Information in Black Hole Radiation

The question we look into is that if Alice throws a k-qubit quantum information M into a black hole H, can Bob tell what the information was by looking at the Hawking radiation, given that Bob holds a quantum memory that is perfectly correlated with the previously emitted Black Hole radiation E. We assume the black hole dynamics to be unitary and rapidly mixing. Also, consider a reference system N which is initially maximally entangled with M. By retrieving information of M from the black hole, we mean extracting a subsystem of size $|M|$ which is maximally entangled with N, so that Bob can do anything with it that he would have been able to do with M before Alice threw it into the black hole.

Just after Alice throws the information into the black hole, the system B $(=HM)$ is transformed by a unitary transformation $V^B$ chosen uniformly with respect to a Haar measure. After some time, the black hole is in state B’ and has radiated system R which Bob observes.

Consider first the entanglement between H and E [2]. When the black hole is just formed, and the radiation E is maximally entangled with the black hole state H. As the evaporation proceeds, $\log |H|$ declines to half it’s value and soon after . The black hole state H is then maximally entangled with radiation E.

We now give a proof of the above [3]. Consider a bipartite pure state $|\psi ⟩$ composed of subsystems A and B. The state is chosen uniformly with respect to a Haar measure. Consider the flip operator $𝔽:𝔽(|{\phi }_i⟩\otimes |{\phi }_j⟩)=|{\phi }_j⟩\otimes |{\phi }_i⟩$.

We use this by creating a copy $A^{\prime }\otimes B^{\prime }$ of the system $A\otimes B$.

We now compute

where U are unitary matrices. For any unitary matrix V, we have

Therefore Z commutes with every unitary matrix $U\in U(n)$ representation. By Schur’s lemma, Z can be decomposed as a sum of projection operators on the invariant subspaces of ${ℂ}^n\otimes {ℂ}^n$.

Noting that $𝔽$ commutes with all $U\otimes U$

and hence, using ${\rho }_0=|{\psi }_0⟩⟨{\psi }_0|\otimes |{\psi }_0⟩⟨{\psi }_0|$,

Therefore,

For $A>>B$, we have ${𝔼}_{\psi }\text{tr}({\rho }_A^2)\approx 1/|A|$. Let $a_i$ be the eigenvalues of ${\rho }_A$, then

Therefore, A is almost maximally mixed. Using Levy’s Lemma we can say that for almost all pure states $|\psi ⟩$, ${\rho }_A$ is close to maximally mixed.

Consider the case when so that H is maximally entangled with E. Just after Alice throws in her information, the black hole system B is maximally entangled with NE. As the information leaks from the black hole through the radiation R, the correlation between N and remaining black hole state B’ weakens, and the information in M goes to Bob. Suppose $s$ qubits have radiated in R, and $n-s$ qubits remain in B’.

Let ${\mathrm{\Psi }}^{\text{BNE}}$ be the pure density operator of BNE, and ${\rho }^{\text{BN}}={\text{Tr}}_E{\mathrm{\Psi }}^{\text{BNE}}$ be the corresponding marginal density operator on BN. The marginal density operator on NB’ is given by

We show that this marginal density operator is very close to a maximally mixed and B’ and N are almost separable. To this end we prove the following inequality

where ${\sigma }^N(V^B)={\text{Tr}}_{B^{\prime }}\left[{\sigma }^{\text{NB’}}(V^B)\right]$ is the marginal density operator on N, and ${\sigma }_{\text{max}}^{B^{\prime }}=I^{B^{\prime }}/|B^{\prime }|$ is the maximally mixed density operator on B’.

As was done in the previous section, and using the cyclic property of the trace

where $\overline{B}$ was an auxiliary copy of the system B. $\alpha$ and $\beta$ are found using the same method as in previous section

Similarly,

Therefore,

From Cauchy-Schwarz inequality,

B is maximally entangled with NE, and therefore BN is maximally mixed on a system of dimension $|B|/|N|$, implying $\text{Tr}[{({\rho }^{NB})}^2]=|N|/|B|$. Thus,

Hence, we see that as soon as Bob receives k qubits from the black hole radiation, he gets hold of almost all information that Alice had.

Even though Bob now hold’s Alice’s information in a subsystem M’ of RE, it may be very diffusely distributed within RE. Bob can then perform a computation that maps M’ to a compact localized system $\widehat{M}$ so that ${\rho }^{\widehat{M}N}$ is a maximally entangled state $|{\mathrm{\Phi }}^{\widehat{M}N}⟩$. The fidelity, which is a measure of correlation of ${\rho }^{\widehat{M}N}$ with $|{\mathrm{\Phi }}^{\widehat{M}N}⟩$, is bounded by

Thus, the state obtained after computation is very close to maximally entangled. To prove the above inequality, we start with the definition of fidelity

Now consider the basis $|i⟩$ that diagonalizes $\sqrt{\varrho }-\sqrt{\varsigma }$ with eigenvalues ${\lambda }_i$ and U the unitary transformation $U={\sum }_i\text{sign}({\lambda }_i)|i⟩⟨i|$.

where in the last line we have used the property that $\varrho$ and $\varsigma$ are positive semi-definite Hermitian operators. Therefore,

${\mathrm{\Psi }}^{B^{\prime }RNE}$ is the purification of NB’ density operator ${\sigma }^{N{B}^{\prime }}$. If N and B’ are decoupled, then ER can be split into two subsystems $E=\widehat{M}\check{M}$ such that $\widehat{M}$ purifies ${\sigma }^N$ and $\check{M}$ purifies ${\sigma }^{B^{\prime }}$

By Uhlmann’s Theorem, $F(\varrho ,\varsigma )=\max {|⟨{\psi }_{\varrho }|{\psi }_{\varsigma }⟩|}^2$ where max is over all purifications of $\varrho$. As $|{\mathrm{\Phi }}^{B^{\prime }RNE}⟩={\mathrm{\Phi }}^{N\widehat{M}}\otimes {\mathrm{\Phi }}^{\check{M}{B}^{\prime }}$ is the purification of ${\sigma }^N(V^B)\otimes {\sigma }_{\text{max}}^{B^{\prime }}$, there exists a purification $|{\rho }^{B^{\prime }RNE}⟩$ of the state ${\sigma }^{\text{NB’}}(V^B)$, such that

As a corollary to Uhlmann’s theorem, we have $F({\varrho }_{AB},{\varsigma }_{AB})\le F({\varrho }_A,{\varsigma }_A)$ because purifications of AB are also purifications of A. Hence, performing a partial trace on the subsystem $\check{M}{B}^{\prime }$ in the above inequality, we obtain

For the case when $|H|>>|E|$, essentially no information is released until the black hole has radiated enough so that $|B^{\prime }|$ equals $|NRE|$, because till this time the radiation is maximally entangled with the black hole and N may be coupled to B’. But soon after this state is reached, the above analysis becomes applicable and black hole starts radiating the information that Alice had.

We observe that if the black hole has radiated more than half of it’s initial state, any k qubit information that is thrown into the black hole gets reflected back in the next k qubits the black hole emits, assuming the black hole mixing happens rapidly. In case the black hole is new and hasn’t radiated enough, it emits the information in the next k qubits past it’s half evaporation.