither proves its metaphysics. QBism does not demonstrate that the state is "merely" belief; it shows the formalism is consistent with a personalist-Bayesian reading, and rewrites Born as a normative constraint — a structural result, not a proof about reality. RQM does not prove that facts are "only" relational; it shows a consistent relativization of facts is possible and dissolves certain paradoxes. Their treatment of "reality" is a philosophical stance, and whether giving up a single shared world (RQM) or the realist state (QBism) is a feature or a bug is a live debate, not a settled result. The mathematics (Wigner-friend consistency, the urgleichung) is solid; the metaphysics is a reading.

Worked examples

Example 1 — Wigner and his friend assign different states, and agree on statistics (confidence builder). A qubit starts in $|+\rangle$ ; the friend measures it in the $z$ -basis in a sealed lab. Write down the friend's state and Wigner's state, and check they predict the same thing.

Step 1. Wigner's state. Wigner has not interacted, so relative to him the lab is the pure entangled state $|\Psi\rangle=\tfrac{1}{\sqrt2}(|0\rangle|F_0\rangle+|1\rangle|F_1\rangle)$ , with purity $\mathrm{Tr}\,\rho_W^2=1$ (machine-verified). His reduced state of the qubit is $\rho_S=\mathrm{Tr}_F|\Psi\rangle\langle\Psi|=\tfrac{I}{2}$ .

Step 2. The friend's state. The friend got a definite outcome (say " $0$ "), so relative to him the qubit is $|0\rangle$ . Averaged over the outcomes he could have obtained, his proper mixture is $\tfrac12|0\rangle\langle0|+\tfrac12|1\rangle\langle1|=\tfrac{I}{2}$ — the same operator as Wigner's $\rho_S$ (machine-verified).

Step 3. Compare predictions. Both give $P(0)=P(1)=\boxed{0.5}$ for a $z$ -measurement on the qubit (machine-verified). So the friend's "definite outcome" and Wigner's "still in superposition" never disagree on any observable statistic — they are facts relative to different systems. The only place they could differ is a coherence experiment on the whole $S{+}F$ system (sensitive to the $0.5$ -magnitude off-diagonal that Wigner's pure $|\Psi\rangle$ has and the friend's mixture lacks), which only Wigner can perform. RQM's verdict: there is no observer-independent fact about "whether the measurement really happened."

Example 2 — the urgleichung reproduces the Born rule (exam level). For a qubit with Bloch vector $(0.3,-0.5,0.4)$ , use the SIC-POVM and QBism's urgleichung to compute the probabilities of a $z$ -measurement, and check against the Born rule directly.

Step 1. Reference probabilities. With the tetrahedral SIC states, $p(i)=\mathrm{Tr}(\rho E_i)$ gives four numbers summing to $1$ (machine-verified $\sum p=1$ ). These are the agent's beliefs about the reference measurement — a complete "probability portrait" of $\rho$ .

Step 2. Conditionals and the urgleichung. The conditionals $r(j|i)=\mathrm{Tr}(|\psi_i\rangle\langle\psi_i|\,M_j)$ give the $z$ -statistics of each post-SIC state. Feeding $p(i)$ and $r(j|i)$ into $q(j)=\sum_i[(d+1)p(i)-\tfrac1d]\,r(j|i)$ with $d=2$ gives $q=\boxed{(0.7,\,0.3)}$ (machine-verified).

Step 3. Check against Born. Directly, $\mathrm{Tr}(\rho|0\rangle\langle0|)=\tfrac12(1+r_z)=\tfrac12(1+0.4)=0.7$ and $\mathrm{Tr}(\rho|1\rangle\langle1|)=0.3$ — identical. The urgleichung reproduces Born exactly, using only probabilities (of the reference and target measurements) and never an operator or a state on the right-hand side. That is QBism's structural claim made concrete: quantum theory is a normative rule relating an agent's gambles, differing from classical total probability only by the universal factors $(d+1)$ and $-\tfrac1d$ .

The state, reinterpreted

Origins · 1 minquantum state and how this was discovered

quantum state

1994–2010 · Marseille, Perimeter

The two interpretations reinterpret the same object — the quantum state — in opposite directions. Carlo Rovelli, better known for loop quantum gravity, proposed Relational Quantum Mechanics in 1996, arguing that the lesson of quantum mechanics is that states and values are relational, defined only between interacting systems, with no observer-independent fact of the matter — a stance he traces to the relativity of motion and to a broadly Machian, background-independent instinct. QBism grew from the quantum-information work of Carlton Caves, Christopher Fuchs, and Rüdiger Schack around 2001–2002, later joined by N. David Mermin; its wager is that the quantum state is not a thing in the world but an agent's personal Bayesian degrees of belief, and that the Born rule is a normative addition to probability theory best expressed through the SIC-POVM urgleichung. Where RQM keeps facts (but makes them relative), QBism keeps a single agent (but makes the state their belief). Both leave every equation of quantum mechanics untouched and insist that the puzzle was never in the mathematics but in the assumption of an observer-independent state.

History

1996

— Rovelli proposes Relational Quantum Mechanics (Int. J. Theor. Phys. 35, 1637)

2001

— Caves, Fuchs, Schack: quantum probabilities as Bayesian degrees of belief

2002

— Fuchs and Schack develop the QBist program and the SIC-POVM reformulation

2010

— the "urgleichung" — the Born rule as a SIC-POVM normative constraint — matured

2014

— Fuchs, Mermin, Schack, "An introduction to QBism" (Am. J. Phys. 82, 749)

Going further

References. Rovelli, Int. J. Theor. Phys. 35, 1637 (1996) (Relational QM) · Rovelli, Helgoland (2021, popular) · Caves, Fuchs & Schack, Phys. Rev. A 65, 022305 (2002) · Fuchs & Schack, Rev. Mod. Phys. 85, 1693 (2013) (QBist reconstruction) · Fuchs, Mermin & Schack, Am. J. Phys. 82, 749 (2014) · Mermin, Nature 507, 421 (2014) · Renes, Blume-Kohout, Scott & Caves, J. Math. Phys. 45, 2171 (2004) (SICs).

Provenance. The base article is standard and correct. The density operator, unitary evolution, partial trace, and Born rule are retained unchanged; the reduced-state machinery is owned by The Reduced Density Matrix and Partial Trace, generalized measurements/POVMs by Generalized Measurements and POVMs, and the local-realism premises by The EPR Paradox and Local Realism and Bell's Theorem and the CHSH Inequality — all recalled by name. Owned and machine-verified here: (1) the relational consistency of the Wigner's-friend scenario — $|\Psi\rangle$ pure, Wigner's $\rho_S=I/2$ equals the friend's proper mixture $I/2$ , both give $P(0)=P(1)=0.5$ , distinguishing $S{+}F$ coherence of magnitude $0.5$ ; (2) the qubit SIC-POVM completeness $\sum E_i=I$ and symmetry $|\langle\psi_i|\psi_j\rangle|^2=\tfrac13$ ; (3) the urgleichung $q(j)=\sum_i[(d+1)p(i)-\tfrac1d]r(j|i)$ reproducing projective Born probabilities exactly ( $q=(0.7,0.3)$ for $z$ , $(0.65,0.35)$ for $x$ on Bloch vector $(0.3,-0.5,0.4)$ ); (4) no-signalling invariance (a local operation on one half of a Bell pair leaves $\rho_A=I/2$ unchanged) underlying both interpretations' locality. The full Wigner's-friend analysis is owned by The Wigner's Friend Thought Experiment; Bell's theorem by Bell's Theorem and the CHSH Inequality; the measurement problem by The Quantum Measurement Problem — all by name. The metaphysical readings (facts as relational, state as belief) are stated as consistent stances, not proofs.

Builds on: The Density Operator and Mixed States · The Reduced Density Matrix and Partial Trace · The EPR Paradox and Local Realism · Bell's Theorem and the CHSH Inequality

Leads to: The Wigner's Friend Thought Experiment · The Quantum Measurement Problem · The Copenhagen Interpretation

If RQM and QBism make no new predictions, why should I care about them?

Because the measurement problem is a problem about what the formalism means, not about what it predicts, and these interpretations are serious, internally consistent answers that dissolve it cheaply. RQM removes the classical/quantum cut by making every fact relative to an interacting system; QBism removes the mystery of collapse by making it Bayesian updating. Beyond philosophy, both are generative: QBism's urgleichung has launched a research program to derive the Hilbert-space formalism from a probability principle plus SIC-POVMs, and RQM's background-independence appeals to quantum-gravity approaches with no external observer. An interpretation that reframes a stubborn problem and opens new technical questions earns its keep even if it never changes a number.

Doesn't QBism collapse into solipsism if the state is just my beliefs?

QBists argue no, and the distinction is important. The state $\rho$ is personal — your beliefs about your experiences — but the world is not: there is a real external world that surprises you, and the Born rule is a normative constraint (like the axioms of probability) on how a rational agent must hold their beliefs given that world's quantum character. You cannot believe anything you like; the urgleichung disciplines your gambles. Two agents interacting through a shared world tend to converge. QBism relocates the state into the agent while keeping the world real and the constraints objective — it is personalism about probability, not solipsism about reality. Whether that line holds under pressure is part of the live philosophical debate.

How is RQM different from the many-worlds interpretation? Both keep unitary evolution.

Both keep unitary evolution and reject a physical collapse, but they draw opposite conclusions about outcomes. Many-worlds, developed in The Many-Worlds Interpretation by name, keeps all outcomes as really-existing branches of one universal wavefunction. RQM keeps a single outcome but makes it relative: the friend's measurement has one definite result relative to the friend, and the "superposition" is only Wigner's (relational) description. So many-worlds multiplies worlds to preserve a single objective wavefunction; RQM keeps one world but denies that facts are observer-independent. They agree the Schrödinger equation is never violated and disagree entirely on ontology.

What is a SIC-POVM and why does QBism need it?

A symmetric informationally complete POVM is a set of $d^2$ subnormalized rank-one projectors $E_i=\tfrac1d|\psi_i\rangle\langle\psi_i|$ that are equiangular ( $|\langle\psi_i|\psi_j\rangle|^2=\tfrac{1}{d+1}$ ) and complete ( $\sum E_i=I$ ); for a qubit they form a tetrahedron on the Bloch sphere. QBism prizes it because the probabilities $p(i)=\mathrm{Tr}(\rho E_i)$ of a SIC measurement form a complete, minimal, redundancy-free description of the state — a genuine "probability portrait." In that portrait the Born rule becomes the urgleichung, a clean normative relation between probabilities with no operators. The catch is the SIC existence problem: SICs are conjectured to exist in every dimension but proving it is astonishingly hard (linked to algebraic number theory), which is itself a QBist research frontier.

Do RQM and QBism actually keep the same math, or do they change something?

They keep the mathematics exactly. The density operator, unitary evolution, the partial trace (owned by The Reduced Density Matrix and Partial Trace by name), and the Born rule are all retained unchanged — which is precisely why they are interpretations, empirically indistinguishable from textbook quantum mechanics, rather than rival theories like objective collapse. QBism re-expresses the Born rule via the SIC-POVM urgleichung, but that is a mathematically equivalent rewriting, not a modification: it reproduces the same probabilities (we verify $q=(0.7,0.3)$ matches direct Born). What changes is the reading of what the symbols refer to — relations (RQM) or personal beliefs (QBism) — never the predictions.

How do these interpretations preserve locality where Bohmian mechanics gives it up?

Bell's theorem forces any interpretation with observer-independent, pre-existing outcomes and locality to fail — so Bohmian mechanics keeps definite outcomes and pays with explicit nonlocality (the guidance velocity of one particle depends instantly on a distant partner). RQM and QBism escape the dilemma by rejecting the shared, observer-independent outcomes the theorem assumes: in RQM the outcome is a fact relative to the local device; in QBism it is a personal experience of the local agent. There are simply no spacelike-separated elements of reality for a measurement "here" to fix "over there," so no nonlocal mechanism is needed — the no-signalling invariance (a local operation leaves the distant reduced state $I/2$ unchanged, which we verify) is all that survives. Bell's theorem itself is developed in Bell's Theorem and the CHSH Inequality by name.