The line from the scratch
Two State Vector Formalism and Quantum Logic
J.E.D’Ulisse
Abstract:
Quantum mechanics refutes classical intuitions about ontology, as exemplified by the Multi-worlds (MW) and Copenhagen Interpretations (CI). Despite their apparent differences, MW and CI share the same mathitcal and logical structures, characterized by a non-distributive quantum logic. In stark contrast, the Two State Vector Formalism (TSVF) suggests a different Hilbert space structure and exhibits the distributive property in its quantum logic.
This raises questions about the interpretation of quantum mechanics and the foundations of logic. Hilary Putnam proposed that logic, like geometry, should be revised to match observed reality. While frequentism (given enough time the probable will appear determined) can explain some aspect of the emergence of the classical from the quantum, it cannot explain how classical logic relates to the quantum.
The TSVF demonstrates a framework through which classical features could emerge by considering both forward and backward-evolving quantum states. This demonstrates how classical-like logic could emerge through system constrictions like post-selection. Weak measurement experiments, such as the Quantum Cheshire Cat, provide empirical support for these counterintuitive quantum behaviors.
The argument here is not the validity of TSVF, (a matter for physists or reality depending on how you see it). It is that an experiment could be designed, at low cost, to explore this fundamental difference. More importantly, that classical characteristics can emerge in a quantum framework, suggests Puttnam was right.
Hillary Putnam controversially suggested that logic is like geometry and should thus be revised to match reality, just as non-euclidean geometry amended classical concepts. The issue is that to revise logic to account for the quantum world it behaves in a way variously described as non-local, non-binary, and non-realistic. This manifests in Quantum Logic as it being non-distributive, meaning that the order of interactions changes its value. That is to say the measurement affects the outcome, rather than objectively describing it.
Classical logic is based on Boolean flow and order, as is classical physics. When you formalize this process you can see that the elements within a framework, such as A, B and C can be reordered like this:
This does not hold in Quantum logic, where waveform collapse, superposition and entanglement, make this impossible. Thus Putnam’s bold demand to throw out Distribution as a universal logical rule, and make it a special case or convention. This would deeply and radically alter our view of the world, so most reject the position, because either our description of QM may be incomplete and when it is it will show itself to exhibit True Logic, or the QM may just represent a fundamentally separate and unbridgeable logic.
Those in Putnam’s camp need to demonstrate how classical logic would have emerged, for their argument to be compelling. While a frequentist reading of probabilities can explain the appearance of Classicality in our mechanics (i.e. given infinite iterations the most probable outcome will appear as the determined outcome), this can not explain how a boolean, classical-like logic could emerge from quantum logic. TSVF(Two-state vector formalism) shows how this could be resolved.
TSVF utilizes the reversible nature of the Schrodinger equation, meaning that it world both forward and backward in time (the forward-evolving state of the waveform is Ψ, and the backward-evolving is Φ. Interestingly, TSV exhibits distribution.
This distribution is only pragmatic and not fundamental. This is because TSVF is post-selective and non-predictive, so that the systemic restraints on the system (the automatic alignment between Ψ and Φ, ensures non-distributive results are excluded. Weak measurements can be used to demonstrate this.
A weak measurement is an experiment which attempts to exhibit aspects of quantum behavior, without the collapse of the waveform. Typically these experiments have been used to demonstrate the possibility of disaggregating quantum properties in a system (ie. the cheshire cat experiment explained below).
This suggests Putnam is correct, and that classical logic is emergent from quantum logic and that the passage from one logic to the other is determined by the establishment of a non-reversible, entropic effect. That is, the cut between the classical and quantum is made with the blade of time, with the hand of a clock.
Projections and Properties
Quantum mechanics relies on mathematical structures that differ significantly from those of classical physics. Central to this framework is the concept of a Hilbert space, a complex vector space that represents the possible states of a quantum system. Projections onto subspaces of this Hilbert space correspond to properties or propositions about the system, such as potential measurement outcomes. This forms the basis of quantum logic, a non-classical logic that accounts for the limitations on simultaneous knowledge and measurement of certain properties.
The state of a quantum system is described by a vector in Hilbert space, known as a wave function or state vector. Physical observables, like position, momentum, and energy, are represented by operators that act on these state vectors. The inner product on Hilbert space enables the calculation of probabilities and expectation values, which are crucial for making predictions about measurement outcomes. The completeness property of Hilbert spaces ensures that all necessary mathematical operations are well-defined.
Von Neumann recognized a special class of observables, called {0, 1}-valued observables, which correspond to propositions about a system's state. These observables are represented by projection operators, self-adjoint operators that square to themselves (P² = P). Each projection operator is associated with a unique closed subspace of the Hilbert space. For a normalized state vector |ψ⟩, the probability of obtaining the outcome "1" when measuring the property associated with projection P is given by ⟨ψ|P|ψ⟩. This mathematical formalism provides a direct link between the abstract structure of Hilbert space and the concrete outcomes of physical measurements in quantum mechanics.
Von Neumann observed that the relationship between projections and physical properties enables a kind of "logical calculus." Unlike classical logic, quantum logic includes the idea of "simultaneous decidability," reflecting the unique nature of quantum measurements. This requires that quantum mechanics, as interpreted by Copenhagen (CI), is non-distributive.
Logical Structure of Projections
The closed subspaces of a Hilbert space H, organized by inclusion, form a structure called a lattice. This lattice has two key operations:
Meet (∧): The greatest lower bound of subspaces is their intersection.
Join (∨): The least upper bound is the closed span (smallest closed space containing their union).
While this lattice has many complementary subspaces, it is not distributive like classical logic. Instead, it is ortho complemented, meaning each subspace M has an orthogonal complement M⊥, consisting of all vectors orthogonal to M.
Using the one-to-one correspondence between projections and subspaces, we can define logical relationships for projections:
Commuting projections are simultaneously measurable, meaning that their associated properties can be tested together. This allows classical logic (with operations like "and," "or," and "not") to apply to commuting projections, even in the quantum world.
Below by Daniel Mader 10. 06 2005 from en.wikipedia; CC BY-SA 3.0
The Cheshire Cat
In quantum mechanics, the "quantum Cheshire cat" phenomenon suggests that a particle's physical properties can take a different trajectory from the particle itself. This concept, named after the Cheshire Cat in Lewis Carroll's Alice's Adventures in Wonderland, was introduced in 2012 by Yakir Aharonov, Daniel Rohrlich, Sandu Popescu, and Paul Skrzypczyk. The cat's ability to disappear, leaving only its grin behind.
In classical physics, an object's properties are inseparable from its trajectory. For instance, a magnet and its magnetic moment follow the same path. However, in quantum mechanics, particles can exist in superpositions of different trajectories prior to measurement. Experiments suggest that a particle, such as a neutron, may traverse one path while a property like spin or polarization travels a different path. This behavior is analyzed using weak measurements, a technique that interprets the particle's pre-measurement history through minor quantum disturbances.
The quantum Cheshire cat has been experimentally demonstrated with electrons and neutrons, for example, detaching a neutron’s magnetic moment from its trajectory shields it from electromagnetic interference
In a typical setup, a neutron beam enters an interferometer, splitting into superpositions along two paths. Modifications, such as magnetic moment-flipping filters, enable postselection, creating distinct states where the neutron’s magnetic moment resides in one path, while the neutron itself predominantly travels another. Experimental results, confirmed by absorbers and magnetic fields, demonstrate that the detected neutrons originate from one path, while their magnetic moments are influenced by the other. This separation effectively illustrates the "cat" (neutron) and its "grin" (magnetic moment) as distinct entities within the quantum realm.
Quantum logic
The Quantum Cheshire Cat phenomenon represents an example of how quantum logic diverges from classical intuitions, illustrating that a particle’s physical properties can seemingly be disjoint from the particle itself. Below is a complete description of the phenomenon in terms of formal quantum logic:
In quantum mechanics, the state of a particle with two orthogonal paths |A⟩ and |B⟩, and two-level physical properties (e.g., spin or polarization |0⟩ and |1⟩), is represented as a Hilbert space state:
Here, |0⟩ corresponds to the chosen property orientation (e.g., polarization along a specific axis). This is the preselected state.
The Post Selected State is after a filtering operation flips the property along path B, the system evolves into:
Here, |A⟩|0⟩ corresponds to the particle in path A with its property unaltered, and |B⟩|1⟩ corresponds to the particle in path B with its property flipped.
In formal quantum logic, propositions about quantum systems correspond to projectors in a Hilbert space. Logical operations (e.g., conjunction, disjunction) are mapped to operations on these projectors. The phenomenon is analyzed via the following:
(a) Projection Operators:
Π_A = |A⟩⟨A|: Measures if the particle is in path A.
Π_B = |B⟩⟨B|: Measures if the particle is in path B.
σ_⊥ = |0⟩⟨1| + |1⟩⟨0|: Represents the polarization or spin-flipping operator.
(b) Weak Values:
Weak measurements probe quantum systems without collapsing the wave function, offering insight into the system's properties in superposition. The weak value of an operator O is defined as:
where |ψ⟩ is the preselected state and |ϕ⟩ is the postselected state.
(c) Key Weak Values:
Path Projectors:
These indicate the particle is found exclusively in path A, as any perturbation to A affects the outcome, while B does not.Polarization on Paths:
These values show the physical property (e.g., polarization) resides exclusively in path B, decoupled from the particle trajectory.
Quantum Cheshire Cat Interpretation
In classical logic, a particle's properties (e.g., trajectory and spin) are inseparably tied. In quantum logic, superposition allows for a separation:
Particle Trajectory (Path A) ∨ Property Location (Path B)
The weak measurements reveal these are distinct components. This demonstrates non-classical correlations between the particle's position and its physical property. Path A has the particle without the property, while path B has the property without the particle.
The measurement results depend on complementary properties (position and polarization), governed by quantum uncertainty. Disturbing one component (e.g., trajectory) disrupts coherence but does not collapse the system entirely.
Despite the particle being predominantly in Path A (according to the weak measurement), its property is predominantly in Path B. This apparent separation of the particle and its property is a manifestation of quantum superposition and the non-classical nature of quantum measurements.
TSVF
The Two-State Vector Formalism (TSVF) in quantum mechanics provides a framework for understanding phenomena like the Quantum Cheshire Cat effect. TSVF describes a quantum system using two state vectors: the preselected state (before measurement) and the post-selected state (after measurement). These two vectors propagate in opposite temporal directions and provide a time-symmetric interpretation of quantum phenomena.
Here’s a reformulation of the Quantum Cheshire Cat effect within the TSVF framework:
Preselected State (|Ψ⟩):
A neutron (or particle) is prepared in a Mach-Zehnder interferometer. Upon entering the system, the particle is in a quantum superposition of two paths |A⟩ and |B⟩, with an associated polarization state |0⟩:
Modification of Path B:
A filter is placed in path |B⟩, flipping the particle's polarization to |1⟩ while leaving the particle in |A⟩ unchanged. The resultant state is:
Post Selected State (|Φ⟩):
After passing through the interferometer, the system is postselected on detecting the particle in a specific final state. This ensures that only a subset of events contributes to the analysis, refining the interpretation of the particle’s trajectory and its properties.
Weak Values in TSVF
The TSVF allows us to calculate weak values for observables using both the preselected and post selected states:
where O is an observable, and ⟨Φ|Ψ⟩ ensures the proper overlap between the two states.
Projectors for Path Detection:
Path |A⟩: Π_A = |A⟩⟨A|
Path |B⟩: Π_B = |B⟩⟨B|
Weak values for these projectors:
⟨Π_A⟩_w = 1, ⟨Π_B⟩_w = 0
These results imply that the particle is predominantly associated with path A, while no weak evidence supports its presence in B.
Polarization Operator: Define the polarization observable σ_⊥:
σ_⊥ = |0⟩⟨1| + |1⟩⟨0|
Weak values for polarization along paths:
⟨Π_Aσ_⊥⟩_w = 0, ⟨Π_Bσ_⊥⟩_w = 1
These indicate that the particle's polarization is localized in path B, even though the particle itself is in path A.
Interpretation of the TSVF Chesire Cat
The particle ("cat") is weakly detected in path A, while its magnetic moment ("grin") is weakly detected in path B. The Two-State Vector Formalism (TSVF) supports this separation by attributing observables to a combination of preselected and post-selected states, rather than assuming intrinsic properties tied to the particle's trajectory.
The preselected state |Ψ⟩ represents the particle’s forward-evolving history.
The postselected state |Φ⟩ provides future measurement constraints that retroactively influence interpretations of the particle's past behavior.
Weak measurements interact with the system without significantly disturbing its quantum state. The weak values obtained in TSVF reveal the particle and its property appearing in different paths.
Lattice comparison
The Quantum Cheshire Cat experiment explores how quantum properties, such as position and spin (or polarization), can appear to "disentangle" in ways that challenge classical intuition. This analysis applies Quantum Logic and the principle of distributivity within both Orthodox Quantum Mechanics (OQM) and TSVF.
1. Orthodox Quantum Mechanics (OQM):
Non-Distributive Lattice: OQM employs a lattice of projections on a Hilbert space, where:
The meet (∧) and join (∨) operations correspond to intersection and closed linear span, respectively. Crucially, this lattice is non-distributive, meaning that the distributive law, A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C), does not always hold.
This non-distributivity arises from the fact that the join operation, B ∨ C, can encompass states orthogonal to A, thereby violating the distributive law. In the experiment, the particle (e.g., a neutron) and its property (e.g., polarization) can be found in seemingly distinct spatial locations or subspaces.
Within OQM, these properties are represented as projections on the Hilbert space. The observed results, including the apparent separation of the particle and its property, stem from the non-commutative nature of the relevant quantum operators.The non-distributive lattice structure underlying OQM directly reflects the entangled nature of quantum systems and the probabilistic interference patterns that characterize quantum phenomena.
Preselection and Postselection Constraint in TSVF
Preselected State |ψ⟩:
The initial state prepared at the beginning of the experiment.
Defines the forward-evolving quantum state.
Postselected State |ϕ⟩:
The final state measured at the end of the experiment.
Defines the backward-evolving quantum state.
Weak Value: Observable properties in TSVF are determined by the overlap between |ψ⟩ and |ϕ⟩, formalized as the "weak value":
This weak value can sometimes yield surprising results, such as values outside the eigenvalue spectrum of the observable.
Example: Quantum Cheshire Cat Experiment
Preselection:
Postselection:
In TSVF, properties like "position" (path) and "polarization" (spin) are examined as weak values, revealing that the neutron's position and polarization can appear to separate. For instance, the neutron might be found in one path, while its polarization is in another.
2. Conditional Distributivity in TSVF
In TSVF, pre- and post selection effectively constrain the Hilbert space to a subspace Hψ,ϕ, defined by the overlap of |ψ⟩ and |ϕ⟩. Within this subspace:
Projections A, B, C that are compatible with both |ψ⟩ and |ϕ⟩ behave more like classical logic operations.
Conditional distributivity holds:
A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
Formal Example: Let |L⟩, |R⟩ represent orthogonal paths.
A = PL: projection onto |L⟩ (neutron in the left path)
B = PH: projection onto horizontal polarization
C = PV: projection onto vertical polarization
Under TSVF: the weak values determine the behavior of PL, PH, and PV.
Pre- and post selection ensure compatibility:
PL ∧ (PH ∨ PV) = (PL ∧ PH) ∨ (PL ∧ PV)
This means the distributive law holds for these projections in the constrained subspace Hψ,ϕ.
3. Quantum Cheshire Cat in TSVF
Position and Property Separation: In TSVF, the "particle" (e.g., neutron) and its "property" (e.g., polarization) can be described as residing in different locations due to the weak values calculated with pre- and postselection. Weak values reveal:
⟨PL⟩w = 1: The neutron is fully in the left path.
⟨PH⟩w = 0: The horizontal polarization is not in the left path.
⟨PV⟩w = 1: The vertical polarization is fully in the left path. This suggests that the neutron is "in one place" (left path), while its polarization "moves elsewhere" (right path).
Logical Structure and Recovery of Distributivity:
Pre- and post selection restrict the logical framework to projections consistent with |ψ⟩ and |ϕ⟩
Distributive operations apply within this restricted framework:
PL ∧ (PH ∨ PV) = (PL ∧ PH) ∨ (PL ∧ PV)
This aligns with classical intuition, as the neutron in the left path can only have one polarization.
Separation of Particle and Property:
Using preselection (initial state) and postselection (final state), scientists can calculate "weak values" that tell us where the particle and its properties are likely to be.
The experiment restricts what we can say logically to be consistent with the particle's quantum states before and after the experiment. Within this restricted logic, classical rules still hold:
For example, if the neutron is in the left path, it can only have one type of polarization (either horizontal or vertical) in that path. This logical structure ensures that the separation of the particle and its property still fits with our classical understanding of "one thing in one place at a time," but in a uniquely quantum way.
Weak Measurement: The weak measurements are represented by the overlaps between these operators, such as ⟨PL⟩w for the Left Path, ⟨PH⟩w for the Horizontal Polarization, and so on. These weak values can yield non-classical results such as observing the particle in one path, while its polarization property appears to be in another.
BaB̅ar and TSVF
This analysis uses the Two-State Vector Formalism (TSVF) to explore results from the BaBar experiment, which investigates CP symmetry and its violation in the behavior of B-mesons. CP symmetry is a key principle in physics that combines two distinct transformations:
Charge Conjugation (C): This operation swaps particles with their corresponding antiparticles, reversing their charges and other quantum properties. For instance, an electron becomes a positron under C.
Parity (P): This operation flips the spatial coordinates of a system, essentially creating a mirror image. For example, if a particle moves to the right in normal space, under parity it would appear to move to the left.
When CP symmetry holds, the laws of physics should remain identical after applying both transformations simultaneously. In essence, any physical process and its CP-transformed counterpart should occur with the same probability.
B-mesons are particles made up of a bottom quark and its corresponding antiquark. These particles are fascinating because they defy CP symmetry in certain interactions. This phenomenon, known as CP violation, means that processes involving B-mesons and their antiparticles do not behave as perfect mirror images of each other under CP transformations.
Studying CP violation in B-mesons is crucial because it helps us understand why there is more matter than antimatter in the universe—a question that standard physics theories cannot fully explain.
In orthodox quantum mechanics the evolution of ΨB and ΨB̅ are two distinct values, fully consistent with the empirical asymmetry. TSVF is also consistent with the BaBar results but by involving ΦB̅ and ΦB, the system again exhibits distribution, and a mathematical symmetry.
Forward-Evolving State
|
where:
Time-dependent coefficients governed by the Schrödinger equation
Backward-Evolving State
The final state, representing decay products, propagates backward in time:
where:
By combining forward- and backward-evolving states, TSVF calculates transition amplitudes and probabilities for various decay processes, enabling detailed analysis of CP violation and B-meson mixing.
B̅-Mesons in TSVF
For Bˉ\bar{B}-mesons, a similar approach applies:
The mass eigenstates (∣BH⟩|B_H) are the same for both B- and B̅-mesons, and the time evolution follows identical mathematical procedures.
This means, mathematically, B̅ΦΨ is identical, symmetrical to BΦΨ when you involve the post selection. Also B̅Φ=BΨ, and B̅Ψ=BΦ, meaning we are observing the distributive property.
Contextual Classism: an Interperatation
This is not an essay on fundamental physics but rather an illustration of two frameworks, using the same empirical data, the same conformal results, but whose diverse visions offer the possibility of falsification. If a two phase distributivity can be experimentally validated, then we have to throw out MW and CI or radically modify their interperatations. If it is disproven then TSVF is shown to be a mathematical convenience.
Orthodox quantum mechanics, whether CI, Many-Worlds, Pilot-Wave, or Objective Collapse, all use the same Hilbert space and share a non-distributive quantum logic. TSVF is unique in that, while its mathematics are orthodox, its logic exhibits classical features. The specific constraints imposed by TSVF lead to a more classical-like structure, where time symmetry and well-defined initial and final states restrict possible quantum operations and measurements.
Projectors represent logical propositions about the system's state. In TSVF, these projectors correspond to the initial and final states, defining boundary conditions. Time-symmetric evolution ensures these projectors remain well-defined and their logical operations are distributive.The time evolution operator U(t) preserves the distributive structure of quantum logic because it is unitary, ensuring the reversibility of evolution. This guarantees consistency in logical operations performed at different times.
While measurement in quantum mechanics introduces randomness and irreversibility, in TSVF, measurements are seen as specific choices of final states. The time-symmetric nature of the formalism ensures that the measurement process does not violate the distributive law.
TSVF constraint logic within specific Hilbert subspaces, enabling conditional distributivity. Visualizing the Hilbert space as a geometric space, where each quantum state corresponds to a vector, we can represent logical operations as vector manipulations. TSVF partitions the Hilbert space into subsections where distribution holds, because the forward and backward states interact to form a structure where outcomes adhere to distributive properties within those partitions. In regions consistent with both states, quantum logic behaves distributively, excluding non-distributive effects seen in other interpretations or when collapse is not assumed.
This clearly separates TSVF from the two main interpretations of QM, for which an experiment could be formulated.
Despite this difference, both interpretations rely on the same underlying mathematical structure and non-distributive quantum logic.TSVF suggests a different configuration of the Hilbert space, allowing for some function of the distributive property. This interpretation may be an artifact of the math, and not of the real but bluntly, TSVF itself is not a distortion of the maths, in fact they are quite elegant, consistent and clear, it is not shocking that its logic shares this classical characteristic.
This is not to say TSVF is correct, but it doespredict a differentiation on which a test can be based. An experiment could be designed to test these genuinely different conceptions of the Hilbert space. The data already exists, for the brave grad student who dares to quest.
As for the implications for logic, TSVF demonstrates it is possible for classical characteristics can emerge from quantum environments, under certain conditions. It would suggest Putnam was right, but that would be redundant.
Sources:
https://plato.stanford.edu/entries/qt-quantlog/
https://en.wikipedia.org/wiki/Quantum_Cheshire_cat
©J.E.D’Ulisse 2024, NYC
