# Clueless: A Tool Characterising Values Leaking as Addresses

Xiaoyue Chen  Uppsala UniversityUppsalaSweden<EMAIL_ADDRESS>,  Pavlos
Aimoniotis  Uppsala UniversityUppsalaSweden<EMAIL_ADDRESS>and
Stefanos Kaxiras  Uppsala UniversityUppsalaSweden<EMAIL_ADDRESS>
###### Abstract.

Clueless is a binary instrumentation tool that characterises explicit cache
side channel vulnerabilities of programs. It detects the transformation of
data values into addresses by tracking dynamic instruction dependencies.
Clueless tags data values in memory if it discovers that they are used in
address calculations to further access other data.

Clueless can report on the amount of data that are used as addresses at each
point during execution. It can also be specifically instructed to track
certain data in memory (e.g., a password) to see if they are turned into
addresses at any point during execution. It returns a trace on how the tracked
data are turned into addresses, if they do.

We demonstrate Clueless on SPEC 2006 and characterise, for the first time, the
amount of data values that are turned into addresses in these programs. We
further demonstrate Clueless on a micro benchmark and on a case study. The
case study is the different implementations of AES in OpenSSL: T-table, Vector
Permutation AES (VPAES), and Intel Advanced Encryption Standard New
Instructions (AES-NI). Clueless shows how the encryption key is transformed
into addresses in the T-table implementation, while explicit cache side
channel vulnerabilities are note detected in the other implementations.

††ccs: Security and privacy Side-channel analysis and countermeasures††ccs:
Security and privacy Information flow control

##  1\. Introduction

Cache side-channel attacks leak information through a microarchitectural
covert channel – the cache. By observing changes in the shared cache state, a
spy process can bypass process isolation and read secret data from a victim
process. Cache side-channel attacks have been demonstrated on processors of
different architectures and on different algorithms, e.g., RSA
(yarom2014flush+, ), AES (aciiccmez2007cache, ; osvik2006cache, ;
bonneau2006cache, ; gullasch2011cache, ), and ElGamal (liu2015lastlevelcache,
). _Speculative_ side-channel attacks such as Spectre (spectre:SP2019, ),
Meltdown (lipp2018meltdown, ) and their variants (smotherspectre:CCS2019, ;
Kiriansky:ARXIV2018, ; storebypass:WWW2018, ; koruyeh2018spectre, ;
maisuradze2018ret2spec, ; Aimoniotis+:CAL2021-fsi, ) have caused major changes
on how the architecture community view security. These attacks exploit
speculative instructions that are to be squashed (e.g., instructions in
mispredicted branches) to access and then transmit secret data over the shared
cache. While non-speculative cache side-channel attacks could usually be
mitigated by improving the implementations of vulnerable algorithms (e.g.,
avoid using secret data to look up in large tables), the speculative variants
of them are difficult to prevent by changing software implementations because
the information leakage happens in speculation.

Fig. 1 shows Spectre Variant 1 where an attacker can exploit the branch
misprediction to access arbitrary program data and transmit the secret over a
shared cache (spectre:SP2019, ). The victim program is correctly implemented
with the appropriate bound check, yet it is still vulnerable due to
speculative execution.

⬇

1uint8 A[10];

2uint8 B[256*64];

3void victim (size_t addr) {

4 if (addr < 10) { // mispredicted branch

5 uint8 val = A[addr]; // secret accessed

6 ... = B[64*val]; // secret transmitted

7 }

8}

Figure 1: Spectre Variant 1.

Speculative side-channel attacks have found to be an enormous security threat.
Different hardware approaches have been proposed to protect against them. For
example, InvisiSpec (invisispec:MICRO2018, ) and GhostMinion
(ghostminion:MICRO2021, ) makes speculation invisible in the data cache
hierarchy using additional speculative buffers so that secrets cannot be
transmitted over cache channels. Delay-on-Miss (DoM) (dom:ISCA2019, ) delays
all speculative loads that miss in data cache and thus prevent the observable
timing differences, while Speculative Taint Tracking (STT) (stt:MICRO2019, )
focuses on blocking only the transmitter instruction. STT uses dynamic
information flow tracking (DIFT) to taint secret data. It allows to forward
the results of speculative instructions if they cannot leak secrets via any
potential covert channels.

This work does not propose new mitigation methods for speculative side-channel
attacks. Instead, we intend to understand how prevalent these vulnerabilities
are in programs from a new perspective. Side-channel attacks rely on a
fundamental programming feature to leak the value of secrets – _the
transformation of data values into memory addresses_. Besides the victim
program in Fig. 1, consider for example sorting, hashing, or many other
algorithms that create addresses based on data values. While we understand the
mechanism that leaks data as addresses, there is no clear indication of how
serious the problem is in our workloads: How many values do “leak” as
addresses in a given application?

This work aims to shed some light on how exposed are we to the potential
vulnerability. _Clueless_ is a tool (based on binary re-writing) that tracks
dynamic instruction dependencies and tags data values in memory if it
discovers that they are used in address calculations to further access other
data.

Clueless can be used in two modes: _aggregating mode_ , where it reports on
the amount of data that are used as addresses at each point during execution,
and _tracking mode_ where the tool is specifically asked to track certain data
in memory (e.g., a password) to see if they are turned into addresses at any
point during execution. Tracking mode returns a trace on how the tracked data
are turned into addresses, if they do.

We demonstrate Clueless in aggregating mode on SPEC 2006 and characterise, for
the first time, the amount of data values that are turned into addresses in
these programs. We further demonstrate Clueless in tracking mode on a micro
benchmark and on a case study. The case study is the different implementations
of AES in OpenSSL: T-table, Vector Permutation AES (VPAES), and Intel Advanced
Encryption Standard New Instructions (AES-NI). The T-table AES implementation
can be easily broken with a cache side-channel attack (e.g., Prime+Probe), but
VPAES and AES-NI are immune to cache-timing attacks. Clueless readily shows
how the encryption key is transformed into addresses in the T-table
implementation and a lack of the corresponding transformations in the other
two implementations.

##  2\. Methods

Clueless is a dynamic instrumentation tool that analyses instructions at run-
time to track _values_ that leak as memory addresses. Values are data that
should not be used, directly or indirectly, as memory addresses, e.g.,
password hashes, private encryption keys. A value can leak as a memory address
when there is information flow from the value to a memory address. The scope
of the tool is limited to detecting data-flow: it tracks data dependences but
disregards control dependences. In other words, Clueless is able to detect
explicit channels (stt:MICRO2019, ), a value that is used as an address on a
load instruction, but not implicit channels (stt:MICRO2019, ), where the value
is leaked through control flow interaction. Assume secret is a value, the
leakage in the code in Fig. 2(a) will not be detected by the tool because
&A[0] and &A[128] only have control dependence on secret. On the other hand,
the tool will detect the leakage in Fig. 2(b) because secret is involved in
the computation of &A[i]. Furthermore, addr will be tagged as a leak point. A
leak point is a memory location where a leaked value resides.

⬇

1secret = *addr;

2if (secret % 2)

3 a = A[0];

4else

5 a = A[128];

(a) Control flow

⬇

1secret = *addr;

2i += 64 * secret;

3a = A[i];

(b) Data flow

Figure 2. Code that leaks secret.

Clueless uses an algorithm based on dynamic information flow tracking (DIFT)
(10.1145/359636.359712, ; suh2004secure, ; clause2007dytan, ). DIFT has been
successfully applied to prevent attacks on software (clause2007dytan, ;
lam2006general, ; suh2004secure, ; haldar2005dynamic, ; kong2006improving, ;
qin2006lift, ) and has been seen in hardware protection proposals against
speculative execution attacks (stt:MICRO2019, ). DIFT tracks information flow
by associating taints with data and propagating the taints according to the
data flow. In addition, Clueless’s algorithm needs to automatically assign
taints to data and maintain the taints.

###  2.1. Taint assignment

A new taint is assigned to a memory location whenever a _value_ (i.e., data
that should not be used to address memory) is loaded from that memory
location. Each taint is associated to the address of a value. In the example
in Fig. 3, suppose that values reside at memory location addrX and addrY, a
new taint $t_{x}$ is assigned to addrX when the load instruction on line 3
executes, and then another taint $t_{y}$ is assigned to addrY when line 4
executes.

Clueless needs to know if the loaded data is a value. Most contemporary
Instruction Set Architectures (ISAs) do not make a distinction between _value_
and _address_ loads nor between _value_ and _address_ registers. As a binary
instrumentation tool, Clueless is clueless about which loads actually load
values. Clueless provides two solutions to this problem.

⬇

1r1 = addrX

2r2 = addrY

3load rX <- (r1)

4load rY <- (r2)

5r3 = rX * 64

6r4 = r3 + rY

7load r5 <- (r4)

Figure 3: Instructions where  addrX and  addrY are leak points.

#### Everything is a value

One solution is to regard all data in the memory initially as values, i.e.,
Clueless assumes nothing in the memory should be used as a memory address. For
every load instruction, a new taint is assigned to the memory address of the
load. Consequently, all memory locations which contain memory addresses will
be considered as leak points. Clueless effectively provides a way to classify
any data in memory into memory addresses or non-addresses based on the past
execution. This provides a new perspective to analyse programs: how much of a
process’s memory is potentially observable by another process through a cache
side channel? We name this model as _aggregation mode_. Aggregation mode
indicate how visible a program memory can be — Section 3 presents its results.

#### Users set watchpoints

Another solution is to let users mark out memory regions that contain values.
A new taint is assigned to the memory address of a load only if that address
is within a marked memory region. Clueless supports this solution by providing
an API that can dynamically register and unregister memory regions to watch.
This requires users to modify the source code of instrumented programs by
inserting Clueless watchpoint API calls. We name this model as _tracking
mode_. Section 4 presents its results.

###  2.2. Taint propagation

Clueless uses bit arrays to store taint sets, where each bit represents a
different taint. With this representation, set union operations are equivalent
to bit-wise or operations, which are efficient to perform. Each bit array is
associated to a register or a memory location. The number of bits in a bit
array is finite and can be configured when compiling the tool. Consequently,
the maximal number of taints is equal to the number of bits in a bit array.

At instruction level, data-flow can be divided into two categories: register-
register flow and register-memory flow. One of the main differences of the two
categories in the context of dependence tracking is that the space required by
register-register flow tracking is upper-bounded by the number of
architectural registers while that of register-memory flow tracking is upper-
bounded by the number of virtual memory locations. For example, pairs of a
load and a store (both cause register-memory flow) can copy some data
throughout the entire virtual memory and result in every memory location being
tainted by the taints of the data, requiring enormous amount of space to store
the taint sets. This might not be an issue when a few pieces of data are
tracked because the data are not likely to flow through a large part of the
memory. When the numbers of tracking points are large, however, the space
overhead makes complete tracking of register-memory flow impractical. This is
the case for Clueless in aggregating mode — it regards everything in the
memory as a value and tracks the entire memory. On the other hand, storing a
taint set for each register requires much less space because the number of
architectural registers is low.

Table 1. Example Taint Propagation Instruction | rX | rY | r3 | r4 | Remark
---|---|---|---|---|---
r1 = addrX | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ |
r2 = addrY | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ |
load rX <\- (r1) | $\\{t_{x}\\}$ | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ |  $t_{x}$ associated to addrX
load rY <\- (r2) | $\\{t_{x}\\}$ | $\\{t_{y}\\}$ | $\\{\\}$ | $\\{\\}$ |  $t_{y}$ associated to addrY
r3 = rX * 64 | $\\{t_{x}\\}$ | $\\{t_{y}\\}$ | $\\{t_{x}\\}$ | $\\{\\}$ |
r4 = r3 + rY | $\\{t_{x}\\}$ | $\\{t_{y}\\}$ | $\\{t_{x}\\}$ | $\\{t_{x},t_{y}\\}$ |
load r5 <\- (r4) | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ | $\\{\\}$ |  addrX, addrY tagged

#### Tracking dependences via registers

Clueless tracks register-register flow by examining instructions, identifying
source, destination, and memory addressing registers and following propagation
rules. Table 1 demonstrates how taints propagate through the registers as
instructions from Fig. 3 are executed. The propagation rules are listed below:

  1. (1)

For each load instruction, the taint set of its destination registers becomes
either a singleton or an empty set. If a value is loaded, the taint set is a
singleton whose element is the new taint associated to the value’s address. If
what is loaded is not a value, the taint set is the empty set.

  2. (2)

For instructions that set their destination registers to a constant (e.g. xor
with two same source registers, mov a constant to a register), the taint sets
of their destination registers become the empty set.

  3. (3)

For instructions whose source and destination operands are all registers
except the instructions in rule 2, the taint sets of their destination
registers become the union of the taint sets of their source registers.

  4. (4)

For load and store instructions, memory addressing registers have their taint
sets emptied. All the memory addresses associated with the emptied taints are
tagged as leak points.

  5. (5)

For store instructions, if the taint sets of all the memory addressing
registers are the empty set, the address is no longer a leak point and is
untagged.

#### Expanding dependence tracking to memory

Using register-register flow tracking alone, the taint sets of data could be
lost because programs often store some data to the memory, use the register
containing the data for something else, and later reload the data from the
memory. These cases require tracking register-memory flow to store and reload
the taint sets. Two additional propagation rules are introduced to expand
dependence tracking to memory:

  1. (6)

For each store instruction, the taint set of the memory address becomes the
taint set of the source register that contains the stored data.

  2. (7)

For each load instruction, in addition to rule 1, the taint set of a
destination register becomes the union of the resulting taint set from rule 1
and the taint set of the memory address.

Although tracking all the register-memory flow using a complete method is
impractical due to the space requirements, it is still important to track
these flows because temporarily storing data to memory is very common. For
this reason, a set-associative cache is used as a best-effort approach to
store the taint sets that are associated with memory addresses. The cache uses
a first-in-first-out replacement policy. The number of sets as well as the
associativity of the cache can be configured when compiling the tool.

###  2.3. Taint maintenance

Clueless has finite number of taints because of the use of statically sized
bit arrays as taint sets. Therefore, taints must be maintained and reused. A
taint can only be reused when it is in none of the taint sets. Propagation
rule 1, 2 and 4 are the rules that can empty taint sets and make taints
reusable. Since the addresses associated with the emptied taints are already
tagged as leak points according to propagation rule 4, the emptied taints no
longer have useful information, thus can be removed from all the taint sets,
resulting in them immediately becoming reusable.

Taints can still be exhausted in spite of the recycling. For example, a
program can have a loop that loads many values from the memory and sums them.
In these cases, Clueless makes the taint assigned by the earliest load
available by removing it from all the taint sets.

###  2.4. Limitations

#### No tracking on speculative execution

Clueless is a binary instrumentation tool. It is not a hardware simulator and
does not obtain micro architectural information such as instructions executed
in speculation. As a result, Clueless cannot track speculative execution.

#### Incomplete tracking

Clueless is a characterisation tool as oppose to a verification tool. The
tracking of Clueless is incomplete. Clueless can track data dependence within
a limited window. The incompleteness is the consequence of our implementation
that uses a finite number of taints and a finite sized cache. When compiling
the tool, users can adjust these parameters to find the desired size of the
tracking window.

#### Dependencies

Clueless depends on Intel Pin (intel, ). Clueless is compiled into a shared
library and needs to be loaded by Intel Pin. The propagation algorithms of
Clueless is implemented in a platform-independent way, but Intel Pin only
supports instrumentation of IA-32, x86-64 and MIC ISAs. As a result, Clueless
currently only supports these ISAs.

###  2.5. Source code

The source code of Clueless is published under the GNU General Public License,
Version 3. Its git repository is accessible at
https://github.com/xiaoyuechen/dift-addr.git.

##  3\. Aggregating mode

Clueless in aggregating mode regards everything in the instrumented program’s
memory as values. Clueless in this mode tags any memory locations whose data
transform into addresses as leak points. In addition, Clueless collects a set
of all memory addresses used by the program, i.e., addresses used in any
memory accessing instructions. With the set of leak points and the set of all
memory addresses, we could introduce a metric that describes the proportion of
data that are used as addresses for a given execution of a program.

###  3.1. The $\Lambda$ metric

Let $L_{i}$ be the set of leak points and $A_{i}$ be the set of all addresses
after the execution of the $i$:th instruction of a program (Trivially,
$L_{i}\subseteq A_{i}$). Let $n$ be the number of instructions of the entire
execution of the program, metric $\Lambda$ defined by

$\Lambda=\frac{\sum_{i=1}^{n}|L_{i}|}{\sum_{i=1}^{n}|A_{i}|}$

indicates the average proportion of data that transform into addresses during
the entire executing of the program. Figuratively, $\Lambda$ is the area under
the $|L_{i}|$ curve divided by the area under the $|A_{i}|$ curve in Fig. 9.

###  3.2. $\Lambda$ of SPEC benchmarks

We used Clueless’s aggregating mode on SPEC 2006 to characterise data
transformation into memory addresses by analysing how $|A_{i}|$ and $|L_{i}|$
change and comparing the $\Lambda$ values of different benchmarks. Since
Clueless uses incomplete methods to track $L_{i}$ while the tracking of
$A_{i}$ is complete, the reported values of $\Lambda$ are lower-bounds of the
actual $\Lambda$.

The prevalence of data-address transformations, indicated by $\Lambda$, is an
innate property of a program. Fig. 8 shows the values of $\Lambda$ of
different benchmarks programs. The astar program and soplex program use more
than one third of their memory to store addresses. In the bwaves program and
sjeng program, on the other hand, such transformations are rarely seen.

Figure 8. $\Lambda$ of SPEC benchmarks.

###  3.3. A closer look

For more insights into data-address transformation, we further study how much
data are transformed into addresses at each point of execution of some
benchmark programs.

(a) astar

(b) bzip2

(c) calculix

(d) soplex

Figure 9. $|A_{i}|$ and $|L_{i}|$ of benchmark programs.

#### Astar

Fig. 9(a) shows how $|A_{i}|$ and $|L_{i}|$ change during an execution of the
astar program. Note that $|A_{i}|$ increases monotonically while $|L_{i}|$
does not. When an address $l\in L_{i}$ is written to by the $(i+1)$:th
instruction, $L_{i+1}=L_{i}\setminus\\{l\\}$. This is very common when $l$ is
a stack address, as the stack memory tends to be rewritten often. In the astar
program, $|L_{i}|$ fluctuates when $i\in[2\times 10^{11},~{}3\times 10^{11}]$.
The cause of such fluctuations is that the same blocks of memory containing
addresses are repeatedly loaded from and written to.

#### Bzip2

Fig. 9(b) shows that the bzip2 program periodically store new addresses to the
same blocks of memory. One common data-address transformation pattern can be
found when $i\in[0,~{}3.9\times 10^{10}]$, $i\in[3.9\times
10^{10},~{}7.9\times 10^{10}]$, and $i\in[7.9\times 10^{10},~{}1.75\times
10^{11}]$ — a rapid increase in $|L_{i}|$ which then fluctuates periodically,
followed by another rapid but smaller increase in $|L_{i}|$ and fluctuates
periodically again. The cause for the repeated pattern could be that the bzip2
program reallocates memory to store memory addresses, but the same algorithm
is used on the reallocated memory.

#### Calculix

Fig. 9(c) shows that the calculix program has an obvious periodic memory
access pattern. After the initial increase of both $|A_{i}|$ and $|L_{i}|$,
$|A_{i}|$ becomes stable while $|L_{i}|$ becomes periodic. The amplitude of
$|L_{i}|$ is relatively large at approximately $2.1\times 10^{6}$, indicating
that blocks containing $2.1\times 10^{6}$ addresses are repeatedly written
with new addresses.

#### Soplex

Fig. 9(d) shows how $|A_{i}|$ and $|L_{i}|$ of the soplex program change.
After the initial increase, both $|A_{i}|$ and $|L_{i}|$ become stable. This
does _not_ mean that data in this program are transformed to memory addresses
only once. After the data are tagged, they may still be transformed into
memory addresses multiple times in different ways, but $L_{i}$ would remain
the same. The stable $|L_{i}|$ only indicates that no new data are tagged, and
no tagged memory location is written to.

##  4\. Tracking mode

Clueless in tracking mode allows users to dynamically register and unregister
watchpoints, i.e., memory blocks that contain values. If data from any
watchpoint are transformed into memory addresses, Clueless will provide a
detailed diagnose on each leakage. The diagnostic information include the leak
point, the memory address that the value in the leak point transforms into, a
trace of instructions that shows the value-address transformation, and the
relevant routine and image names where the leakage happens. This mode could be
used to test the side-channel vulnerability of programs and help understand
where and how secrets are leaked if such vulnerability exists.

###  4.1. The micro benchmark

We demonstrate Clueless in tracking mode on a micro benchmark program in Fig.
10. Array T[] and function foo are defined in a shared library the victim
program links against. The victim program has a secret stored in the s[]
array. The victim calls foo with the secret as its parameter. Function foo
loads each byte of its parameter array, multiplies the byte value by 64, and
uses the result as the index of a constant array T[] to do some lookup.

This program is vulnerable to side channel attacks such as Flush + Reload
(yarom2014flush+, ). The attacking program may mmap the shared library and
flush the cache lines containing T[], wait for the victim to call the foo
function, and measure the time to reload the cache lines to find out which
lines are accessed by foo. Assuming that the victim’s machine has 64-byte
cache lines, the attacker can recover the secret completely — each access of
T[s[i]*64] will be on a different cache line, so each byte of the secret can
be computed using (l-T)/64 where l is the address of an accessed line. The
offset of T can also be found trivially because it is just a symbol in a
shared library. In our example, T has an offset of 0x2020.

⬇

1// The shared library:

2const uint8 T[256 * 64];

3int foo (const uint8 *s, size_t n) {

4 for (size_t i = 0; i < n; ++i) {

5 uint8 r = T[s[i] * 64];

6 ...

7 }

8}

9

10// The victim program:

11uint8 s[] = "cLUe";

12foo (s, sizeof (s) - 1);

Figure 10: The micro benchmark program.

###  4.2. Pinpointing the leakage

Clueless’s aggregation mode can be used to characterise the micro benchmark
program. Fig. 11 shows how its $|A_{i}|$ and $|L_{i}|$ change. With Clueless
in tracking mode, we can pinpoint which increase of $|L_{i}|$ results in the
tracked secrets being leaked. The annotations in Fig. 11 reveal the content of
the leaked secrets at the point of their leakage. For example, 0x38e0 is
leaked when $i=20$, i.e., &T[s[0]*64] evaluates to 0x38e0 (after subtracting
the image load offset), and is used as a memory address in a load. To recover
s[0], we compute (0x38e0-0x2020)/64 and yield 0x63, which is the ASCII code
for ‘c’.

Figure 11. $|A_{i}|$ and $|L_{i}|$ of the micro benchmark program.

###  4.3. Tracing the transformation

⬇

10x7fff41801683 { 0 } -> %rax

2%rax { 0 } -> %rax

3%rax { 0 } -> %rax

4%rax { 0 } -> %rax

5%rdx %rax { 0 } -> %rdx

6%rdx { 0 } -> %rcx

7[ %rcx { 0 } ] = 0x55baae46d8e0 ->

(a) Propagation

⬇

movzbl (%rax),%eax

movsbl %al,%eax

shl $0x6,%eax

cltq

add %rax,%rdx

mov %rdx,%rcx

mov (%rcx),%rax

(b) Instruction

Figure 12. Tracing the micro benchmark.

For further understanding on the leakage, Clueless gives a trace of
propagation that causes it. Fig. 12(a) shows the part the propagation trace
that causes s[0] to leak, and Fig. 12(b) shows the corresponding instructions.

The propagation trace has the following syntax:

  * •

{ NUM, .. } is a taint set, e.g., %rax { 0, 3 } means register rax has taint
set $\\{t_{0},t_{3}\\}$.

  * •

-> represents data flow and taint propagation, e.g., %rdx { 0 } -> %rcx means data flow from register rdx to register rcx, and taint $t_{0}$ propagates to register rcx.

  * •

[ REG { NUM, .. } ] represents a register being used as a memory address,
followed by an equal sign and the effective address, e.g., [ %rcx { 0 } ] =
0x55baae46d8e0 -> means that register rcx whose taint set is ${t_{0}}$ is used
as a memory address in a load.

By analysing the trace of the micro benchmark, we find that the instruction at
line 1 loads &s[0] which is 0x7fff41801683 with taint set $\\{t_{0}\\}$ to
register rax. The following 3 instructions propagate $\\{t_{0}\\}$ from
register rax to itself. Then the instruction at line 5 propagates
$\\{t_{0}\\}$ to register rdx by adding register rax to it. $\\{t_{0}\\}$ is
further propagated to register rcx which is eventually used as the memory
address 0x55baae46d8e0.

##  5\. Case study: AES

We have seen how the micro benchmark in Section 4 could leak secrets due to
its value-address transformations. Some implementations of Advanced Encryption
Standard (AES) (aes2001, ) are susceptible to cache side-channel attacks for
the same reason. These implementations often depend on large tables to speed
up the encryption process (hamburg2009accelerating, ). If encryption keys are
transformed into indices of large tables for lookups, attackers may partially
or completely recover the keys by observing the corresponding cache state
changes. Numerous attacks on AES exploiting this class of vulnerability have
been demonstrated in the past (aciiccmez2007cache, ; gullasch2011cache, ;
osvik2006cache, ; bonneau2006cache, ). Different implementations of AES have
also been proposed to protect against such attacks while retaining or
improving the speed of encryption (osvik2006cache, ; hamburg2009accelerating,
; gueron2010intel, ).

In this case study, we use Clueless in tracking mode to analyse three
different implementations of AES present in OpenSSL 3.0.3 — T-table, Vector
Permutation AES (VPAES), and Intel Advanced Encryption Standard New
Instructions (AES-NI). The expanded encryption key is set as the watchpoint in
order to observe if it is transformed into memory addresses.

###  5.1. T-table

The T-table implementation of AES in OpenSSL uses 9 T-tables, i.e., pre-
computed lookup tables, with 8 of them being 8kiB each and 1 of them being
2kiB. The encryption key is first expanded to round keys. The first round key
is combined with the 16-byte plaintext using xor to form the initial state
vector. The elements in the state vector are then used as indices of the
T-tables to look up values which are combined with the next round key to form
the next state vector. This implementation could be easily broken using
Prime+Probe, with the 128-bit encryption key fully recovered after only 300
encryptions (osvik2006cache, ).

Clueless detects the potential leak and marks all the bytes of the key as leak
points. In addition, Clueless gives a propagation trace that shows how the key
is transformed into memory addresses. Fig. 13(a) shows the propagation trace
of the first round of the encryption (all rounds are similar) while Fig. 13(b)
shows the corresponding instructions. The traces shows that the first round
key is leaked as follows:

  1. (1)

Instructions from line 1 to 4 load the first round key and xor them with the
plaintext in register rax, rbx, rcx, and rax to form the initial state vector.
Different taints are assigned to the watched memory locations and propagated
to the destination registers.

  2. (2)

Instructions from line 5 to 8 extracts some bytes from the initial state
vector and copy them to a different set of registers (r10, r11, r12 and r8).
Taints propagate from the source registers to the destination registers.

  3. (3)

Instructions from line 13 to 16 use the extracted bytes to perform lookup in
one of the T-tables. The tainted registers are used to form effective
addresses. At this point, Clueless marks the memory locations of the first
round key as leak points.

⬇

10x7ffc4afe86b0 { 1 } -> %rax

20x7ffc4afe86b4 { 2 } -> %rbx

30x7ffc4afe86b8 { 3 } -> %rcx

40x7ffc4afe86bc { 4 } -> %rdx

5%rax { 1 } -> %r10

6%rbx { 2 } -> %r11

7%rcx { 3 } -> %r12

8%rdx { 4 } -> %r8

9%rbx { 2 } -> %rsi

10%rcx { 3 } -> %rdi

11%rcx { 3 } -> %rcx

12%rdx { 4 } -> %rbp

13[ %r10 { 1 } ] = 0x555e939ad533 ->

14[ %r11 { 2 } ] = 0x555e939ad527 ->

15[ %r12 { 3 } ] = 0x555e939ad4c0 ->

16[ %r8 { 4 } ] = 0x555e939ad4c0 ->

(a) Propagation

⬇

xor (%r15),%eax

xor 0x4(%r15),%ebx

xor 0x8(%r15),%ecx

xor 0xc(%r15),%edx

movzbl %al,%r10d

movzbl %bl,%r11d

movzbl %cl,%r12d

movzbl %dl,%r8d

movzbl %bh,%esi

movzbl %ch,%edi

shr $0x10,%ecx

movzbl %dh,%ebp

movzbl (%r14,%r10,1),%r10d

movzbl (%r14,%r11,1),%r11d

movzbl (%r14,%r12,1),%r12d

movzbl (%r14,%r8,1),%r8d

(b) Instruction

Figure 13. Tracing the T-table implementation.

###  5.2. VPAES

VPAES is a technique for accelerating AES using vector permute instructions.
It avoids key-dependent memory references thus being immune to known cache-
timing attacks (hamburg2009accelerating, ). The hardware must support vector
permutation instructions to use VPAES.

Clueless in tracking mode shows that no part of the key is leaked as memory
addresses in OpenSSL’s VPAES implementation. The reported propagation trace in
Fig. 14 indicates that the key is loaded (e.g., by instruction at 86e1) and
combined in different rounds, but no part of the key is used to reference
memory.

⬇

0x7fff609ec660 { 1 } -> %xmm5

%xmm2 %xmm5 { 1 } -> %xmm2

%xmm0 %xmm2 { 1 } -> %xmm0

%xmm1 %xmm0 { 1 } -> %xmm1

%xmm1 { 1 } -> %xmm1

%xmm0 { 1 } -> %xmm0

%xmm5 %xmm0 { 1 } -> %xmm5

%xmm0 %xmm1 { 1 } -> %xmm0

%xmm3 %xmm1 { 1 } -> %xmm3

%xmm3 %xmm5 { 1 } -> %xmm3

%xmm4 %xmm0 { 1 } -> %xmm4

%xmm4 %xmm5 { 1 } -> %xmm4

%xmm2 %xmm3 { 1 } -> %xmm2

%xmm2 %xmm0 { 1 } -> %xmm2

%xmm3 %xmm4 { 1 } -> %xmm3

...

%xmm3 %xmm1 { 1 2 ... 10 } -> %xmm3

%xmm4 %xmm2 { 1 2 ... 10 } -> %xmm4

%xmm4 %xmm5 { 1 2 ... 11 } -> %xmm4

%xmm0 %xmm3 { 1 2 ... 10 } -> %xmm0

%xmm0 %xmm4 { 1 2 ... 11 } -> %xmm0

%xmm0 { 1 2 ... 11 } -> %xmm0

Figure 14: Tracing the VPAES implementation.

###  5.3. AES-NI

Intel introduced AES-NI instruction sets in 2010 to provide direct hardware
support for AES. These new instructions run in data-independent time and do
not use tables (gueron2010intel, ). AES-NI can encrypt an entire round with a
single instruction. AES implementations that properly use AES-NI should be
immune to cache side-channel attacks as no cache is involved in these
instructions.

The AES implementation in OpenSSL that uses AES-NI has not been found to
transform the key into memory addresses. The propagation trace in Fig. 15
shows that no part of the key is used to reference memory.

⬇

0x7ffd85426fd0 { 1 } -> %xmm0

0x7ffd85426fe0 { 2 } -> %xmm1

%xmm2 %xmm0 { 1 } -> %xmm2

%xmm2 %xmm1 { 1 2 } -> %xmm2

0x7ffd85426ff0 { 3 } -> %xmm1

%xmm2 %xmm1 { 1 2 3 } -> %xmm2

0x7ffd85427000 { 4 } -> %xmm1

...

%xmm2 %xmm1 { 1 2 3 ... 10 } -> %xmm2

0x7ffd85427070 { 11 } -> %xmm1

%xmm2 %xmm1 { 1 2 ... 11 } -> %xmm2

%xmm0 { 1 } -> %xmm0

%xmm1 { 11 } -> %xmm1

%xmm2 { 1 2 ... 11 } -> 0x55feae1dd130

Figure 15: Tracing the AES-NI implementation.

##  6\. Conclusion and future work

We have presented Clueless: a tool characterising values leaking as addresses.
Using Clueless in aggregating mode, we have characterised, for the first time,
the amount of data values that transformed into memory addresses in SPEC 2006
benchmark programs. Some benchmark programs use more than one third of
accessed memory to reference memory. Clueless in tracking mode has provided
the traces of how secrets propagate and leak in a micro benchmark and AES
implementations in OpenSSL. The T-table implementation of AES exhibits
potential vulnerabilities to cache side-channel attacks while the VPAES and
AES-NI implementations are immune to such attacks.

The “leaks” reported by Clueless are to be further studied. We hope to
identify the value-address transformations that would lead to the danger of
leaking sensitive information from the false positives (e.g. secrets
transforming to addresses on the same cache lines). We are interested in
applying similar dynamic information flow tracking techniques on hardware
models to mitigate cache side-channel attacks such as Specture. The high
frequencies of data-address transformations in some programs also indicate
optimisation opportunities in cache systems: data that would transform to
memory addresses may be associated to the data the transformed addresses point
to. This may be a focus of our future work.

###### Acknowledgements.

This work was supported by Microsoft Research through its EMEA PhD Scholarship
Programme grant no. 2021-020, the Swedish Research Council (VR) grant
2018-05254, the VINNOVA grant 2021-02422, the SSF grant FUS21-0067, and
Uppsala University funds for Cybersecurity.

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#  Single phase and correlated phase estimation with multi-photon annihilated
squeezed vacuum states: An energy balancing scenario

N. Samantaray  Email<EMAIL_ADDRESS>Quantum Engineering Technology
Labs, H. H. Wills Physics Laboratory and Department of Electrical and
Electronic Engineering, University of Bristol, BS8 1FD, UK  I. Ruo-Berchera
INRIM, Strada delle Cacce 91, I-10135 Torino, Italy  I.P. Degiovanni  INRIM,
Strada delle Cacce 91, I-10135 Torino, Italy  INFN, via P. Giuria 1, I-10125
Torino, Italy

###### Abstract

In the last years, several works have demonstrated the advantage of photon
subtracted Gaussian states for various quantum optics and information
protocols. In most of these works, it was not clearly investigated the
relation between the advantages and the usual increasing energy of the quantum
state related to photon subtraction. In this paper, we study the performance
of an interferometer injected with multi photon annihilated squeezed vacuum
states mixed with coherent states for both single and correlated phase
estimation. For single phase estimation, albeit the use of multi-photon
annihilated squeezed vacuum states at low mean photons per mode provide
advantage compared to classical strategy, when the total input energies is
held fixed, the advantage due to photon subtraction is completely lost.
However, for the correlated case in analogous scenario, some advantage appears
to come from both the energy rise and improvement in photon statistics. In
particular quantum enhanced sensitivity with photon subtracted states appears
more robust to losses, showing an advantage of about 30$\%$ with respect to
squeezed vacuum state in case of realistic value of the detection efficiency.

##  I Introduction

Non-Gaussian states have been recognized as a valuable resources for many
quantum information processing protocols Genoni:2010 , for example to enhance
the fidelity of quantum teleportation Olivares:2003 ; Optarny:2000 ;
Cochrane:2002 ; Dell'Anno:2010 , improving secret key rate in quantum key
distribution Borelli:2016 ; Liao:2018 , and in quantum cloning of coherent
states Cerf:2005 . These exotic states are required in Gaussian entanglement
distillation Eisert:2002 ; Cirac:2002 ; Fiurasek:2002 , error correction
Niset:2009 , noiseless amplification Ferreyrol:2010 ; Xiang:2010 , and
fundamental loop-hole free Bell tests in continuous variables Banaszek:1998 ;
Nha:2004 . More recently, resource theories quantifying the importance of
Wigner negativity and non-Gaussianity for continuous variable quantum
computation have been reported Takagi:2018 ; Albarelli:2018 . Because of
higher potential distinguishability of non-Gaussian states from their original
Gaussian states, they have been proven useful for more precise parameter
estimation in quantum optics Genoni:2009 ; raul:2012 . Photon addition or
photon subtraction can transform a Gaussian in a non-Gaussian state kim:2008 ;
kim1:2008 ; Bellini:2007 . Tara and Agarwal tara:1990  were the first to
propose the transformation of a classical like coherent state into a non-
classical state through photon addition and this operation was experimentally
implemented for the first time on coherent and thermal state in Bellini:2004 ;
Bellini1:2007 . Furthermore, photon addition and subtraction have been
reported to enhance the entanglement in two mode squeezed vacuum state (TSV)
Carlos:2012 ; Tim:2013 ; ourjoumtsev:2007 . It is well known that each mode of
the TSV has super-poissonian photon statistics. In the experimental
workchekova:2016 , it has been reported that multi-photon subtraction makes
the TSV less noisy and helps in shifting the most probable distribution to
higher mean photon number, thereby it increases the mean energy of the
resulting state. In the last years, photon subtracted TSV states have been
theoretically investigated for other applications, e.g. in Zhang:2014  these
states were proved to be advantageous with respect to TSVs for target
detection in noisy environment, a scheme dubbed "quantum illumination"
Lloyd:2008 ; Lopaeva:2013 . Their advantage has also been demonstrated in
single interferometry with parity measurementsraul:2012 ; Birritella:2014 ;
yoa:2016 .

Since the seminal Caves work Caves:1981 , it is well known that single mode
squeezed vacuum (SSV) mixed with intense coherent state provides substantial
advantage in practical phase estimation and very recently that scheme has been
applied by LIGO and VIRGO collaborations to further improve the sensitivity of
gravitational waves detectors Tse:2019 ; Acernese:2019 . It has also been
shown that squeezed vacuum mixed with a intense coherent beam allow to
approach the optimal sensitivity achievable by a linear interferometer
operated with large photon number and non negligible losses Rafal:2013 . In
that context, single photon subtracted SSV brings advantage in phase
estimation and allow reaching Heisenberg’s limit (HL) olivares:2016 . In that
case olivares:2016 , the mean energy increasing of the photon subtracted state
has been compensated by a reduction of the coherent beam energy in order to
keep constant the total input photon number. Photon subtracted SSV mixed with
coherent states lead to improved phase shift sensitivity in parity measurement
Birritella:2014 . In this work, because of non-linear increase of mean number
of photons with the number of subtracted photons, the total average number of
photons is fixed by choosing fixed squeezing parameter and increased coherent
energy. This approach is little different than the energy balancing scenario
reported in olivares:2016 .

However, in most of the literature with some exception that we will point out
later raul:2012 ; Dell'Anno:2007 , it was not clear whether the advantages
come from energy shifts of the photon subtracted states, or from their
potentially improved photon noise properties. In fact, most of the time, the
comparison between the performance of Gaussian quantum states and the
correspondent photon subtracted states has been considered at fixed squeezed
parameter, that means generally that the energy put into the quantum resource
is not fixed. Photon subtraction is a complex operation that can be
experimentally realized probabilistically or with low efficiency Marek:2018 .
So, if the advantage comes mainly from the increased energy, it can
practically more convenient to increase the squeezing parameter of the
Gaussian state, rather than performing photon subtraction. This is the reason
why we consider of great importance understanding if the advantage relies
solely on increase of the energy, or there are more fundamental reasons that
justify operation such as photon subtraction. Answering to this question is
the main motivation of this paper.

Specifically, given the importance for practical interferometry Rafal:2013 ,
here we study in detail multi-photon subtracted single mode and two mode
squeezed vacuum state for single phase and correlated phase estimation
respectively by combining them with coherent states on a beam splitters. On
one side, we show that a multi-photon subtracted (one-) two-mode squeezed
states is formally equivalent to a state obtained by a (one-) two- mode
squeezing operator applied to a certain class of finite superposition states
in the photon number basis. This class of states have been investigated
earlier Wodkiewicz:1987 ; figurny:1993  and they show quadrature squeezing
their-self. One could expect that this initial squeezing could bring benefit
in phase estimation. We have therefore investigated this possibility.

In order to properly understand the origin of the improvement in phase
measurement uncertainties if any, we think that the proper procedure requires
that the total energy should be fixed by balancing the energies of the
subtracted and un-subtracted states keeping the coherent pump energy constant.
We will consider this energy balancing condition for both single and
correlated phase estimations. Similar analysis has been done in raul:2012 ,
where an advantage in phase estimation with parity measurement at fixed energy
has been reported, but that scheme does not involve the mixing with a coherent
state, and parity measurement is quite far from realistic applications. In
Dell'Anno:2007 , while a precise comparison with two-mode squeezed states with
the same energy has not been carried out they showed that the larger is the
"affinity" of a non-Gaussian state with a two-mode squeezed vacuum (with
larger energy), the larger is the teleportation fidelity.

This paper is organized as follows. In Sec. II, we will introduce multi-photon
annihilated single mode squeezed vacuum (PASSV), discussing their properties
and their usefulness, for single phase estimation by the conventional
measurement strategy II.1 and in the more general framework of the fisher
information II.2. In Sec. III, we will describe multi-photon symmetrically
annihilated two mode squeezed vacuum (SPATSV) state. In particular, in section
III.1, we will analyze squeezing and photon statistical properties of SPATSV.
In section III.2 we study the problem of correlated phase estimation. We
present results up to four and three photons subtraction for single and
correlated phase estimations respectively. Finally, we will summarize and
discuss the main results in Sec. IV.

##  II PASSVS

PASSV states are defined as:

$|\Psi^{(m)}_{PASSV}\rangle=N^{m}_{-}(\lambda)\hat{a}^{m}\hat{S}(\lambda
e^{i\chi})|0\rangle,$ (1)

where $\hat{S}(re^{i\chi})=e^{re^{i\chi}a^{2}-h.c.}$ is the single mode
squeezing operator, $r$ being the squeezing parameter and $\chi$ is the
squeezing angle. The squeezing operator applied to the vacuum state originates
SSV with energy (mean number of photon) equal to with $\lambda=\sinh^{2}r$.
The number of subtracted photon is $m$, obtained by $m$ consecutive action of
the annihilation operator $\hat{a}$. Since the photon subtraction is not a
unitary operation, it is necessary to introduce the normalization constant
$N^{m}_{-}(\lambda)$. Its explicit form is
$N^{m}_{-}(\lambda)=m!(-i\sqrt{\lambda})^{m}P_{m}(i\sqrt{\lambda})$ nigam:2017
, with $P_{m}$ being the $m$-th order Legendre’s polynomial. A known effect of
the photon subtraction is the increasing of the mean energy of the state. This
is intuitively explained because it is more likely to subtract a photon from a
highly populated state corresponding to a selection of the more energetic
components of the state. For example, the mean photon number of PASSV state
$\mathcal{N}_{m}(\lambda)$, for $m=0-3$ which correspond to zero, one, two and
three photon subtraction from the SSV state, becomes
$\mathcal{N}_{0}=\lambda$, $\mathcal{N}_{1}=3\lambda+1$,
$\mathcal{N}_{2}=3\lambda(3+5\lambda)/(1+3\lambda)$, and
$\mathcal{N}_{3}=(3+30\lambda+35\lambda^{2})/(3+5\lambda)$ respectively.

We have found a new representation for PASSV states (exploiting integration
within an ordered product (IWOP) technique Fan:2003 ) which is equivalent to
seeding squeezing operator with photon number superposition state
$|\Theta^{s}_{m}(\lambda,\chi)\rangle$ in input as follows:

$|\Psi^{(m)}_{PASSV}\rangle=\hat{S}|\Theta^{s}_{m}(\lambda,\chi)\rangle,$
$\displaystyle|\Theta^{s}_{m}(\lambda,\chi)\rangle=N^{m}_{-}(\lambda)m!\left(e^{i\chi}\sqrt{\lambda}\right)^{m}\times$
(2)
$\displaystyle\sum_{l=0}^{[m/2]}\frac{1}{l!\sqrt{(m-2l)!}}\left(\frac{e^{-i\chi}}{2}\sqrt{\frac{1+\lambda}{\lambda}}\right)^{l}|m-2l\rangle,$

where the upper bound of the summation $[m/2]$ stand for the integer part of
$m/2$. Without any loss of generality, hereinafter we set the squeezing angle
to $\chi=0$. For $m=0$, the input simplifies to vacuum state as expected. For
$m=1$, it becomes a single photon state as reported in olivares:2016 . Note
that, for other values of $m$, it becomes a $(m+1)/2$ components superposition
state for odd $m$, and a $(m+2)/2$ components superposition state for even
$m$. The energy increasing with $m$ of the PASSV state can now be understood
in terms of increasing of the mean number of photons of the seeding states.

Figure 1: (Color online) Quadrature squeezing of PASSV states and phase
measurement uncertainty at working point $\phi=\pi/2$ with $\mu=100$,
$\eta=0.98$, and $\psi=0$ for different number of photon subtraction $m$:
$m=0$ (solid red line), $m=1$ (solid black line), $m=2$ ( solid yellow line),
$m=3$ (solid cyan line), and $m=4$ (solid green line). Dashed lines represents
the coherent state: (a.) quadrature squeezing, (b.) phase uncertainty, and
(c.) phase measurement as per balancing condition.

The states $|\Theta^{s}_{m}(\lambda,\chi)\rangle$ are known to show quadrature
squeezing Wodkiewicz:1987 ; figurny:1993 , even though they can not be
obtained by any unitary transformation on vacuum state, like standard squeezed
state. However, these states do not always have lower quadrature noise
compared to vacuum state. In particular, for $m=1$, the state
$|\Theta^{s}_{1}(\lambda,\chi)\rangle$ is a single photon state having more
quadrature noise than the vacuum state. We checked for subsequent odd values
of $m$, although the quadrature noise of the seeding states decrease with
respect to single photon state, its value still remain above the vacuum noise.
This can be appreciated in Fig 1a, when considering small value of the
squeezing parameter that actually means $\hat{S}\approx\mathbb{I}$, with
$\mathbb{I}$ the identity operator. In fact, in Fig 1 we plot the variance of
the quadrature $\hat{Y}=\left(\hat{a}-\hat{a}^{\dagger}\right)/i\sqrt{2}$ of
the PASSV states in Eq. (2). In general, we observe that for odd values of $m$
the quadrature noise of the PASSV state is worse than the one of the SSV
(corresponding to PASSV with $m=0$), while for even values of $m$ the
quadrature noise is better than SSV for low value of $\lambda$. Detectors are
not ideal in realistic scenarios. The effect of a non-unit quantum efficiency
$\eta$ can be modelled as the evolution of the input field passing through a
beam splitter (BS) with transmission equal to $\eta$, while the other free
port of the BS is in vacuum stateMandel:1995 . This approach has been used
throughout the paper to account for the optical/detection losses.

In the next subsection, we will discuss the performance of PASSV states in
phase estimation in connection to the quadrature squeezing.

###  II.1 Single phase estimation with PASSV states

Let us consider the Mach Zehnder interferometer (MZI) sketched in Fig.2, where
one ports of the first beam splitter is injected with coherent light and the
other port with a PASSV state. Thus, the total input state is
$|\Psi^{(m)}_{PASSV}\rangle_{1}\otimes|\alpha\rangle_{2}$, with
$\alpha=|\alpha|e^{i\psi}$, where $|\alpha|$ ($\mu=|\alpha|^{2}$) and $\psi$
are the amplitude, the mean photon number and the phase of the coherent pump
respectively.

Figure 2: Schematic of mixing single mode squeezed vacuum and coherent state
in MachZehnder interferometer for phase estimation $\phi$.

The uncertainty in measuring the phase $\phi$, in the configuration of Fig.2,
is expressed by

$U(\phi)=\frac{\sqrt{\Delta^{2}\hat{o}}}{|\frac{\partial\hat{o}}{\partial\phi}|},$
(3)

where $\hat{o}$ is the photon number difference operator at the output port of
the interferometer and $\Delta^{2}\hat{o}$ is its variance. For zero-mean
quadrature field such as SSV, $\langle\hat{o}\rangle=(\mu-\lambda)\cos(\phi)$.
For SSV, it can be shown that the lowest uncertainty is reached for
$\phi=\pi/2$ and in the limit of $\lambda\gg\mu$, the uncertainty is shot-
noise limited, scaling as $\lambda^{-1/2}$. Whereas in case of
$\lambda\ll\mu$, the uncertainty is
$U(\phi)=\left(\Delta^{2}X_{\theta=\psi+\frac{\pi}{2}}/\mu\right)^{1/2}$,
proportional to the noise of the rotated quadrature
$\hat{X}_{\theta=\psi+\frac{\pi}{2}}=\left(\hat{a}e^{-i\theta}+\hat{a}^{\dagger}e^{i\theta}\right)/\sqrt{2}$.
In our case and for the choice of $\psi=0$, the sub-shot noise sensitivity is
related to the squeezing of the $X_{\frac{\pi}{2}}\equiv Y$ quadrature. For
the sake of completeness we mention that for a more specific repartition of
the total energy ($\mu+\lambda$) between squeezing and coherent input states,
a more efficient scaling of the uncertainty with can be in principle achieved
($\propto(\mu+\lambda)^{-3/4}$) Demkowicz:2015 , and different more
sophisticated detection scheme could allow to approach the Heisenberg limit
(ideal decoherence-free scenario) Pezze:2008

We have derived analytically the uncertainty on the phase estimation according
to Eq. 3, when PASSV states are injected. The results are shown graphically in
Fig1 b, compared with the shot-noise limit (SNL) at equivalent total energy
(dotted lines). The last is obtained considering the performance of a coherent
state with mean number of photon equal to the sum of $\mu$ and the mean number
of photons of the PASSV state which varies with $m$. It is easy to check that
for PASSV the uncertainty always approach asymptotically the SNL when
$\lambda\gg\mu$. For $\lambda\ll\mu$, the uncertainty is basically determined
by the variance of the $Y$ quadrature, reported in Fig1 a, as expected. Indeed
the advantage over the SNL is present only in the region of quadrature
squeezing, and PASSV($m>0$) performs better than SSV only for even $m$.
However we are going to show that this apparent improvement is only due to the
energy increasing of the state due to photon subtraction. For that purpose, we
have renormalized the energy of the initial SSV state before the photon
subtraction, so that the mean number of photon of the subtracted states
($m=0-4$) are all equal to $\lambda$. In this way, also the total input
energies to the interferometer is fixed to $N_{tot}=\mu+\lambda$. With the
energy balancing, SSV outperforms PASSV regardless of the values of $\lambda$,
as presented in Fig1c.

Figure 3: (Color online) Phase measurement uncertainty in energy balancing
scenario for $\phi\neq\pi/2\approx\pi/2-1$ with $\mu=10000$, and $\eta=0.98$
for different number of photon subtraction $m$: $m=0$ (solid red line), $m=1$
(solid black line), $m=2$ ( solid yellow line), $m=3$ (solid cyan line), and
$m=4$ (solid green line). Dotted line is the classical strategy.

The diverging behaviour of uncertainty in Fig.1 b-c for $\lambda=100$ comes
from the singularity in the denominator of Eq.3 that is when the mean number
of photons of the PASSV state equals the one of the coherent state, i.e.
$N_{m}(\lambda)=\mu$. Because of energy increment as a result of photon
subtraction, relatively lower value of $\lambda$ is required for fulfilling
the singularity condition as evident from Fig1 b. Incidentally, we have
observed that far from the optimal working point $\pi/2$, PASSV can still
provide some advantage, even under energy balancing conditions as shown in
Fig. 3, anyway without surpassing the sensitivity obtained by SSV in the
optimal point. Typically, this happens from value of $\lambda$ in a middle
range (namely from $\mu$/ 100 <$\lambda$ < $\mu/10$).

Next we will see the advantage if any in QFI perspective.

###  II.2 Quantum fisher information

Quantum Fisher information (QFI), $F_{Q}$, can be used to identify the lower
uncertainty attainable in a parameter estimation problem according to the
expression

$U(\phi)\geq\frac{1}{\sqrt{F_{Q}(\phi)}}.$ (4)

For class of pure states Demkowicz:2015 , QFI takes the following compact form

$F_{Q}(\phi)=4\langle(\Delta\hat{H})^{2}\rangle_{|\psi\rangle_{1,2}},$ (5)

where $\hat{H}$ is the generator of the unitary transformation associated with
the parameter $\phi$, i.e $\hat{U}(\phi)=e^{i\hat{H}\phi}$ and
$|\psi\rangle_{1,2}$ being the input states injected into the interferometer.
In the case of the MZI of Fig.2, the generator is the photon number operator
$\hat{n}_{3}=\hat{a}^{\dagger}_{3}\hat{a}_{3}$ where
$\hat{a}_{3}=(\hat{a}_{1}+\hat{a}_{2})/\sqrt{2}$. As per Eq. (5), we shall
evaluate QFI by considering PASSVs and coherent state injected into the
interferometer. The complete expressions of QFI are cumbersome: graphical
representation can help to understand its peculiar features.

Figure 4: (Color online) Quantum fisher information versus mean number of
photon $\lambda$ with $\mu=100$ for different number of photon subtraction
$m$: $m=0$ (solid red line), $m=1$ (solid black line), $m=2$ ( solid yellow
line), $m=3$ (solid cyan line), and $m=4$ (solid green line). Dotted line
represents classical bound obtained when only a coherent state with the same
total energy ($\lambda+\mu$) is in input: (a.) without energy balance (b.)
balanced condition.

Specifically, Fig.4(a) shows a general increasing of the QFI for PASSVs at the
increasing of $m$, for both low and high values of $\lambda$, i.e. photon
subtraction is always advantageous with respect to best classical strategies
(dotted line). This advantage in QFI due to photon subtraction anyway can be
due to the increase of the mean number of photons of the input state. Indeed,
for energy balancing condition, the advantage is completely lost as evident
from Fig.4(b) and also from the expression of QFI in the limit
$N_{tot}\rightarrow\infty$ (at a finite fixed coherent energy) reported here:

$\displaystyle F_{Q(m=0)}$ $\displaystyle=$ $\displaystyle 2N^{2}_{Tot},$ (6)
$\displaystyle F_{Q(m=1)}$ $\displaystyle=$
$\displaystyle\frac{2N^{2}_{Tot}}{3},$ $\displaystyle F_{Q(m=2)}$
$\displaystyle=$ $\displaystyle\frac{2N^{2}_{Tot}}{5},$ $\displaystyle
F_{Q(m=3)}$ $\displaystyle=$ $\displaystyle\frac{2N^{2}_{Tot}}{7}$
$\displaystyle F_{Q(m=4)}$ $\displaystyle=$
$\displaystyle\frac{2N^{2}_{Tot}}{9}.$

This confirms that the advantage in phase parameter estimation in a MZI
provided by photon subtracted of squeezed state is exclusively due to the
increasing in the energy of the field. Using a simple SSV state with the same
energy provides similar sensitivity.

##  III  SPATSV states

Starting from the definition of the TSV as the two mode squeezing operator
$\hat{S}_{1,2}\left(r_{12}e^{i\chi}\right)=e^{r_{12}e^{i\chi}a_{1}a_{2}-h.c.}$
applied to the vacuum, the SPATSV can be obtained by the non Hermitian
operation represented as

$\displaystyle|\Psi^{(m)}_{SPATSV}\rangle_{1,2}=\textbf{N}^{-}_{m}\left(\lambda\right)(\hat{a_{1}})^{m}(\hat{a_{2}})^{m}\hat{S}_{1,2}|0,0\rangle_{1,2},$
(7)

where $\textbf{N}^{-}_{m}$ is the normalization constant,
$\lambda=\sinh^{2}r_{12}$ is the mean energy (mean photon number) per mode for
the TSV, $\chi$ is squeezing angle and $m$ is the number of subtracted
photons. It is possible to express the state in Fock basis as follows

$|\Psi^{(m)}_{SPATSV}\rangle=\frac{\textbf{N}^{-}_{m}\left(\lambda\right)}{\sqrt{1+\lambda}}\sum_{n=0}^{\infty}\left(\frac{\lambda
e^{i\chi}}{1+\lambda}\right)^{\frac{n+m}{2}}\frac{\left(n+m\right)!}{n!}|n,n\rangle_{1,2}.$
(8)

The normalization constant is of the form
$\textbf{N}^{-}_{m}\left(\lambda\right)=\left[\left(m!\right)^{2}\lambda^{m}P_{m}\left(2\lambda+1\right)\right]^{-1/2}$,
where $P_{m}$ is the $m$-th order Legendre’s polynomial. Furthermore, using
squeezed transformation of mode operators $a_{j}$ (and IWOP technique Fan:2003
) it is possible to generate the state in eq $(\ref{SPATSV})$ by applying
squeezing operator to a $m+1$ components superposition of photon number
states, $|\Theta^{m}(\lambda,\chi)\rangle_{1,2}$, as it follows:

$\displaystyle|\Psi^{(m)}_{SPATSV}\rangle_{1,2}$
$\displaystyle=\hat{S_{1,2}}|\Theta^{m}(\lambda,\chi)\rangle_{1,2},$ (9)

where

$\displaystyle|\Theta^{m}(\lambda,\chi)\rangle_{1,2}$
$\displaystyle=\sum_{k=0}^{m}C^{m}_{k}(\lambda,\chi)|k,k\rangle_{1,2},$ (10)

and

$\displaystyle C^{m}_{k}(\lambda,\chi)=e^{i\chi
m}\sqrt{\frac{(1+\lambda)^{m}}{P_{m}(2\lambda+1)}}\times$ (11)
$\displaystyle\times e^{i\chi k}{m\choose
k}\left(\sqrt{\frac{\lambda}{\lambda+1}}\right)^{k}$

with $\sum_{k}\left|C^{m}_{k}(\lambda,\chi)\right|^{2}=1$. Interestingly,
$|\Theta^{m}(\lambda,\chi)\rangle_{1,2}$ is similar to a truncated TSV up to
the components with $k\leq m$. As it can be seen by a careful inspection of
$C^{m}_{k}(\lambda,\chi)$, they differs only by a binomial coefficient and a
normalization factor. For $m=0$ the state $|\Psi^{(0)}_{SPATSV}\rangle_{1,2}$
coincides obviously with TSV. For $m=1$, namely one photon subtraction, the
corresponding normalized two components photon number superposition state is

$|\Theta^{1}(\lambda,\chi)\rangle_{1,2}=\frac{1}{\sqrt{2\lambda+1}}\left(\sqrt{1+\lambda}|0,0\rangle+e^{i\chi}\sqrt{\lambda}|1,1\rangle\right).$
(12)

This superposition state is entangled for non-zero value of $\lambda$ and
resembles to the state Dell'Anno:2007 , and for $\lambda>>1$ it becomes
asymptotically a maximally entangled state. In general, Eq.s (9),(10) and (11)
suggest that a SPATSV state can be generated by seeding the input modes of a
non-linear two-mode-squeezing interaction by an opportune superposition state
in the photon number basis, see Fig. 5 b. This, represent an alternative way
to generate the photon subtracted states in contrast to common approach
depicted Fig. 5 a, consisting in a post selection of the state, conditioned to
double click events at the detectors placed in the two arms, experimentally
realized through unbalanced beam splitter (BS).

Figure 5: Generation scheme for SPATSV state with symmetric photon subtraction
of one photon $m=1$:(a) Probabilistic process by two beam splitter of high
transmittance $T_{1}=T_{2}\approx 1$ placed in two arms of the PDC source.
Simultaneous clicks on the two single photon detectors confirms the generation
of SPATSV. (b) Alternative approach to the generation of SPATSV consists in
injecting seeding state of the general form in Eq. (10) into the non-linear
crystal (NL), in particular the one reported in Eq. (12) for the case $m=1$.

###  III.1 Squeezing properties and photon statistics of the SPATSV state

Analogously to what has been already discussed in Sec. II for single mode
states Wodkiewicz:1987 ; figurny:1993 , also two mode superposition states
$|\Theta(\lambda,\chi)\rangle_{m}$ are squeezed in the quadrature difference
even though they do not minimize the uncertainty principle. The maximum
squeezing in reached for

$\hat{X}_{\chi}^{-}=\hat{X}_{1,\chi}-\hat{X}_{2,\chi}$ (13)

where
$\hat{X}_{j,\chi}=(\hat{a}_{j}e^{-i\chi}+\hat{a}^{\dagger}_{j}e^{i\chi})/\sqrt{2}$
is the quadratures of the $j$-th individual input modes. This is reported in
Fig. 6 (a). Note that, for $m\geq 1$ non-classical correlations are always
present, becoming stronger at the increasing of $m$ in the region of small
$\lambda$. When $m\geq 2$ the squeezing level overcomes the TSV limit (dotted
purple line). The quadrature noise behavior of the seeding state has a direct
effect on the squeezing properties of the SPATSV as shown in Fig. 6(b),
basically leading to a further noise reduction especially for $\lambda<1$ with
respect to the TSV state. This effect is not trivially related to an energy
shift and it can bring beneficial when using SPATSV state for specific
interferometric schemes, as studied in Sec. III.2.

Figure 6: (Color online) Non-classical amplitude quadrature correlation (0.5
is the classical bound) versus $\lambda$ for different number of photon
subtraction $m$: $m=0$ (solid red line), $m=1$ (solid black line), $m=2$ (
solid orange line), and $m=3$ (solid green line). We lavel a) for
superposition state $|\Theta^{m}(\lambda,\chi)\rangle_{1,2}$, except the
dotted curve which corresponds to TSV and b) for the photon subtracted SPATSV
state.  Figure 7: (Color online) 3D plots showing joint photon number
distribution in SPASTV state. $j$ and $k$ are the photon number in the mode
"1" and "2", respectively. The parameters value chosen are $\lambda=0.6$, (a)
$m=0$, (b) $m=1$, (c) $m=3$. Figure 8: (Color online) Mandels $Q$ of the
SPATSV state in function of the mean photon number per mode ($\lambda$) for
different number of photon subtraction $m$: $m=0$ (solid red line), $m=1$
(solid black line), $m=2$ ( solid orange line), and $m=3$ (solid green line).
Here, we have chosen $\eta=0.98$

Aside quadrature squeezing, other statistical properties of the field can be
improved in terms of noise reduction and turned to non-classical regime by the
photon subtraction operation. Indeed, by post-selecting on the components of
the state with at least one photon, induces a shrinking of the photon-number
distribution because of the elimination of the vacuum-component. On one side
it leads to a shift to an higher value of the mean photon number per mode
$\mathcal{L}_{m}(\lambda)\geq\lambda$, as showed in Fig. 7. On the other side
it induces a sub-shot noise behavior in each of the two modes.

Non-classicality in the photon number statistics is usually described by the
Mandels $Q$ parameter Mandel:1995 :

$Q=\frac{Var(\hat{N})-\langle\hat{N}\rangle}{\langle\hat{N}\rangle},$ (14)

where $\hat{N}=\hat{a}^{\dagger}\hat{a}$ is the photon number operator. For
classical light Mandel parameter is bouded by $Q\geq 0$. It is worth noting
that the individual modes of TSV have thermal statistics but, once we apply
the subtraction operation with $m>1$ they become non classical for low mean
photons number, as evident from the negative value of Mandel’s parameter
reported in Fig. 8. This non-classical behavior induced by photon subtraction
is clearly not related to an energy shift of the single mode (which would
conserve a thermal statistics), rather it is a more fundamental uncertainty
reduction of the photon number distribution.

###  III.2 Correlated phase estimation with SPASTV states

Figure 9: Correlated interferometric scheme: The modes of the bipartite input
state $|\psi\rangle$ are mixed with two identical coherent states
$|\alpha\rangle=|\mu e^{i\psi}\rangle$ in two interferometers
$I_{1}(\phi_{1})$ and $I_{2}(\phi_{2})$. A joint detection is performed and
the observable $\hat{C}(\phi_{1},\phi_{2})$ is measured.The losses are
accounted by considering two identical detectors in both channels with the
same quantum efficiency, i.e, $\eta_{5}=\eta_{7}=\eta$

The interferometry system we will consider in this section is presented in
Fig.9. It is composed by two linear interferometers, for instance a pair of
MZIs in the figure, whose photo-currents at the read-out ports are jointly
measured. This is an elegant and powerful scheme in the detection of extremely
faint phase signals whose magnitude can be much smaller than other sources of
noise, including the shot noise. The advantage of this scheme comes from the
fact that the same signal shared by the two interferometers, even if hidden in
the noise in the single device, can emerges by correlating their outputs. This
strategy has been already considered in several highly demanding applications,
in general related to the research of stochastic fundamental backgrounds, such
as gravitational wave background Nishizawa:2008 ; Abadie:2012 ; Chou:2017a ;
Chou:2017  and quantum gravity effects at the Plank scale Hogan:2012 ;
Chou:2016 .

The advantage of using quantum state of light in such a configuration has been
analyzed in ref. Berchera:2013 ; noi:2015 . It has been shown that injection
of quantum state of light in the classically unused input ports (labeled as 1
and 2 in Fig. 9), either as two independent squeezed states or as a TSV allow
to achieve better sensitivities. In case of TSV, for specific working
conditions, i.e. very close to the dark fringes and for high quantum
efficiency, the quantum advantage is dramatic even with respect to the double
squeezing. Here our purpose is to investigate if and to what extent photon
subtracted TVS allows to obtain better performance in virtue of their improved
non-classical properties discussed in Sec. III.1.

####  III.2.1 Noise reduction factor at the read-out ports

Let start considering the correlation properties of the read-out signals at
the output ports (labeled as 5 and 7 in Fig. 9). In particular we are
interested into photo-current subtraction, proportional to the photon number
difference $\hat{N}_{5}-\hat{N}_{7}$.

Figure 10: (Color online) Noise reduction factor at the output ports 5 and 7
of the interferometers in function of the transmittance parameter
$1-\tau=\sin^{2}{\phi/2}$ for different number of photon subtraction $m$:
$m=0$ (red line), $m=1$ (black line), and $m=2$ (orange line). Solid thick
lines are for $\lambda=0.05$, dashed lines stand for $\lambda=2$. Asymptotic
limits for $\lambda\gg 1$ and for $\lambda\ll 1$ ($m=0$) are the dotted and
the dot-dashed lines, respectively. The other parameters are: $\eta=1$,
$\psi=\pi/2$, $\mu=10^{6}$.

Here we consider the noise reduction factor, a standard measure of non-
classical correlation for a bipartite state defined as Brida:2009

$\textrm{NRF}=\frac{\langle\Delta^{2}\left(\hat{N}_{5}-\hat{N}_{7}\right)\rangle}{\langle\hat{N}_{5}\rangle+\langle\hat{N}_{7}\rangle}$
(15)

The numerator is the variance of photon number difference and the denominator
represents the standard quantum limit. Thus, $\textrm{NRF}<1$ indicates non-
classical correlation. It is convenient to introduce the factor
$\tau=\cos^{2}(\phi/2)$, representing the fraction of the power at the input
port 1(2), transmitted to the read-out port 5(7). Consequently, $1-\tau$ is
the equivalent loss experienced by the quantum modes due to the interference
fringe position and, at the same time, the fraction of coherent power injected
in ports 3(4) and transmitted to the output port 5(7). The NRF has been
evaluate analytically, and reported in Fig. 10 as a function of $1-\tau$. In
order to analyze its behavior let us distinguish between two regimes. When the
output signal is dominated by photon coming from coherent beam, i.e
$\mathcal{L}_{m}(\lambda)\tau\ll\mu(1-\tau)$ (we remind
$\mathcal{L}_{m}(\lambda)$ is the SPATSV mean photon number), each
interferometer acts similarly to a homodyne detectors. This represents the
typical working condition, since usually the coherent beam is order of
magnitude brighter than the quantum light, and corresponds roughly to the
region $1-\tau>10^{-4}$ in Fig. 10. In this case, the difference in the photon
number at the output ports 5 and 7 becomes approximatively proportional to the
difference between the quadrature of the input modes at the ports 1 and 2:

$\hat{N}_{5}-\hat{N}_{7}\propto\sqrt{\frac{\mu}{2}}\sin(\phi)\hat{X}^{-}_{\theta=\psi+\pi/2}$
(16)

where $\psi$ can be chosen to match the angle of the squeezed quadrature
difference in Eq. (13), in particular $\psi=\chi-\pi/2$. Therefore, the non-
classical correlation of the input state reported in Fig. 6 b, immediately
traduces in the non-classical properties of the NRF. In particular, for
$\lambda\gg 1$ the NRF is well approximated by
$\textrm{NRF}_{m}(\lambda)\approx 1-\tau+\tau/(4\lambda)$ (dashed line in Fig.
10) for all number of subtracted photon $m$ (for $1-\tau>10^{-4}$ ); Also, for
$\lambda\ll 1$, i.e. when quadrature squeezing of the SPATSV states increases
with the number of subtracted photons, the NRFs follows the same behaviour,
demonstrating the advantage in using photon subtracted states, as clearly
shown in Fig. 10. Analytically, it can be found that in this asymptotic limit
of $\lambda\ll 1$ the expression of the NFR at the different orders of photon
subtraction can be approximated as:

$\textrm{NRF}_{m=0}\approx 1-2\tau\left(\sqrt{\lambda}-\lambda\right)$ (17)
$\textrm{NRF}_{m=1}\approx 1-4\tau\left(\sqrt{\lambda}-2\lambda\right)$ (18)
$\textrm{NRF}_{m=2}\approx 1-6\tau\left(\sqrt{\lambda}-3\lambda\right),$ (19)

where the first of these equation is reported in Fig. 10 as the dotted-dashed
line.

In the opposite scenario, when the coherent beam does not contribute
significantly to the outputs and the two interferometers, i.e.
$\mathcal{L}_{m}\tau\gg\mu(1-\tau)$, the interferometers can be seen as
attenuators with transmission $\tau$ of the input state. The photon number
entanglement between the two modes of the SPATSV input state are then
preserved at the output ports for $\tau\sim 1$. Indeed, in the ideal case of
$\phi=0(\tau=1)$ and unit detection efficiency, the photon number correlation
is perfect, independently on the energy $\lambda$ of the input quantum state.
This explains the sudden dropping down of the NRF observed in Fig. 10 for
$1-\tau<10^{-4}$. However, the condition $\mathcal{L}_{m}\tau\gg\mu(1-\tau)$
is reached for smaller value of $\tau$ (higher value of $\phi$) when the input
energy $\mathcal{L}_{m}(\lambda)$ is larger. So, reminding that
$\mathcal{L}_{m+1}(\lambda)>\mathcal{L}_{m}(\lambda)$, if the energy $\lambda$
of the TSV before the photon subtraction is fixed, subtracting more photons
makes easier to reach the region in which entanglement determine a dramatic
reduction of the uncertainty. In the next section we shall show that the
characteristics of the NRF are strictly related with the sensitivity of the
double interferometric set up.

####  III.2.2 Phase correlation estimation

In setup of Fig. 9 rather than the magnitude of phase noise in the single MZI,
the quantity under estimation is the covariance between the phase fluctuations
in two interferometers. This estimate can be somehow related to a joint
measurement of the read-out signals $N_{5}$ and $N_{7}$. In the limit of faint
signal, any joint observable
$\widehat{C}(\phi_{1},\phi_{2})=\widehat{C}(N_{5}(\phi_{1}),N_{7}(\phi_{2}))$
with local non-null double partial derivative
$\partial_{\phi_{1},\phi_{2}}^{2}C(\phi_{1},\phi_{2})$, can be exploited for a
phase-noise covariance estimation noi:2015 . Here, the goal is to investigate
whether SPATSV can lead to some sensitivity advantage with respect to TSV
state in that scheme. The uncertainty in the phase covariance measurement is
Berchera:2013

$\mathcal{U}=\frac{\sqrt{2\,\mathrm{Var}\left[\widehat{C}(\phi_{1},\phi_{2})\right]}}{\left|\partial_{\phi_{1},\phi_{2}}^{2}C(\phi_{1},\phi_{2})\right|}.$
(20)

A good choice for the joint measurement operator is
$\widehat{C}(\phi_{1},\phi_{2})=\left(N_{5}(\phi_{1})-N_{7}(\phi_{2})\right)^{2}=N_{5}^{2}+N_{7}^{2}-2N_{5}N_{7}$.
On one side, according to the results on the NRF discussed in Sec. III.2.1, it
has fluctuation below the classical limit. On the other, it satisfies the
condition $\partial_{\phi_{1},\phi_{2}}^{2}C(\phi_{1},\phi_{2})\neq 0$. The
classical bound, obtained with coherent states at ports "3" and "4" and vacuum
at the ports "1" and "2", is given by
$\mathcal{U}_{cl}=\sqrt{2}\left(\eta\mu\cos^{2}[\phi/2]\right)^{-1}$ noi:2015
, where we have introduced the detection efficiency $\eta$, assumed to be
equal in the two channels. Hereinafter, we present the uncertainties $U_{m}$
for $m$-th SPATSV state, as normalized to the coherent classical limit, namely
$U_{m}=\mathcal{U}_{m}/\mathcal{U}_{cl}$.

Figure 11: (Color online) Normalized uncertainty as a function of $\phi$ with
$\mu=10^{12}$ for different number of photon subtraction $m$: $m=0$ (solid red
line), $m=1$ (solid black line), $m=2$ ( solid orange line), and $m=3$ (solid
green line). Dashed purple line represents two independent squeezed state:
(a.) for $\lambda=2$, $\eta=0.98$ (b.) $\lambda=0.05$, $\eta=0.98$ (c.) energy
balancing scenario for $\lambda=2$, $\eta=0.96$.

Analytical results of uncertainties as a function of the working central phase
$\phi_{1}=\phi_{2}=\phi$ are plotted in Fig.11. Similarly to the case of the
NRF analyzed in the previous section, one can distinguish two different
regions; one laying roughly in the range $10^{-5}<\phi<\pi$ (showed up to
$\phi\approx 10^{-3}$ in figure) and the other for smaller value of phase,
$\phi<<10^{-6}$, separated by a short transient. The range $10^{-5}<\phi<\pi$
corresponds to the situation in which the mean number of coherent photon at
the read-out ports is much larger than the transmitted SPATSV photons ( i.e.
$\mathcal{L}_{m}(\lambda)\tau\ll\mu(1-\tau)$). In this case, the quadrature
non-classical correlation of the input modes are responsible for the read-out
signal correlation. To provide compact expressions we have reported analytical
results in relevant regimes. In the limit of high coherent power, $\mu\gg 1$,
and low squeezing, $\lambda\ll 1$, one gets:

$U_{m=0}\approx\sqrt{2}\left[1-\tau\eta\left(2\sqrt{\lambda}-2\lambda\right)\right]$
(21)
$U_{m=1}\approx\sqrt{2}\left[1-\tau\eta\left(4\sqrt{\lambda}+\frac{1}{2}\lambda(3\eta\tau-16)\right)\right]$
(22)
$U_{m=2}\approx\sqrt{2}\left[1-\tau\eta\left(6\sqrt{\lambda}+\frac{9}{2}\lambda(\eta\tau-4)\right)\right]$
(23)
$U_{m=3}\approx\sqrt{2}\left[1-\tau\eta\left(8\sqrt{\lambda}+\lambda(9\eta\tau-32)\right)\right].$
(24)

Note that these expressions follow the NRF behavior reported in Eq.s
(17,18,19) up to the terms in $\sqrt{\lambda}$. It comes out that
asymptotically, for $\lambda\ll 1$, there is an advantage which increases with
the number of photon subtracted $m$. However, when the asymptotic condition is
not fully fulfilled, i.e. for finite values of the SPATSV energy
$\mathcal{L}_{m}(\lambda)$, it can happens that at higher values of $m$ do not
always correspond lower uncertainties, as reported in Fig. 11b in the range
$\phi>10^{-5}$, e.g. for $\lambda=0.05$. Moreover, Eq.s (21-24) show that the
detection efficiency $\eta$ plays the same role as the interferometer
transmission $\tau$ and both of them should be high enough to ensure a
significant quantum advantage.

In the case of strong squeezing, $\mu\gg\lambda\gg 1$, provided the condition
$\mathcal{L}_{m}(\lambda)\tau\ll\mu(1-\tau)$ is still fulfilled, it turns out
that the $U_{m}$’s respective expressions for different values of $m$ do not
differ much, in fact we have:

$U_{m=0,1,2,3}\approx\sqrt{2}\left(1-\tau\eta-\frac{\tau\eta}{4\lambda}\right)$
(25)

Also in this case, some deviation from the asymptotic behavior can emerge when
finite values of the parameters are considered. For instance the case
$\lambda=2$ is reported Fig. 11a.

The opposite situation, when number of SPATSV photons are dominant with
respect to the coherent ones at the read-out ports 5 and 7
($\mathcal{L}_{m}\tau\gg\mu(1-\tau)$), corresponds in Fig.11 to the range
$\phi<<10^{-6}$. Perfect photon number correlation of the SPATSV entangled
state at the input ports 2 and 3 are preserved between $N_{5}$ and $N_{7}$.
For $\mu\gg 1$ (and $\phi\rightarrow 0$ ) we obtaining the following
asymptotic behavior:

$U_{m=0,1,2,3}\approx\sqrt{2}\sqrt{(1-\eta)/\eta}\qquad\lambda\ll 1$ (26)

$\displaystyle U_{m=0}$ $\displaystyle\approx 2$
$\displaystyle\sqrt{5}(1-\eta)\qquad\lambda\gg 1$ $\displaystyle U_{m=1}$
$\displaystyle\approx 2$ $\displaystyle\sqrt{3}(1-\eta)$ $\displaystyle
U_{m=2}$ $\displaystyle\approx 2$ $\displaystyle\sqrt{13/5}(1-\eta))$
$\displaystyle U_{m=3}$ $\displaystyle\approx 2$
$\displaystyle\sqrt{17/7}(1-\eta).$ (27)

Eq.s (26, III.2.2) shows that in this regime of perfect photon number
correlations the uncertainty reduction is mainly limited by the detection
efficiency. This means that there exists always a value of the efficiency high
enough to make this regime more advantageous with respect to the one
exploiting quadrature correlation. For example in Fig.11, $\eta=0.98$
guarantees a stronger advantage for $\phi<<10^{-6}$. Only or $\lambda\gg 1$,
the uncertainty reduction depends on the number of subtracted photons $m$. In
addition, photon subtraction brings a further improvement: the increasing of
the energy $\mathcal{L}_{m}$ with $m$ (at fixed $\lambda$) extends the range
of validity of photon number correlation advantage,
$\mathcal{L}_{m}\tau\gg\mu(1-\tau)$, towards higher values of $\phi$, as shown
in Fig.11.

Figure 12: (Color online) Normalized uncertainty versus the detection
efficiency $\eta$ with $\mu=10^{12}$, $\lambda=2$, and $\phi=10^{-8}$ for
different number of photon subtraction $m$: $m=0$ (solid red line), $m=1$
(solid black line), $m=2$ ( solid orange line), and $m=3$ (solid green line),
(a) without energy balancing (b) balanced energy condition.

It is relevant to understand whether the uncertainty reduction observed for
SPATSV states can be explained only in terms of the mean energy increasing due
to the photon subtraction operation, or the advantage comes from other
properties of these states. Also in this case, we consider the energy
balancing approach as in Sec. II.1, where the energies of two mode photon
subtracted states ($m=0,1,2,3$) are made equivalent to the energy of TSV, and
we observe that the uncertainty reduction advantage in the high detection
efficiency cases (Fig. 11a-b) almost disappears (see Fig. 11c). However, in
case of realistic value of the detection efficiency and optical losses, we
observe that the improvement of uncertainty reduction is still present (Fig.
12). For instance, in this scenario SPATSV with $m=3$ present around 26 % of
uncertainty reduction advantage compared to TSV at a detection efficiency of
0.8. Thus, the uncertainty reduction obtained with SPATSV states is, in
general, not only due to the energy shifts, but it also comes from the
enhancement in mode correlation and statistics.

##  IV Summary and conclusions

We have studied in detail multi photon subtracted one- and and two-mode
squeezed vacuum state, in relation to phase estimation in both single and
correlated interferometry. The squeezing of the single mode PASSV state not
necessarily improves with the number of subtracted photons. In the case of odd
number of photon subtraction, it is definitely worse than SSV, while for even
photon subtraction, it is better than the SSV only for relatively small
brightness. The phase estimation uncertainty in a single interferometer
reflects this behavior, as expected. Moreover, by comparing the phase
sensitivity after re-adjusting the energy of the PASSVs to match the one of
SSV, the advantage of the photon subtraction disappears, at least at the
optimal working point of $\phi=\pi/2$. For other values of the central
operating phase we have different behavior and in some cases, as shown in Fig.
3, the advantage of photon subtraction is preserved even when energies are
balanced.

In terms of QFI, we have found improvements with the number of photon
subtraction, but for energy balancing condition this advantage disappears.
However, Heisenberg limit can be reached for an asymptotically large number of
photons in a lossless interferometer.

We also investigated SPATSV for correlated interferometry Berchera:2013 ;
noi:2015 . Usually such SPATSV states are generated by probabilistic events
with low success rate. We showed analytically how symmetric photon subtraction
from two mode squeezed vacuum is equivalent to the squeezing of a finite
component superposition state, suggesting an alternative way for the
generation of SPATSV states. We found that, these SPATSV states always show
quadrature squeezing and their strength increases with the number of
symmetrical photon subtraction for small energy of the state. Various
statistical properties including photon number distribution, Mandel’s $Q$
function, and noise reduction factor shows higher non-classical signature of
SPATSV with respect to TSV suggesting its potential advantages in correlated
phase estimation. In fact, concerning the phase correlations measurement among
two interferometers, we observed that SPATSV are able to achieve a smaller
uncertainty than TSV for an operation point close to the dark fringe
($\phi\approx 0$). In the low losses scenario, SPATSV apparently provide
substantial advantage in uncertainty reduction with respect to TVS state, but
this is essentially explained by the energy increasing of the states due to
the photon-subtraction. In fact, re-normalizing the energies of the SPATSV to
the one of the initial TSV state, the uncertainty reduction is lost. However,
SPATSV conserve an advantage of about 30$\%$ with respect to TVS in the high
loss scenario and this can be attributed to their improved statistical
properties.

##  V ACKNOWLEDGMENTS

This work has been sponsored by the EU via ”Quantum readout techniques and
technologies” (QUARTET, Grant Agreement No. 862644) and by the EMPIR
Participating States in the context of the project 17FUN01 BeCOMe.

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