Gleason theorem: Every probability measure is generated by a quantum state

Hilbert lattices

We consider a separable Hilbert space $H$ and let $L$ be the corresponding Hilbert lattice. We have already established that $L$  is the  orthomodular  lattice of the set ${\bf C}(H)$ of all closed linear subspaces of $H$ which  is isomorphic to the set ${\bf P}(H)$ of all projectors on $H$. We will for simplicity write $L={\bf C}(H)$.

Quantum events are experimental propositions and they are elements of ${\bf C}(H)$ or equivalently ${\bf P}(H)$. A subalgebra of $L$ is a subset of ${\bf C}(H)$ which is closed under the logical operations $\wedge=\cap$, $\vee=\oplus$ and $\neg=\perp$ and which contains the identity elements ${\bf 1}$ and  ${\bf 0}$. A block $B$ of $L$ is a maximal Boolean subalgebra of $L$ ("maximal" means that the subalgebra contains a maximum numbers of atoms which are those elements above ${\bf 0}$ directly and "Boolean" means that the distributive law holds).

 The elements of ${\bf C}(H)$ and ${\bf P}(H)$ will be denoted by $M_p$ and $P$ respectively whereas the corresponding propositions will be denoted $p$.

Kolmogorov probability theory

 The Kolmogorov axioms of probability theory can only be applied to a classical Boolean algebra $B$ of events.  A probability measure ${\cal P}$ is a function from the set of classical events $B$ into the unit interaval, i.e. ${\cal P}:B\longrightarrow [0,1]$ which satisfies i) ${\cal P}(p)\geq 0$, $\forall p\in B$, ii) ${\cal P}(B)=1$, iii) ${\cal P}(p\vee q)={\cal P}(p)\vee {\cal P}(q)$ if $p$ and $q$ are mutually disjoint events.

An important example is a maximal Boolean sublagbera $B$ of the Hilbert lattice $L$ with $n$ atoms corresponding to $n$ comeasurable obserbales. Any positive function ${\cal P}$  such that 1) ${\cal P}(p_i)\geq 0$, $\forall i$, and 2) $\sum_{i=1}^n{\cal P}(p_i)=1$ will satisfy  Kolmogorov axioms. Two important examples are:

  1. The equidistribution ${\cal P}(p_i)=1/n$ (which represents maximum uncertainty or maximum entropy).

  3. The two-valued dispersion free probability measure ${\cal P}(p_i)=\{0,1\}$ (which represents maximum knowledge and $0$ entropy).

An important result is the fact that on Boolean algebras $B$ with $n$ atoms $p_1$,...,$p_n$ every probability measure is actually a linear combination (a convex sum) of the $n$ two-valued measures ${\cal P}_i(p_j)=\delta_{ij}$, $i=1,...,n$. Both Gleason theorem \cite{gleason} and the Kochen-Specker theorem  state in essense that on a Hilbert lattice, for Hilbert spaces of dimension greater than two, no two-valued probability measure and thus no representation of probability measures as a convex sum exist.

 A much more refined notion of classicality is the "embeddibility"  of a quantum logic in a classical propositional Boolean structure (more refined than the notion of a "Boolean" lattice structure).  Embeddibility means the existence of a two-valued probability measure on the set of quantum events. Indeed, Kochen and Specker showed  that the set of quantum events ${\bf C}(H)$ can be embedded into a Boolean algebra $B$ if and only if a two-valued probability measure ${\cal P}:{\bf C}(H)\longrightarrow\{0,1\}$ exists such that ${\cal P}(p)\neq {\cal P}(q)$ if $p\neq q$. We say then that the set of probability measures is "separating" which is a better characterization  of classicality than the "distributivity" of a Boolean structure.

Gleason theorem

Next we will substitute the Hilbert lattice of quantum events given by the closed subspaces of the Hilbert space  for the Boolean algebra of classical events.

The  Kolmogorov axioms of probability theory can be applied to the Hilbert lattice $L$ consistently by applying it to each and every Boolean subalgebra $B$ of $L$ as follows.  A probability measure ${\cal P}$ is a mapping from the set of quantum events or experimental propositions ${\bf C}(H)$ into the unit interval, i.e. ${\cal P}: {\bf C}(H)\longrightarrow [0,1]$ which satisfies, in each Boolean subalgebra or block $B$ corresponding to comeasurable propositions or events, the following quasi-classical conditions:

  1. The probability of each proposition $p$ associated with the projection operator $P$ on the subspace $M_p$ is positive, i.e.
         {\cal P}(M_p)\geq 0~,~\forall M_p\in {\bf C}(H).

  3. The sum of all probabilities is one (the Hilbert space corresponds to the  tautology ${\bf 1}$), viz
         {\cal P}(H)=1.

  5. The probability measure is countably additive, i.e. for mutually disjoint events $p$ and $q$ represented by orthogonal projection operators $P$ and $Q$ respectively we have
          M_{p\vee q}= M_p\oplus M_q.
         {\cal P}(M_{p\vee q})= {\cal P}(M_p)+ {\cal P}(M_q).
    The propositions $p$ and $q$  belong to the same Boolean subalgebra and therefore they correspond to comeasurable events associated with compatible observables which are represented by commuting self-adjoint operators (projectors) on the Hilbert space.

  7. But a projector can belong to different blocks or Boolean algebras representing the different contexts of the measurement of the corresponding proposition (equivalently the blocks represent the different perspectives of the various observers).  Thus, at the intersection of two Boolean subalgebras $B_1$ and $B_2$ pasted together we must also impose the non-contextuality requirement
      {\cal P}_{B_1}(M_p)= {\cal P}_{B_2}(M_p)~,~M_p\in B_1\cap B_2.

A probability measure on the subspaces of the Hilbert space satisfying the above requirements can be constructed as follows. Let $|\phi\rangle$ be an arbitrary normalized vector in the Hilbert space $H$ and let $P_{\phi}$ be the corresponding one-dimensional projector, i.e. $P_{\phi}=|\phi\rangle\langle\phi|$. The probability of the projector $P=|\psi\rangle\langle \psi|$ is set to be given by the Born's rule
  {\cal P}_{\phi}(M_p)=tr PP_{\phi}= \langle \phi|P|\phi\rangle=|\langle\psi|\phi\rangle|^2.\label{br}
This is the probability that the projector $P$ have the value $1$ in the state $|\phi\rangle$ which is indeed the statistical algorithm of quantum mechanics or Born's rule.

If ${\cal P}_{\phi_i}$ are different probability measures on the Hilbert lattice $L={\bf C}(H)$ associated with the normalized vectors $|\phi_i\rangle$ then the convex sum (mixture) ${\cal P}=\sum_{i=1}^nt_i{\cal P}_i$ where $0\leq t_i\leq 1$ and $\sum_{i=1}^nt_i=1$ is also a probability measure on $L$. We can immediately compute
  {\cal P}(M_p)=tr P\rho~,~\rho=\sum_{i=1}^nt_iP_{\phi_i}.
The convex sum $\rho$ of the one-dimensional projectors $P_{\phi_i}$ (pure states) is called the density operator (which is generally a mixed state).

Thus, every density operator $\rho$ yields a countably additive probability measure on the subspaces of the Hilbert space. The converse is precisely Gleason theorem which is a highly non-trivial result in quantum mechanics and its foundations.

Gleason's theorem: 

Every countably additive probability measure on the subspaces of a separable Hilbert space of dimension greater than two is i) characterized by a unique non-negative self-adjoint operator $\rho$ satisfying $tr\rho=1$ (trace class operator) and ii) is necessarily of the form
  {\cal P}_{\rho}(M_p)=tr P\rho.
The density operator represents of course the quantum state of the physical system. Gleason's theorem shows therefore that all probability measures on ${\bf C}(H)$ are generated by quantum mechanical states.

Gleason's theorem is a substitute to  Born's rule. Indeed, if the physical system is prepared in the pure state $|\phi\rangle$ then the density operator which is generally of the form $\rho=\sum_iP_i|\phi_i\rangle\langle\phi_i|$ reduces to the pure state $\rho=|\phi\rangle\langle\phi|=P_{\phi}$ and we end up with the Born's rule (\ref{br}). Thus, we can derive the statistical algorithm of quantum mechanics or Born's rule in a straightforward way from Gleason theorem.

One immediate corollary of Gleason's theorem is effectively the Kochen-Specker theorem (see next sections for more detail). Indeed, in Gleason theorem we are attempting to assign probabilities to closed subspaces of a Hilbert space $H$ such that the probabilities assigned to orthogonal projection operators are additive. This assignment is found to be provided by a density operator $\rho$. In the Kochen-Specker theorem we will deal instead with the special case in which we attempt to assign only the values $0$ and $1$ (two-valued probability measures) to the closed subspaces of $H$ in a consistent way (non-contextuality). This assignment, as we will show, is actually impossible. In other words, the set of quantum events ${\bf C}(H)$ does not admit a global two-valued probability measure.

This fundamental result can be derived from Gleason' theorem as follows.  Every probability measure on ${\bf C}(H)$ must be necessarily a continuous  mapping into the interval $[0,1]$. Indeed, for any density operator $\rho$ the mapping $|\psi\rangle\longrightarrow   {\cal P}(M_p)=\langle \psi|\rho|\psi\rangle$ (with $P=|\psi\rangle\langle\psi|$) is a continuous function on the unit sphere in $H$. Obviously, there is no  continuous function defined on the unit sphere (because it is a connected space) which takes only the two values $0$ and $1$. Hence, the set of closed subspaces of the Hilbert space  does not admit a two-valued probability measure.

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