Finite tensor density, observable and measurement language

For every integer $d\ge 1$ and Hermitian matrix $A\in M_d(\mathbb{C})$, there is an orthonormal basis of ${\mathbb{C}}^d$ consisting of eigenvectors of $A$, and all eigenvalues are real. If in addition $x^{\ast }Ax\ge 0$ for every $x\in {\mathbb{C}}^d$, every eigenvalue is nonnegative and the matrix $A^{1\mathrm{/}2}$ obtained by replacing each eigenvalue by its nonnegative square root is well defined.

Fix an integer $q\ge 1$, let $V={\mathbb{C}}^q$ with its standard ordered orthonormal basis $(\mathrm{\mid }j⟩{)}_{j=1}^q$, and use the inner-product convention that is linear in the first argument. For each integer $N\ge 0$, write $V_N=V^{\otimes N}$ with the displayed factor order; $V_0=\mathbb{C}$. The lexicographically ordered product vectors form the chosen basis of $V_N$, as in Ordered tensor registers, their inner product, and wire permutations.

A finite tensor density on $V_N$ is a density operator in the exact root-library sense of Density operator: a nonnegative endomorphism $\rho$ with $\mathrm{Tr}\rho =1$. In the standard column coordinates this means $\rho ={\rho }^{\ast }$ and $x^{\ast }\rho x\ge 0$ for every $x$.

A bounded observable matrix on $V_N$ is a Hermitian matrix $A=A^{\ast }$, with operator norm $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}={\sup }_{\mathrm{\parallel }x\mathrm{\parallel }=1}\mathrm{\parallel }Ax\mathrm{\parallel }$. Its mathematical expectation functional in $\rho$ is ${\omega }_{\rho }(A)=\mathrm{Tr}(\rho A)$. This is a finite matrix trace pairing; its physical interpretation is supplied separately by a measurement postulate.

A finite POVM with $r\ge 1$ outcomes is a family $M_1,\dots ,M_r$ of nonnegative matrices on $V_N$ with ${\sum }_{a=1}^r{M}_a=I$. Each $M_a$ is an effect, meaning $0\le M_a\le I$; the upper bound follows because $I-M_a={\sum }_{b\mathrm{\ne }a}{M}_b\ge 0$. The numbers $\mathrm{Tr}(\rho M_a)$ are its finite trace pairings. They are nonnegative and normalized when read as outcome probabilities under the adopted physical Born rule. Here $M\ge 0$ means $M=M^{\ast }$ and $x^{\ast }Mx\ge 0$ for every column vector $x$. This definition alone does not adopt the physical probability rule.

For a finite tensor density $\rho$ on $V_N$, list its eigenvalues with multiplicity as ${\lambda }_1,\dots ,{\lambda }_{q^N}$. The finite Hermitian diagonalization in Finite Hermitian diagonalization and positivity of $\rho$ give ${\lambda }_i\ge 0$ and ${\sum }_i{\lambda }_i=1$. Define its dimensionless spectral entropy, measured in nats, by $s_N(\rho ):=-{\sum }_{i:{\lambda }_i>0}{\lambda }_i\log {\lambda }_i,$ with $0\log 0:=0$. This is independent of the ordering of the eigenvalues, including when eigenvalues are repeated. It contains no physical entropy unit; here $\log$ is the natural logarithm of The natural logarithm as the inverse of the exponential function, and a later physical postulate may assign $S_N=k_B{s}_N$.

Fix $q\ge 1$ and real level parameters $e_1,\dots ,e_q$. Use the real exponential of The real exponential function and the number $e$ by a power series and natural logarithm of The natural logarithm as the inverse of the exponential function. For $N\ge 1$, define the one-site diagonal matrix $h\mathrm{\mid }j⟩=e_j\mathrm{\mid }j⟩$ and, for $a=1,\dots ,N$, $h^{(a)}=I^{\otimes (a-1)}\otimes h\otimes I^{\otimes (N-a)},H_N={\sum }_{a=1}^N{h}^{(a)}\mathrm{.}$ Thus the product basis vector $\mathrm{\mid }{j}_1,\dots ,j_N⟩$ has eigenvalue $E(j_1,\dots ,j_N)={\sum }_{a=1}^N{e}_{j_a}$; repeated level values remain distinct basis labels and therefore count as degeneracy.

For finite $\beta \in \mathbb{R}$, put $Z_1={\sum }_{j=1}^q{e}^{-\beta e_j},p_j=\frac{e^{-\beta e_j}}{Z_1},E={\sum }_{j=1}^q{p}_j{e}_j\mathrm{.}$ All terms in $Z_1$ are positive, so $Z_1>0$. Define the one-site canonical density $\gamma =\mathrm{diag}(p_1,\dots ,p_q)$. For $N$ sites put $Z_N={\sum }_{(j_1,\dots ,j_N)\in \{1,\dots ,q{\}}^N}{e}^{-\beta {\sum }_{a=1}^N{e}_{j_a}},{\gamma }_N=\mathrm{diag}{\left(\frac{e^{-\beta {\sum }_{a=1}^N{e}_{j_a}}}{Z_N}\right)}_{(j_1,\dots ,j_N)\in \{1,\dots ,q{\}}^N}\mathrm{.}$ The mathematical definition permits $\beta =0$ and negative $\beta$ as well as positive $\beta$; a physical positive-temperature interpretation is a separate assumption.

For a width ${\delta }_N\ge 0$, define the spectral band label set and its projection by ${\mathrm{\Gamma }}_N=\{(j_1,\dots ,j_N):\mid {\sum }_{a=1}^N{e}_{j_a}-NE\mid \le {\delta }_N\},{\mathrm{\Pi }}_N={\sum }_{\mathbf{j}\in {\mathrm{\Gamma }}_N}\mathrm{\mid }\mathbf{j}⟩⟨\mathbf{j}\mathrm{\mid }\mathrm{.}$ Its rank is ${\mathrm{\Omega }}_N=\mathrm{Tr}{\mathrm{\Pi }}_N=\mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$, which counts all basis vectors in the band, including degeneracy. When ${\mathrm{\Omega }}_N>0$, define the normalized band density ${\sigma }_N={\mathrm{\Pi }}_N\mathrm{/}{\mathrm{\Omega }}_N$. If ${\mathrm{\Omega }}_N=0$, this normalized density is undefined.

Let $V_A,V_B$ be nonzero finite-dimensional ordered tensor registers as in Ordered tensor registers, their inner product, and wire permutations, with the product orthonormal bases and linear-first inner products specified there. On any such register $W$, use finite tensor density to mean a matrix $R$ with $R=R^{\ast }$, $x^{\ast }Rx\ge 0$ for every coordinate vector $x$, and $\mathrm{Tr}R=1$. Here $R^{\ast }$ is the conjugate transpose and $\mathrm{Tr}R$ is the sum of its diagonal entries. A finite POVM on $W$ is a family $(M_a{)}_{a=1}^r$, $r\ge 1$, of Hermitian matrices with $x^{\ast }{M}_{a}x\ge 0$ for every $x$ and ${\sum }_a{M}_a=I_W$; an effect is a Hermitian matrix $M$ with $0\le M\le I_W$, meaning both $M$ and $I_W-M$ are nonnegative in this sense. These conventions extend the identical coordinate definitions of Finite tensor density, observable and measurement language from $W=({\mathbb{C}}^q{)}^{\otimes N}$ to arbitrary nonzero finite ordered registers. Let $\rho$ be a finite tensor density on $V_A\otimes V_B$. Then ${\rho }_A:={\mathrm{Tr}}_B\rho$ is a finite tensor density on $V_A$. For every matrix $A$ on $V_A$, $\mathrm{Tr}({\rho }_{A}A)=\mathrm{Tr}(\rho (A\otimes I_B))\mathrm{.}$ For a finite POVM $(M_a{)}_{a=1}^r$ on $V_A$, the trace pairings of its outcomes in ${\rho }_A$ equal the trace pairings of the lifted effects $M_a\otimes I_B$ in $\rho$.

Fix $q\ge 1$ and the ordered registers $V_N=({\mathbb{C}}^q{)}^{\otimes N}$, with $V_0=\mathbb{C}$. Here $\mathrm{End}(V_N)$ is the space of linear maps $V_N\to V_N$ as in Linear map between vector spaces over the same field. For such a map define its operator norm by $\mathrm{\parallel }A\mathrm{\parallel }:={\sup }_{\mathrm{\parallel }x\mathrm{\parallel }=1}\mathrm{\parallel }Ax\mathrm{\parallel }$, using the ordered tensor inner product of Ordered tensor registers, their inner product, and wire permutations. In its ordered orthonormal product basis define the adjoint by $(A^{\ast }{)}_{ij}=\overline{A_{ji}}$. For the linear-first inner product this gives $⟨Ax,y⟩=⟨x,A^{\ast }y⟩$ for all $x,y\in V_N$. Embed ${\mathcal{A}}_N=\mathrm{End}(V_N)$ into ${\mathcal{A}}_{N+1}$ by ${\iota }_N(A)=A\otimes I_q$. Write a vector in $V_N\otimes {\mathbb{C}}^q$ as $x={\sum }_{j=1}^q{x}_j\otimes \mathrm{\mid }j⟩$. Then $\mathrm{\parallel }(A\otimes I_q)x{\mathrm{\parallel }}^2={\sum }_{j=1}^q\mathrm{\parallel }A{x}_j{\mathrm{\parallel }}^2\le \mathrm{\parallel }A{\mathrm{\parallel }}^2{\sum }_{j=1}^q\mathrm{\parallel }{x}_j{\mathrm{\parallel }}^2=\mathrm{\parallel }A{\mathrm{\parallel }}^2\mathrm{\parallel }x{\mathrm{\parallel }}^2\mathrm{.}$ Testing on unit simple tensors $v\otimes \mathrm{\mid }1⟩$ and taking the supremum over unit $v$ gives the reverse norm inequality, hence $\mathrm{\parallel }A\otimes I_q\mathrm{\parallel }=\mathrm{\parallel }A\mathrm{\parallel }$. The embeddings are isometric unital *-homomorphisms. For $m\ge N$, define ${\iota }_{N,m}(A)=A\otimes I_q^{\otimes (m-N)}$ and identify $A\in {\mathcal{A}}_N$ with its image at every larger register. For a two-step inclusion the maps fit into this commutative diagram:

The curved arrow tensors with two identities, and the upper path tensors with one identity twice; both send $A$ to $A\otimes I_q\otimes I_q$. Let ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}={\bigcup }_{N\ge 0}{\mathcal{A}}_N$ with the induced norm, product and adjoint. Define $\mathcal{A}=\overline{{\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}}$ in this norm. This is the quasilocal matrix algebra for the one-dimensional ordered register chain; it is a unital C*-algebra with unit the common image of $1\in {\mathcal{A}}_0$.

The sequence $({\mathcal{A}}_N,{\iota }_N)$ is the specified inductive system. This definition concerns one fixed local matrix size $q$ and makes no claim about a physical lattice geometry or dynamics.

For the finite $q$-level tensor model of Additive spectral Hamiltonian and normalized ensembles, adopt the following preparation rules in addition to the state and outcome assumptions P1 and P2 of State and composition postulate P1: Hilbert spaces, preparations and composite systems and Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes. The quantities in this item are model assignments; they are not inferred from the cited texts or from data.

1. For declared temperature $T>0$ kelvin and Boltzmann constant $k_B=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, the fixed exact SI Boltzmann constant, use $\beta =(k_{B}T{)}^{-1}>0$ joules${}^{-1}$. In this formulation $k_B$ is the primitive SI conversion constant between temperature and energy; it is not derived from this finite-level model or fitted from its data. The real scalar levels $e_j$ in Additive spectral Hamiltonian and normalized ensembles are primitive model energy parameters in joules. The total label energy $E(j_1,\dots ,j_N)={\sum }_{a=1}^N{e}_{j_a}$ and the one-site canonical mean $E={\sum }_{j=1}^q{p}_j{e}_j$ defined there are derived scalar energies in joules. Adopt the canonical finite density ${\gamma }_N$ of that definition.
2. For declared band width ${\delta }_N\ge 0$ in joules, if its label set ${\mathrm{\Gamma }}_N$ is nonempty, the uniform band preparation is the density ${\sigma }_N={\mathrm{\Pi }}_N\mathrm{/}\mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$ from that definition. If the set is empty, this normalized preparation is undefined and no band state is assigned.
3. For a stated finite POVM $(M_a)$ on the register, the outcome law is the adopted Born law $p_a=\mathrm{Tr}(\rho M_a)$ of P2. A prediction is the resulting model probability, not a reported detector frequency.

The positive-temperature interpretation uses SI units as fixed in Dimensions, SI unit conventions and quantity types of the framework. No preparation mechanism, thermalization process, sampling law, apparatus error, or observed data is asserted. A statement that a physical preparation realizes one of these densities requires a separate empirical identification and uncertainty analysis.

Use the finite additive spectral data of Additive spectral Hamiltonian and normalized ensembles. For every $N\ge 1$, $Z_N=Z_1^N$. For each product label $\mathbf{j}=(j_1,\dots ,j_N)$, the diagonal entry of ${\gamma }_N$ is ${\prod }_{a=1}^N{p}_{j_a}$ and its off-diagonal entries are zero. For each integer $k$ with $1\le k\le N$, write $x,x^{\mathrm{\prime }}\in \{1,\dots ,q{\}}^k$ and $y\in \{1,\dots ,q{\}}^{N-k}$, and define the first-$k$ reduced matrix $R_{N,k}$ by $(R_{N,k}{)}_{x,x^{\mathrm{\prime }}}:={\sum }_{y\in \{1,\dots ,q{\}}^{N-k}}({\gamma }_N{)}_{(x,y),(x^{\mathrm{\prime }},y)}\mathrm{.}$ When $k=N$, the tail-label set consists of the single empty tuple. Then $R_{N,k}={\gamma }_k$, where ${\gamma }_k$ is the diagonal matrix on $V_k=({\mathbb{C}}^q{)}^{\otimes k}$ with entries $({\gamma }_k{)}_{x,x}={\prod }_{a=1}^k{p}_{x_a}$, and the canonical state has expected additive energy $NE$.

Let $q\ge 1$ and, for every $N\ge 1$, let ${\mu }_N:\{1,\dots ,q{\}}^N\to [0,1]$ satisfy ${\sum }_{x\in \{1,\dots ,q{\}}^N}{\mu }_N(x)=1.$ Assume consistency under deleting the last coordinate in the explicit sense that, for every $N\ge 1$ and $x\in \{1,\dots ,q{\}}^N$, ${\mu }_N(x)={\sum }_{j=1}^q{\mu }_{N+1}(x,j)\mathrm{.}$ For a local matrix $A\in {\mathcal{A}}_N$, $N\ge 1$, define ${\omega }_N(A)={\sum }_{x\in \{1,\dots ,q{\}}^N}{\mu }_N(x)A_{xx}\mathrm{.}$ For each $N\ge 1$, if $f:\{1,\dots ,q{\}}^N\to \mathbb{C}$ and $D_f$ is the diagonal matrix with entries $(D_f{)}_{xx}=f(x)$, then ${\omega }_N(D_f)={\sum }_x{\mu }_N(x)f(x)$. This rule extends uniquely from the local algebra ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ to a state on its norm completion $\mathcal{A}$ of Quasilocal completion of finite matrix registers. The theorem constructs a classical diagonal state; it does not say that every state on $\mathcal{A}$ is diagonal or classical.

Fix $q\ge 1$, real one-site levels $e_j$, finite $\beta \in \mathbb{R}$, and the ensembles of Additive spectral Hamiltonian and normalized ensembles. Suppose for every $N$ the band width obeys ${\delta }_N\ge q{\sum }_{j=1}^q\mathrm{\mid }{e}_j\mathrm{\mid },{\delta }_N\mathrm{/}N⟶0.$ Then every band is nonempty. For each fixed integer $k\ge 1$ and $N\ge k$, define ${\sigma }_{N,k}$ by its product-basis entries, for $x,x^{\mathrm{\prime }}\in \{1,\dots ,q{\}}^k$, $({\sigma }_{N,k}{)}_{x,x^{\mathrm{\prime }}}={\sum }_{y\in \{1,\dots ,q{\}}^{N-k}}({\sigma }_N{)}_{(x,y),(x^{\mathrm{\prime }},y)}$, with a single empty tail label when $N=k$. Define ${\gamma }_k$ to be the diagonal matrix on $V_k=({\mathbb{C}}^q{)}^{\otimes k}$ with entries $({\gamma }_k{)}_{x,x}={\prod }_{a=1}^k{p}_{x_a}$ and zero off the diagonal. Then $\mathrm{\parallel }{\sigma }_{N,k}-{\gamma }_k{\mathrm{\parallel }}_1\to 0,\frac{1}{N}{s}_N({\sigma }_N)\to -{\sum }_{j=1}^q{p}_j\log p_j\mathrm{.}$ Here $\mathrm{\parallel }D{\mathrm{\parallel }}_1$ for a Hermitian finite matrix is the sum of absolute values of its eigenvalues, and $s_N$ is dimensionless spectral entropy. This is an independent finite-alphabet result; it asserts neither global trace-norm convergence nor an interacting or continuous-model equivalence.

Fix a finite integer $r\ge 1$, positive coefficients $a_j$, a real polynomial $V$ on ${\mathbb{R}}^r$, and constants $c>0,C_0\ge 0$ such that $V(q)\ge c\mathrm{\mid }q{\mathrm{\mid }}^2-C_0$. Use the QFT supplier's form realization of Finite-lattice polynomial Hamiltonian data and Self-adjoint unbounded polynomial interaction at fixed regulator on $L^2({\mathbb{R}}^r,dq)$, and denote its self-adjoint operator by $h$. This item's regulated polynomial model is the tuple of these finite data, the selected form realization, and an energy scale $E_{\ast }>0$ joules, with $H=E_{\ast }h$. The coordinate $q$ and coefficients are dimensionless rescaled model parameters unless another dimensional map is explicitly supplied. The class includes $V(q)=\mathrm{\mid }q{\mathrm{\mid }}^2$ as well as potentials with higher-degree terms or terms coupling coordinates; neither higher-degree terms nor coupling is required. No limit in $r$ is included.

For each $b>0$, Finite-regulator Gibbs trace and bounded expectations proves $\mathrm{Tr}(e^{-bh})<\mathrm{\infty }$ and positivity. Thus for physical $\beta >0$ in joules${}^{-1}$, $b=\beta E_{\ast }>0$ and ${\rho }_{\beta }=\frac{e^{-\beta H}}{Z_{\beta }},Z_{\beta }=\mathrm{Tr}(e^{-\beta H})\in (0,\mathrm{\infty })$ is a trace-one positive trace-class density. Its interpretation as a normal quantum measurement state uses the current batch-36 draft carrier Normal-state quantum measurement interpretation, which adopts a trace-class density on a specified separable Hilbert space and the PVM Born law ${\mathbb{P}}_{\rho }\{A\in C\}=\mathrm{Tr}(\rho E_A(C))$. This interface matches the mathematical state constructed here. The mathematical probability and bounded trace-pairing inputs now have exact canonical suppliers through the trace-state calculus. The normal-state interpretation remains an adopted physical premise; source adoption is separate from native readiness and empirical validation. The QFT mathematical trace theorem remains the mathematical supplier.

This selected finite-regulator realization does not assert essential self-adjointness of the bare compact-support differential expression, a unique realization among all self-adjoint extensions, regulator removal, or an interacting continuum theory.

Assume the physical preparation postulate Adopted finite canonical and band preparation rules, and the exact independent-shell hypotheses and notation of Fixed-subsystem equivalence for independent finite energy shells. In the physical interpretation fixed by the preparation postulate, the primitive scalar levels $e_j$ and total label energies are in joules, the band widths ${\delta }_N$ are in joules, $T$ is in kelvin, and $\beta =(k_{B}T{)}^{-1}>0$ is in joules${}^{-1}$. For each fixed $k$ and each fixed $r$-outcome POVM $(M_a{)}_{a=1}^r$ on the $k$-site register, define the conditional model probabilities $p_{N,a}^{\mathrm{s}\mathrm{h}}:=\mathrm{Tr}({\sigma }_{N,k}{M}_a)$ and $p_{N,a}^{\mathrm{c}\mathrm{a}\mathrm{n}}:=\mathrm{Tr}({\gamma }_k{M}_a)$ for $N\ge k$. They satisfy ${\sum }_{a=1}^r\mathrm{\mid }{p}_{N,a}^{\mathrm{s}\mathrm{h}}-p_{N,a}^{\mathrm{c}\mathrm{a}\mathrm{n}}\mathrm{\mid }⟶0.$ This is a conditional prediction for the adopted states and specified measurement; it supplies no empirical frequencies or sampling inference.

Assume the Axiom of Choice and the finite regulated model of Finite coercive regulated polynomial model, the current draft thermal preparation interface Canonical quantum thermal preparation, the current draft normal-state interface Normal-state quantum measurement interpretation, and the current draft mathematical theorem Self-adjoint trace-Gibbs normalization and form-energy optimization. Its inspected exact interface is: for a nonzero separable complex Hilbert space and a densely defined self-adjoint $A$ bounded below, if $e^{-\beta A}$ is trace class for every $\beta$ in an open interval $I\subset (0,\mathrm{\infty })$, then $Z_{\beta }\in (0,\mathrm{\infty })$ and ${\rho }_{\beta }=e^{-\beta A}\mathrm{/}{Z}_{\beta }$ is a faithful normal density; every normal density of finite absolute spectral form energy satisfies ${\mathcal{E}}_A(\rho )-TS(\rho )\ge F_{\beta }$, with equality exactly at $\rho ={\rho }_{\beta }$. The theorem assumes AC. Under these conditions, for each $\beta >0$ J${}^{-1}$ the model state is ${\rho }_{\beta }=e^{-\beta E_{\ast }h}\mathrm{/}{Z}_{\beta }$, with $0<Z_{\beta }<\mathrm{\infty }$, and the stated free-energy inequality holds for every normal density $\rho$ whose spectral law ${\mu }_{\rho }(C)=\mathrm{Tr}(\rho E_H(C))$ has finite first absolute moment $\int \mathrm{\mid }\lambda \mathrm{\mid }d{\mu }_{\rho }(\lambda )<\mathrm{\infty }$. The energy ${\mathcal{E}}_H(\rho )=\int \lambda d{\mu }_{\rho }(\lambda )$ and free energy are in joules, entropy $S(\rho )$ is in joules per kelvin, and $T>0$ is in kelvin. The conditional result is for the selected fixed finite-mode form realization. It asserts no cutoff removal or empirical realization.