Holographic Interpretation of Shannon Entropy of Coherence of Quantum Pure States

TL;DR

This paper proposes a holographic interpretation of the Shannon entropy of quantum coherence in pure states within conformal field theory, linking it to geometric quantities in the AdS/CFT correspondence.

Contribution

It introduces a conjectured differential geometric formula for the Shannon entropy of coherence, connecting it to holographic complexity and action in the bulk space.

Findings

Conjectured a geometric formula relating Shannon entropy to holographic complexity and action.

Provided a holographic interpretation of quantum coherence entropy in AdS/CFT context.

Suggested a new way to define bulk qubit model actions at thermal and momentum equilibrium.

Abstract

For a quantum pure state in conformal field theory, we generate the Shannon entropy of its coherence, that is, the von Neumann entropy obtained by introducing quantum measurement errors. We give a holographic interpretation of this Shannon entropy, based on Swingle's interpretation of anti-de Sitter space/conformal field theory (AdS/CFT) correspondence in the context of AdS$_3$/CFT$_2$. As a result of this interpretation, we conjecture a differential geometrical formula for the Shannon entropy of the coherence of a quantum pure or purified state in CFT$_2$ at thermal and momentum equilibrium as the sum of the holographic complexity and the abbreviated action, divided by $\pi\hbar$, in the bulk domain enclosed by the Ryu--Takayanagi curve. This result offers a definition of the action of a bulk model of qubits dual to the boundary CFT$_2$ at this equilibrium.

Equations

$$H_{\mathrm{bdy}}^{\mathrm{bit}}=-b^{-1}\frac{I_{\mathrm{bulk}}}{\hslash }$$

$$H_{\mathrm{bdy}}^{\mathrm{bit}}=A_{\mathrm{TN}},$$

$$\frac{1}{N}{\log }_2{w}_N$$

$$\to 1(N\to \infty )$$

$$H_{\mathrm{bdy}}^{\mathrm{bit}}$$

$$=A_{\mathrm{TN}}\frac{1}{N}{\log }_2{w}_N$$

$$\to A_{\mathrm{TN}}(N\to \infty )$$

$$I_{\mathrm{bulk}}=-\hslash A_{\mathrm{TN}}$$

$$\mathrm{sgn}(\mathcal{T})$$

$$=-\mathrm{sgn}(\Lambda ).$$

Full text

In the original paperKonishi1 , we gave a holographic interpretation of the information lost by the quantum measurement (i.e., classicalization a la von NeumannNeumann ) of a strongly coupled quantum many-body system at a quantum critical point.

This result can be formulated by the following two informatical axioms.

1. I)

Time evolution in the holographic boundary obeys the Schr${\ddot{{\rm o}}}$dinger equation of the boundary quantum pure state. No loss or acquisition of information accompanies this mechanical process. 2. II)

Once a quantum measurement (i.e., classicalization) is performed in a spatial subregion $A$ in the boundary, information with amount $H_{A}^{\rm bit}$ in bits, given by (16) in the original paper, is lost.

In this addendum, we have three purposes: i) to formulate a statement of the holographic principle; ii) to explicitly write out some calculations in the original paper; and iii) to point out consequences of negativity of the cosmological constant from the formulation in i). This addendum does not report any error in the original paper.

i) First, we formulate a statement of the holographic principle in the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence: the degrees of freedom in the bulk space are given by the information stored in a boundary quantum many-body system. We denote by $H_{\rm bdy}^{\rm bit}$ the binary Shannon entropy of the coherence of the boundary quantum pure state, which is the binary von Neumann entropy of the quantum mixed state obtained by classicalization of this quantum pure state. We now call this the measurement entropy. We also denote by $I_{\rm bulk}$ the negatively valued action of the classically stochastic bulk system, with negative degrees of freedom, obtained by classicalization111Here, classical probabilities mean probabilities with no interference and generate negative degrees of freedom.. Here, $-I_{\rm bulk}$ is lost by the transverse relaxation processSlichter (i.e., classicalization) of the boundary quantum pure state with a time constant $T_{2}$, that is, the lifetime of purity of the boundary quantum state (of course, $T_{2}$ is positively valued); $T_{2}$ can be determined by the consistency of the correspondence222In the AdS3/CFT2 correspondence, $T_{2}$ is determined by $1/\sqrt{-\Lambda}$ in the Planck units for a negative cosmological constant $\Lambda$.. Then, the statement is formulated as

[TABLE]

for the bit factor $b=\ln 2$. Here, the left-hand side of (A1) is the information in bits lost by classicalization in the boundary quantum many-body system. The right-hand side of (A1) defines the degrees of freedom lost by classicalization as those of spins $s$ in the bulk space; note that the denominator is the Dirac constant $\hbar=s_{\uparrow}-s_{\downarrow}$, and spin is independent of the notion of mass.

ii) For the ground state $|\Omega\rangle_{\rm s.c.}$ of the strongly coupled quantum critical many-body system on the one-dimensional boundary, the result of the original paperKonishi1 is

[TABLE]

where $A_{\rm TN}$ denotes the area of the bulk space discretized by the sites of spins. Here, this is based on the multi-scale entanglement renormalization ansatz (MERA) of the boundary quantum pure stateSwingle .

In the following, we explicitly write out the calculations by which we derive (A2). By using the Stirling formula $\ln N!\simeq N\ln N-N$ for the number $N(\gg 1)$ of the identical systems in the classical mixed ensemble of product spin eigenstates after classicalization, we obtain that, for the multiplicative change of the number of microstates $w_{N}$ per site in the MERA,

[TABLE]

holds, where we introduce the number of combinations for the classicalized Bell state C_{\rm Bell}(N)=\left(\begin{array}[]{c}N\\ \frac{N}{2}\end{array}\right) and that for the bipartite product spin eigenstate (i.e., the disentangled Bell state by a disentangler in the MERA) C_{\rm product}(N)=\left(\begin{array}[]{c}N\\ N\end{array}\right)=1. Then, for the number of microstates $W_{\rm bdy}$, which defines the measurement entropy, we obtain that

[TABLE]

holds. Here, (A3) holds because we consider a strongly coupled boundary quantum many-body system.

iii) Finally, combining (A1) with (A2) and dropping the translation factor $b^{-1}$ from the right-hand side of (A1), we obtain that

[TABLE]

holds in the quantum pure state $|\Omega\rangle_{\rm s.c.}$.

Formula (A8) shows that the action $I_{\rm bulk}$ of the classically stochastic bulk system is equivalent to the worldvolume action of a relativistic membrane with time duration $T_{2}$ and tension (i.e., mass per unit area) ${\cal T}$Nastase . Now, we invoke the action $I_{\rm bulk}$ as the action of the bulk space. Then, the tension ${\cal T}$ is generated by the cosmological term, and the possible contributions from spins higher than 2 are dropped because we consider the ground state $|\Omega\rangle_{\rm s.c.}$ of the strongly coupled quantum many-body system on the boundary. For this tension ${\cal T}$ of the bulk space and its cosmological constant $\Lambda$, we obtain the relations

[TABLE]

For this consequence $\Lambda<0$, the crucial fact is that any process (here, classicalization) from a quantum pure state to a quantum mixed state always loses information, here meaning that the measurement entropy is always positively valued.