SVD Entanglement Entropy

TL;DR

The paper introduces SVD entanglement entropy, a real, nonnegative generalization of entanglement entropy that depends on a pair of pre- and post-selected states through the reduced transition matrix $\tau_A^{1|2}$. It establishes its interpretation in terms of distillable Bell pairs via singular values, develops Rényi extensions, and analyzes foundational properties, including invariances, additivity, and weak concavity, while highlighting the failure of subadditivity and strong subadditivity in general. The authors compute and illustrate the quantity across two-qubit systems, integrable 2d CFTs, holographic CFTs, Chern-Simons theory, and quantum spin chains, revealing how SVD entropy can reflect phase structure and dynamical evolution, and comparing it to pseudo entropy. The work demonstrates both the utility and the limitations of SVD entropy as a tool for understanding post-selection in quantum systems and their field-theoretic and holographic contexts, and it outlines future directions for inequalities, Rényi extensions, and gravity connections. Overall, SVD entanglement entropy provides a versatile, real-valued measure for post-selected quantum information in many-body and field-theoretic settings with potential applications as a diagnostic for quantum phases and dynamics.

Abstract

In this paper, we introduce a new quantity called SVD entanglement entropy. This is a generalization of entanglement entropy in that it depends on two different states, as in pre- and post-selection processes. This SVD entanglement entropy takes non-negative real values and is bounded by the logarithm of the Hilbert space dimensions. The SVD entanglement entropy can be interpreted as the average number of Bell pairs distillable from intermediates states. We observe that the SVD entanglement entropy gets enhanced when the two states are in the different quantum phases in an explicit example of the transverse-field Ising model. Moreover, we calculate the Rényi SVD entropy in various field theories and examine holographic calculations using the AdS/CFT correspondence.

Figures (23)

• Figure 1: Unlike what happens for ordinary entanglement entropy or pseudo entropy, the SVD entanglement entropy values $S(\rho^{1|2}_{A})$ and $S(\rho^{1|2}_{B})$ are not equal in general. These scatter plots show the values of $(S(\rho^{1|2}_{A}),S(\rho^{1|2}_{B}))$ for Haar-random samples of $N=10000$ pairs of unit vectors $|\psi_{1}\rangle,|\psi_{2}\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ for two different dimensions $d_{A}=d_{B}$. If the two SVD entanglement entropies were equal, they would all lie along the diagonal line. Nevertheless, the distribution seems symmetric about the diagonal line suggesting that the subregion SVD entanglement entropies might be equal on average. This property also seems to hold for higher values of $d_{A}=d_{B}$. As the dimensions increase, the average SVD entanglement entropy increases (see also Section \ref{['sec:EVSVDEE']}).
• Figure 2: Given the matrix $A=\mathrm{diag}(x,1-x)$, with $x\in\mathbb{R}$, the different entropy formulas, $S_{\mathrm{SVD}}(A)$, $S_{\mathrm{ABB}}(A)$, $S_{\mathrm{JSK}}(A)$, and $S_{\mathrm{FP}}(A)$ are compared as functions of $x$. Note that they all agree for $x\in[0,1]$ except for $S_{\mathrm{ABB}}(A)$, the latter of which is therefore not an extension of the standard entropy.
• Figure 3: Two scatter plots of 10000 Haar random states $|\psi_{1}\rangle,|\psi_{2}\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}$, where the dimensions are given by $d_{A}=d_{B}=2$ and $d_{E}=4$, are used to illustrate the failure of subadditivity of SVD entropy and the plausibility of the Araki-Lieb (AL) inequality, respectively. However, the AL inequality does not hold for all states.
• Figure 4: The figure on the left illustrates a histogram of values for the values of SVD entropy $S(\rho^{1|2}_{A})$ (in orange). The curve is well-approximated by a Gaussian distribution (the curve is drawn in orange on the zoomed-in figure). The figure on the right compares this to the standard entanglement entropy $S(\rho^{1}_{A})$ (in blue). In particular, the standard entanglement entropy histogram is shifted slightly to the left of the histogram for the SVD entanglement entropy, indicating a lower average value of entropy as compared to the SVD entanglement entropy.
• Figure 5: In this plot, $M=5$, so that the horizontal axis variable $t$ takes values in $\{0,1,\dots,10\}$. A total of 100 states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ were chosen randomly according to the Haar measure on $\mathcal{H}=\mathbb{C}^{2^{10}}=\mathbb{C}^{1024}$. Three types of averages are computed. These are EE1A, SVDEA, and SVDEB denote the usual entanglement entropy for state $|\psi_{1}\rangle$ on region A, the SVD entropy of the transition matrix on region A, and the SVD entropy of the transition matrix on region B, respectively.

Theorems & Definitions (1)

• proof